Differential and Difference Dimension Polynomials

Differential and Difference Dimension Polynomials

Mathematics and Its Applications

Managing Editor:

M.HAZEWINKEL

Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Volume 461

Differential and Difference Dimension Polynomials

by

M. V. Kondratieva

Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

A. B. Levin Department of Mathematics, The Catholic University of America, Washington, D.C., U.S.A.

A. V. Mikhalev Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

and

E. V. Pankratiev Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

II SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. CataIogue record for tiris book is available from the Library of Congress.

ISBN 978-90-481-5141-7 ISBN 978-94-017-1257-6 (eBook) DOI 10.1007/978-94-017-1257-6

Printed on acid-free paper

Al1 Rights Reserved ©1999 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1999 No part of the material protected by tiris copyright notice may be reproduced or uti1ized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

CONTENTS

Preface vii

Chapter I. Preliminaries 1 1.1. Notation and Conventions 1 1.2. Some Basic Notions and Results of the Theory of Com-

mutative Rings 5 1.3. Graded and Filtered Rings and Modules 19 1.4. Conservative Systems 32 1.5. Derivations and Differentials 37

Chapter II. Numerical Polynomials 45 2.1. Definition and some Properties of Numerical Polynomi-~ %

2.2. Subsets on~m and their Dimension Polynomials. Dimen-sion Polynomials of a Matrix 53

2.3. Algorithms for Computation of Dimension Polynomials 64 2.4. Ordering of Kolchin Dimension Polynomials 92 2.5. Dimension Polynomials of Subsets of zm 107

Chapter III. Basic Notion of Differential and Difference Algebra 123

3.1. Rings with Operators 123 3.2. Basis Notions of Differential Algebra 125 3.3. Basic Notions of Difference Algebra 135 3.4. Inversive Difference Rings and Modules 158 3.5. Differential-Difference Structures 174

Chapter IV. Grabner Bases 191 4.1. Grobner Bases for Polynomial, Differential and Differ-

ence Modules 183 4.2. Basic Algorithms of Computation of Grobner Bases 203 4.3. Application of Grobner Bases to the Computation of

Characteristic Polynomials 209

Chapter V. Differential Dimension Polynomials 223 5.1. Characteristic Polynomials of Excellently Filtered Dif-

ferential Modules 223 5.2. Differential Dimension 228 5.3. Autoreduced Sets of Differential Polynomials. Charac-

teristic Sets 231

v

vi CONTENTS

5.4. Differential Dimension Polynomial of a Finitely Gener-ated Differential Field Extension 237

5.5. Coherent Autoreduced Sets. Ritt-Kolchin's Algorithm 244 5.6. Invariants of Differential Dimension Polynomials 257 5.7. Minimal Differential Dimension Polynomial 267 5.8. Jacobi's Bound for a System of Algebraic Differential

Equations 273

Chapter VI. Dimension Polynomials in Difference and Difference-Differential Algebra 281

6.1. Characteristic Polynomials of Graded Difference Module 281 6.2. Dimension Polynomials of Filtered Difference Modules.

Difference Dimension 284 6.3. Characteristic Polynomials of Inversive Difference Mod-

ules and their Invariants 290 6.4. Dimension Polynomials of Extensions of Difference and

Inversive Difference Fields 304 6.5. Linear (1'* -Ideals and their Dimension Polynomials 320 6.6. Computation of Dimension Polynomials in the Case when

the Basic Set Consists of Two Translations 332 6.7. Characteristic Polynomials of Finitely Generated Diffe-

rence-Differential Modules and their Invariants 344

Chapter VII. Some Application of Dimension Polyno-mials in Difference-Differential Algebra 355

7.1. Type and Dimension of Difference-Differential Vector Spaces 355

7.2. Type and Dimension of Finitely Generated Difference-Differential Algebras 361

7.3. Difference-Differential Local Algebras 370

Chapter VIII. Dimension Polynomials of Filtered Gmodules and Finitely Generated G-fields Exten-sions 377

8.1. Rings with a Group of Operators. G-modules 377 8.2. Dimension Polynomials of Excellently Filtered

G-modules 380 8.3. Some Generalizations for Differential G-structures 387

Chapter IX. Computation of Dimension Polynomials 397 9.1. Description of the Program Complex 397 9.2. Computation of Dimension Polynomials for some Sys-

tems of Differential Equations 398

References 405

Index 417

PREFACE

The role of Hilbert polynomials in commutative and homological algebra as well as in algebraic geometry and combinatorics is well known. A similar role in differential algebra is played by the differential dimension polynomials. The notion of differential dimension polynomial was introduced by E. Kolchin in 1964 [KoI64]' but the problems and ideas that had led to this notion (and that are reflected in this book) have essentially more long history. Actually, one can say that the differential dimension polynomial describes in exact terms the freedom degree of a dynamic system as well as the number of arbitrary constants in the general solution of a system of algebraic differential equations. The first attempts of such description were made at the end of 19th century by Jacobi [Ja890] who estimated the number of algebraically independent constants in the general solution of a system of linear ordinary differential equations. Later on, Jacobi's results were extended to some cases of nonlinear systems, but in general case the problem of such estimation (that is known as the problem of Jacobi's bound) remains open. There are some generalization of the problem of Jacobi's bound to the partial differential equations, but the results in this area are just appearing.

At the beginning of the 20th century algebraic methods in the theory of differential equations were actively developed by F. Riquier [RiqlO] and M. Janet, [Jan21] and [J an29] , whose works can be considered as the first stones in the basement of differential algebra. It should be noted that many of the results obtained in this period were connected with the concept of differential dimension. In particular, we mention the Janet conjecture whose formulation in modern differential algebra describes some properties of the differential dimension polynomial.

Differential algebra as a separate area of mathematics is largely due to its foundator J.F. Ritt (1893-1951) and E.R. Kolchin (1916-1991). In the preface to his book An Introduction to Differential Algebra (1957) I. Kaplansky wrote: "Differential algebra is easily described: it is (99 percent or more) the work of Ritt and Kolchin". In 30th J. Ritt published a series of articles (see [RR39], [Rit31], [Rit32], [Rit35aJ[Rit39], and [Rit45]) where he introduced the basic notions of differential algebra (such as the algebra of differential polynomials, autoreduced sets (chains), perfect differential ideals, and differential manifolds), proved some fundamental theorems (including an important case of the basis theorem that was later generalized by H. Raudenbuch) and outlined the main directions of research in this area. In 1950 J. Ritt published the first monograph [Rit50] on differential algebra where he summarized all essential results obtained in this field by that time and formulated a series of problems for the further study many of which remain unsolved up to now.

J. Ritt's research was continued and extended by E. Kolchin who not only created and developed new fruitful theories in differential algebra but recasted the whole

vii

viii PREFACE

subject in the style of modern algebraic geometry with the additional presence of derivation operators. In particular, Ellis Kolchin developed the theory of differential fields and created the differential Galois theory (see [KoI39-KoI55] and [KoISO]) where finite dimensional algebraic groups played the same role as finite groups play in the theory of algebraic equations. In this connection he developed general theory of algebraic groups that became later a field of intensive research with various and deep applications. We should mention here Kolchin's monograph [KoIS5] and works by A. Buium [Bui92], Ph. Cassidy [Cass72-CassS9], J. Kovacic [Kov69-KovS6], E. Pankrat'ev [Pa71]' J.-F. Pommaret [PoS3], C. Mitschi and M. Singer [MS96a], [MS96b], H. Umimura [Um96a], [Um96b], and some other mathematicians that have formed the contemporary differential Galois theory and the modern theory of differential algebraic groups. Among the other fields of Kolchin's activity in differential algebra one should mention the study of singular solutions of algebraic differential equations [KoI65], extensions of differential specializations, rational approximation [KoI59]' [KoI92]' and, of course, the theory of differential dimension where the central role is played by Kolchin's differential dimension polynomials. At the International Congress of Mathematicians in Moscow (1966) Kolchin formulated the main problems and outlined the most perspective directions of research connected with the differential dimension. Later on the results obtained in this area were included into his famous monograph Differential Algebra and Algebraic Groups that hitherto remains the most fundamental work on differential algebra. The role of Kolchin in the development of differential algebra cannot be understood completely if we do not mention his Differential Algebra Seminar at Columbia University that has been working for over thirty years. Various and deep results in the classical differential algebra and in the related fields were obtained by Ph. Cassidy, J. Johnson, W. Keigher, J. Kovacic, S. Morrison, W. Sit and many other Kolchin's disciples.

Discussing the history of creation of the differential dimension theory, it should be noted that in 1953 A. Einstein [Ei53] introduced a notion of the 'strength' of a system of differential equations that is a certain function of integer argument associated with the system. Another work that should be mentioned in connection with the history of differential dimension polynomial is the paper by I. Zuckerman [Zu65] who (independently of Kolchin and at almost the same time) found two invariants of a partial differential field extension that turned out to be what we now call the differential type and typical differential dimension (they define the degree and the leading coefficient of the differential dimension polynomial, respectively). In 1980 A. Mikhalev and E. Pankrat'ev [MP80] showed that this function actually coincides with the appropriate differential dimension polynomial and found the strength of some well-known systems of partial differential equations using methods of differential algebra.

The intensive study of Kolchin's differential dimension polynomials began at the end of 60th with the series of works by J. Johnson [Jo69a-Jo78] who showed that the differential dimension polynomial of a differential field extension coincides with the Hilbert characteristic polynomial of the filtered module of Kahler differentials associated with the extension. In this connection J. Johnson developed the technique of dimension polynomials for differential modules and applied it to the study of some classical problems of differential algebra. In particular, he characterized

PREFACE ix

the Krull dimension of finitely generated differential algebras, developed the theory of local differential algebras, and proved a special case of Janet conjecture. The study of differential dimension polynomials was continued by M. Kondrat'eva [Kon88], [Kon89], A. Levin, A. Mikhalev, and Pankrat'ev [LM87], [LM92a]' [MP73], [MP80], W. Sit [Si78], [Si92] and some other mathematicians. One of the most important directions of this study was the search for new invariants of the differential dimension polynomial. Here we should mention the results of W. Sit [Si75] who showed that the set of all differential dimension polynomials is well-ordered with respect to some natural ordering and introduced the notion of the minimal differential dimension polynomial associated with a differential field extension. This important invariant was studied by M. Kondrat'eva [Kon88], [Kon89]; in particular, she proved that every differential dimension polynomial is minimal for some differential field extension. There were some other results on invariants of a differential dimension polynomial, but many important questions in this area are still open.

Another direction of activity that is related to the differential dimension polynomials and that caught a second wind in 70th is the study of the problem of Jacobi's bound. Some new results in this area were obtained by B. Lando [Land70], S. Tomasovic [T076], R. Cohn [CoR80], [CoR83], and M. Kondrat 'eva, A. Mikhalev and E. Pankrat'ev [KMP82]. At present the interest in the problem of Jacobi's bound increases because the information on this bound leads to the estimates of the complexity of the algorithms that are used for computation of differential dimension polynomials.

In much the same way as differential algebra arises from the study of algebraic differential equations, difference algebra arises from the study of algebraic difference equations and therefore bears a considerable resemblance to its "differential counterpart". Difference algebra was born in the same period as differential algebra and it had the same foundator, J. Ritt, whose works [Rit34], [Rit41], [RD33] and [RR39], as well as works of H. Raudenbush [RR39] and F. Herzog [Herz35], laid the groundwork for the theory of algebraic difference equations. (It should be noted that the first attempts of the study of equations in finite differences from the algebraic standpoint were made at the end of nineteenth century by F. Casorati [Cas882].) In 1934-1939 J. Ritt and H. Raudenbush introduced the basic concepts of difference algebra (such as the ring of difference polynomials, perfect difference ideals and difference manifolds) and described some properties of difference ideals and manifolds in the spirit of differential algebra they were developing at that time. Difference algebra as a separate mathematical area with its own ideas and methods is mainly due to R. Cohn whose works [CoR4S-CoRS6aJ not only raised the difference algebra to the level comparable with the level of development of differential algebra, but also clarified why many ideas that are fruitful in differential algebra cannot be successfully applied in difference case, as well as many methods of difference algebra cannot have differential analogs. R. Cohn's book [CoR65] hitherto remains the only fundamental monograph on difference algebra. Since 60th various problems of difference algebra were developed by A. Babbitt [Bab62], 1. Balaba [BaS4], [Ba87], 1. Bentsen [Be71], A. Bialynicki-Birula [Bia62]' P. Evanovich [Ev73), [Ev84]' C. Franke [Fr63], B. Greenspan [Gre59], R. Infante [InSO], [InSl]' B. Lando [Land72]' and the authors [KPSS5], [Le7S-LeS5b], [MPS7], [MP89], [Pa72a-PaS9].

The first steps in constructing the difference dimension theory were made by

x PREFACE

R. Cohn who introduced the notion of limit degree of an ordinary difference field extension and found some interesting properties of this characteristic. Some generalizations of the concept of limit degree for partial difference extensions were made by P. Evanovich [EvS4], but the intensive study of the problems of dimension in difference algebra began in SOth when the technique of dimension polynomials were applied to the difference algebraic structures. The concept of difference dimension polynomial was introduced by A. Levin first for difference field extension [Le7S] and then for inversive difference field extensions and for difference and inversive difference modules [LeSOa-LeS5a]. As it should be expected, difference dimension polynomials play the same role in difference algebra as Hilbert polynomials in commutative algebra or differential dimension polynomials in differential algebra. In particular, a difference dimension polynomial of a difference field extension f{ C; L carries some important invariants of the extensions (such as the difference transcendence degree, type, and typical transcendence degree of Lover K). The properties of difference dimension polynomials and their invariants were studied in the works of I. Balaba [BaS4], [BaS7], M. Kondrat 'eva, E. Pankrat 'ev, and R. Serov [KPSS5], A. Levin [LeSOa-LeS5b], E. Pankrat'ev [PaS9] and some other authors.

In the last ten years there were obtained some generalizations of the results on difference dimension polynomials to the case of difference-differential algebraic structures (see, for example, works [LMSSa], [LM91b]) and to the case of algebraic and differential algebraic structures on which a finitely generated commutative group acts (see [LMSSb], [LMS9a], [LM91a-LM92b]). These generalizations allow to develop an algebraic approach to study the systems of algebraic differential equations with delay (in particular, they allow to determine the strength of such a system in the sense of A. Einstein [Ei53]).

The increasing role of dimension polynomials in the study of differential and difference algebraic structures stimulated the development of methods of computation of such polynomials and their invariants. It turned out that the processes of computation of differential and difference dimension polynomials are similar; such a process can be divided into two steps: constructing a characteristic set of a prime differential (difference) ideal and solving a certain combinatorial problem for a subset of Nm . The development of the constructive methods in differential and difference algebra was also stimulated by the recent development of computers and their applications to the problems of algebraic computations. One can say that a new science, computer algebra, has arisen on the junction of the mathematics and computer science.

One of the main problems of computer algebra is the problem of construction of "good" bases of polynomial ideals. Such bases (i.e., finite systems of generators of an ideal that satisfy certain conditions) arouse in the works of F.S. Macaulay [Mac16] and G. Herman [He26]. The interest to this kind of problems greatly increased after the work of B. Buchberger [Bu65], [Bu70], [Bu79] who introduced the notion of Grabner basis of a polynomial ideal and proposed an algorithm for its computation. The explicit presentations of Grabner bases allows to answer many questions of the theory of polynomial ideals. In particular, one can algorithmically verify whether a polynomial belongs to a given polynomial ideal, solve the problem of compatibility of a system of algebraic equations, explicitly construct the free resolutions and so on. It should be noted that there are many monographs on

PREFACE xi

the theory of Grobner bases, see, for example, books [AL94]' [BW93], [Mi93], and [Stu96].

Some applications of this technic can be found in [Stu96] (in the domain of integer programming and computational statistics), in [Diop93] (in system theory), and in [CAPR91] (in physical research).

There is an intimate relation between the theory of characteristic sets in differential algebra and the theory of Grobner bases in commutative algebra. The notion of characteristic set can be considered as a generalization of the notion of Grobner basis, however some essential properties of Grobner bases do not hold for characteristic sets. In particular, a characteristic set, as a rule, does not generate the corresponding ideal. The problem of constructing of a more adequate analog of the Grobner bases in differential algebra was considered by G. Carra-Ferro [CF89a], [CF97], F. Ollivier [Olliv91a], E. Mansfield [Mans91] and some other authors, but all the concepts of such analogs had some essential deficiencies. Because of high computational complexity, the algorithm for the construction of the characteristic set of a differential ideal proposed by E. Kolchin in [KoI73] exists hitherto only on paper. The problems that arise in the process of its realization include, in particular, the problem of primary decomposition of a polynomial ideal, which is solved only 'in principle' (see, for example, [BW93]). It should be noted that in 80th W.T. Wu specialized the constructive methods of differential algebra to commutative algebra and obtained a power tool for mechanical geometrical theorem proving. His results [Wu78], [Wu84], as well as the works of G. Gallo and B. Mishra [GM91a]' [GM91b], has greatly increased the interest in the algorithms for computing characteristic sets. Another theory which appeared in differential algebra and afterwards was applied in commutative algebra is connected wich so-called involitive bases. For the first time involutive bases appeared in works of Janet, Thomas, Riquier for studying differential equations, and in 1990th these methods were applied by Zharkov, Gerdt, Blinkov and others to investigation of polynomial ideals [Ap95], [GB96], [Ger97], and [GB97].

Concerning the combinatorial problem that appears in the process of computation of differential (difference) dimension polynomial, we should mention that a general formula for its solution was obtained more then twenty years ago in Buchberger's paper [Bu65]. However, the time complexity of computations with the use of this formula depends on the number s of elements of the appropriate Grobner basis as (~ 2"), that is why the corresponding algorithm is inadequate. The analysis of Buchberger's formula made by H.M. Moller and F. Mora [MM83] allowed them to propose some new methods that essentially diminish the computations for large s. Some connections of the Hilbert polynomials with the Newton poly tops can be found in [Kh92]' [Kh95], and [Stu96].

The authors of this monograph proposed in [KP88], [KP90] other more effective algorithms that were implemented and successively applied to some problems of differential algebra and algebraic geometry. Some examples of employment of these algorithms can be found, for example, in [Pa89].

The book is almost self-contained, the reader is just expected to be familiar with elementary facts concerning groups, rings, and fields, as given, for example in the classical texts of B. Van der Warden [VdW71] and S. Lang [Lang71].

The contents of the book is the following.

xii PREFACE

Chapter 1 is introductory, it contains the basic notions and definitions that are used throughout the book. The chapter also contains a number of classical results of general and commutative algebra that are employed in the subsequent chapters.

In Chapter 2 the multivariate numerical polynomials are studied. For any subset E of the set Nm , a family of so-called dimension numerical polynomials is introduced. These polynomials describe the number of the points in the intersection of the set E with some subsets of Nm • The main attention is given to the univariate polynomials (such polynomials arise in differential algebra, we call them the Kolchin polynomials). Different algorithms for computation of the dimension polynomials are discussed and a general formula, that describes the variation ofthe dimension polynomials after adding one more element to the set E, is obtained. In Section 2.4 we prove a constructive theorem that describes the set W of the Kolchin polynomials and introduce a total order on the set W. In Section 2.5 the notion of the dimension polynomial is extended to the subsets of the set IZm. It is proven that the set of the "new" dimension polynomials coincides with W. In this chapter the reader can find many examples of computations that can be performed without use of computers.

In Chapter 3 the basic concepts of differential and difference algebra are introduced and discussed. The main objects of this consideration are the rings of differential and difference polynomials and their ideals. The chapter contains a number of classical results of differential and difference algebraic geometry.

Chapter 4 deals with the Grobner bases. We present here some algorithms for constructing Grobner bases and give some applications of these algorithms to the problem of computation of Hilbert polynomials of filtered modules.

In Chapter 5 we give two different proofs of existence of the differential dimension polynomial associated with a finitely generated differential field extension. We also introduce the notion of invariants of a differential dimension polynomial and present some results on such invariants as well as a series of conjectures on their estimates. Among the other important topics of this chapter we should mention the RittKolchin algorithm for constructing the characteristic set of a prime differential ideal and results on the problem of Jacobi's bound.

Chapter 6 is devoted to difference dimension polynomials. We prove here the existence theorems for difference and inversive difference dimension polynomials and describe properties of their invariants. A series of examples of computating of difference and inversive difference dimension polynomials is given. In the last sections of the chapter the results on difference dimension polynomials are generalized to the difference-differential case.

In Chapter 7 some applications of the dimension polynomials technique in difference-differential algebra are considered. In the first two sections the type and dimension of the difference-differential vector space and finitely generated differencedifferential algebra are defined and computed. In Section 7.3 we apply the properties of the difference-differential dimension polynomials to study local differencedifferential algebras.

In Chapter 8 some of the results of the previous chapters are generalized to the case when a commutative group G acts on the given field (ring, module). In particular, the existence theorem for the dimension polynomial of an excellently filtered A-G-module over an artinian G-ring A is proven. The results of this chapter

PREFACE xiii

allow to study 'G-equations', i.e., algebraic equations with respect to indeterminates and their images under the action of the elements of the group G.

Chapter 9 contains description of a program complex for computating of the differential dimension polynomial associated with a system of algebraic differential equations. We give here a number of examples where the programs developed are applied for computation of dimension polynomials of some classical systems of equations. (One should note that such a computation is almost impossible without the computer technique.) In particular, we present an example of changing of variables, which demonstrates the difficulty of computation of the minimal differential dimension polynomial.

At the beginning of the work on the manuscript the authors had an honor to discuss the content of the future book with E. Kolchin when he and his wife Kate visited Russia. We will never forget the wonderful atmosphere of that meeting, as well as Kolchin's remarks and suggestions that were extremely helpful in our work.

The authors are deeply indebted to K. Beidar, A. Buium, R. Cohn, Ph. Cassidy, V. Gerdt, E. Golod, J. Johnson, W. Keigher, J. Kovacic, V. Latyshev, V. Markov, and W. Sit for valuable comments and criticism. We are also grateful to all participants of the differential algebra seminars at the Moscow State University and at the City University of New York where many of the topics of this book were discussed. We are also very thankfull to Kate Kolchin for some valuable bibliographical comments and wonderfull hospitality program during our visits of New York. Last, but not least, we wish to thank M. Hazewinkel and the staff of the Kluwer Academic Publishers for their cooperation and patience.

Finally, we are very obliged to INTAS (International Association for the Promotion of Cooperation with Scientists from the Independent States of the Former Soviet Union) and RFBR (Russian Foundation for Basic Research) for financial support of this work.

CHAPTER I

PRELIMIN ARIES

In this chapter we describe the notions and conventions that are in force throughout this book and provide the reader with the relevant background from some parts of algebra for understanding the subsequent chapters. The reader may read the chapter as a whole, or use its appropriate parts for references while reading the latter text (as for those readers who have mastered the whole material covered by the book [Lang71)).

1.1. Notation and Conventions

Sets, mappings and relations. We keep to the following usual notations and conventions of the set theory. The

union, the intersection and the difference of sets A and B are denoted by Au B, An B and A \ B, respectively; the union Al U ... U Ak is also denoted by U7=1 Ai and the intersection Al n··· n Ak by n7=IA i . The empty set is denoted by 0. We write x E A when x is an element of A and x tf. A when x does not belong to A. If Xl E A, ... , Xn E A we also write Xl, ... , Xn E A. The notation A ~ B or B ;2 A means that A is contained in B, i.e., that every element of A belongs to B. When A C B we say that A is a subset of B, and when A C B and A # B we say that A ~ a proper subset of B and write A C B or A ~ 13. The set of elements of a set X that satisfy a condition <p( x) is denoted by {x E X I <p( x)}. A set consisting of finitely many elements Xl, ... , xk is denoted by {Xl, ... , Xk} (sometimes we do not distinguish the set {x} and the element x). The set of all natural numbers (including 0) will be denoted by N, and Nm (m E N, m ~ 1) will denote the subset {I, 2, ... , m} of N.

An ordered pair (x,y) is the set {{X}, {x,y}}. Two ordered pairs (xI,yd and (X2, Y2) are equal if and only if Xl = X2 and YI = Y2. A Cartesian product A x B of sets A and B is the set of ordered pairs (a, b) with a E A, b E B. Any subset of A x B is called a relation. The set of all the first coordinates of a relation is called the domain and the set of all its second coordinates is called the range of the relation. A relation R on a set A (i.e. a subset of Ax A) is called reflexive (respectively, symmetric) if (a, a) E R for all a E A (respectively, if the inclusion (a, b) E R implies the inclusion (b, a) E R for all a, bE R). A relation R on the set A is called transitive if the inclusions (a, b) E Rand (b, c) E R imply the inclusion (a, c) E R for all a, b, c E A. A reflexive, symmetric and transitive relation R on the set A is called an equivalence relation on A. Note, that if R is a relation on a set A, then the inclusion a, bE R (a, bE A) is also written as aRb.

Let R be a relation on a set A. A relation R' on A such that aR'b (a, b E A) if and only if aRb or a = b, is called the reflexive closure of R (by a = b we

M. V. Kondratieva et al., Differential and Difference Dimension Polynomials© Springer Science+Business Media Dordrecht 1999

2 1. PRELIMINARIES

mean that a and b denote the same element of A). A relation R" on A, such that aR"b (a, bE A) if and only if aRb or bRa is called the symmetric closure of R. A relation R+ on A is said to be a transitive closure of R if the following condition holds: aR+b (a, bE A) if and only if there exist elements a1, ... , an E A (where n is some positive integer) such that aRa1,a1Ra2," .,an_1Ran,anRb. It is easy to verify that R', R" and R+ are reflexive, symmetric and transitive relations on A, respectively.

Combining the above constructions one can receive the notions of reflexivesymmetric, reflexive-transitive, symmetric-transitive and reflexive-symmetric-transitive closures of R. For example, the reflexive-symmetric closure R of R is determined by the following condition: aRb (a, b E A) if and only if aR'b or bR'a (so that aRb or bRa or a = b).

1.1.1. EXERCISE. Describe the other types of closures mentioned above.

Let A and B be two sets. A relation f C A x B is called a function from A to B, or a mapping from A to B, if for every a E A there exists a unique element b E B such that (a, b) E f. In this case we write f : A -+ Band b = f(a) or f : a -+ b and say that b is the image of a under the mapping f or the value of f at a. The image of a set Ao ~ A under f is the set f(Ao) = {b E Bib = f(a) for some a E Ao}. If Bo ~ B, then the inverse image of the set Bo under f is the set f-l(Bo) = {a E A I I(a) E Bo}.

1.1.2. EXERCISE. Let f be a mapping from A into B and let Ao ~ A, Bo ~ B. Prove the following elementary facts concerning images and inverse images:

(i) f(J-1 (Bo)) = Bo n f(A); (ii) f-1(J(Ao) n Bo) ;2 Ao n 1-1(Bo);

(iii) I(Ao n f-1(Bo)) = f(Ao) n Bo; (iv) f(Ao) n Bo = 0 if and only if Ao n rl(Bo) = 0; (v) f-l(B \ Bo) = A \ l-l(Bo).

If f is a mapping from A to Band g is a mapping from B to C, then we can define a mapping from A to C such that a t--+ g(J(a)). This mapping is called the composition of f and g; it is denoted by go I (or by gJ). One easily sees that (gJ)-I(CO) = f-l(g-I(Co)) for any subset Co ~ C.

A mapping f: A -+ B is called injective (or one-to-one) if for any al,a2 E A, the equality f(aI} = f(a2) implies al = a2. If a mapping f : A -+ B satisfies the condition f(A) = B, then I is said to be surjective (or a mapping of A onto B); an injective and surjective mapping is called bijective. The identical mapping from a set A onto itself is denoted by idA; it is defined by the equality idA (a) = a for every a EA. Clearly, for any bijective mapping I: A -+ B, there exists an inverse mapping 1-1 : B -+ A such that 10/-1 = idE, 1-1 0 f = idA (if b E Band b = I(a) for some a E A, then 1-1 (b) = a).

As usual, by a sequence we mean a function defined on the set of all positive integers; the value of a sequence at n is usually denoted by Xn (or by some other letter with index n) and the sequence itself is denoted by Xl, X2, .. ' .

Let A be a set and let there exist a function that associates with every element ). E A some set A>.. Then we say that we have an indexed family (or an indexed set) of sets {A>.hEA; the set A is called the index set of the family. The union and

1.1. NOTATION AND CONVENTIONS 3

the intersection of a family {A).hEA of sets are denoted by U).EAA). and n).EAA)., respectively; in the case of sequence AI, A2, ... of sets we use the symbols U~l A; and n~IAi. A Cartesian product of a family {A).hEA of sets, i.e., the set of all functions from A to U).EAA). such that f()..) E A). for !iny ).. E A, is denoted by TI).EA A)., or by TI~1 Ai in the case of a sequence AI, A 2 , ... of sets. If the index set A is finite, A = {I, 2, ... , n}, the Cartesian product is denoted by TI?=1 Ai. If f E TI).EA A)., then the element f()..) E A). (,\ E A) is called the '\-th coordinate of

f· An element of a Cartesian product TI).EA A). whose '\-th coordinate is an element

a). E A). will be denoted by {a).} ).EA or by {a). I ,\ E A} and will be called an indexing (with the index set A). Clearly, {a).hEA = {b).hEA implies a). = h for all ,\ E A. If Ao ~ A, then an element {a).hEAo of TI).EAo A). is called a subindexing of the indexing {a).hEA. If A = {I, ... , n} for some positive integer n, then the element of TI?:1 Ai whose i-th coordinate is ai E Ai (1 ~ i ~ n), is denoted by (aI, ... , an) and is called an n-tuple. In particular, if Al = A2 = ... = An = A for some set A, then elements of TI?=1 Ai are called n-tuples over A and the direct product itself is denoted by An.

Let ~ be a transitive relation on a set A (i.e. a ~ band b ~ c imply a ~ c for any elements a, b, c E A), and let the following property hold: a ~ band b ~ a if and only if a = b (a,b E A). In this case we say that A is a partially ordered set relative to the order ~ and designate it as (A, ~). If ~ is a partial order on A and for every pair of distinct elements a, b E A, precisely one of a ~ b, b ~ a is valid, then A is said to be linearly (or totally) ordered by ~. (In this case A is called a linearly (or totally) ordered set relative to ~, the order ~ is called a linear (or total) order on A.) A linearly ordered sets are also called chains. If A is a partially (linearly) ordered set relative to some order ~, then (if otherwise is not indicated) any non-empty subset Ao C A will be also considered as a partially (respectively, linearly) ordered set relative to the same order ~.

Let A be a partially ordered set (relative to some order ~) and let Ao be a subset of A. An element b E A is called an upper bound for Ao, if for any a E Ao we have either a ~ b or a = b. An element ao E Ao is called maximal (respectively, minimal) in Ao if Ao contains no element a i- ao such that ao ~ a, (respectively, a ~ ao). The proof of the following well-known fact can be found, for example, in [Bou57].

1.1.3. THEOREM. The following properties of a partially ordered non-empty set A with an order ~ are equivalent.

(1) (The condition of minimality). Every non-empty subset Ao ~ A has a minimal element.

(2) (The induction condition). If all minimal elements of A have some property P and if for any a E A the fact that all elements x E A, x ~ a have the property P implies that the element a also has this property, then all elements of A have the property P.

(3) (Ascending chain condition). If al ~ a2 ~ ... ~ ak ... is any chain of elements of A, then there exists a positive integer n such that an = an+! = an +2 = ....

A linearly ordered set (A,~) which satisfies the condition of minimality (and,

4 1. PRELIMINARIES

therefore, the two other conditions of the last theorem) is called well-ordered (in this case the linear order::; is called a well-order).

The axiom of choice (which claims that for every non-empty set M there exists a function from the set of all subsets of Minto M such that I(A) E A for any A ~ M) and its alternative forms are often used in the book. We give here the following two such forms.

1.1.4. THEOREM (ZERMELO THEOREM ON WELL-ORDERING). On every set A there exists a relation < which well-orders A.

1.1.5. THEOREM (ZORN'S LEMMA). If every linearly ordered subset of a partially ordered non-empty set A has an upper bound, then there exists a maximal element in A.

Note that in the typical applications of Zorn's Lemma elements of the partially ordered set (A,::;) are also sets and the relation::; is the inclusion of sets.

The cardinality of A will be denoted by Card A. In particular, if A is a finite set, then Card A denotes the number of elements of A (we shall mostly deal with finite and countable sets, so we do not formulate here the fundamental results of the set theory on cardinal numbers; they can be found in any course of the set theory, (see, for example, [Bou74]).

Algebraic structures. Throughout the book, by a ring we always mean an associative ring with a unit.

Every ring homomorphism is unitary (maps unit onto unit), every subring of a ring contains the unit of the ring, every module over a ring is unitary (mUltiplication by the unit of the ring is the identity mapping of the module), and every algebra over a commutative ring is also unitary. As usual, !Z denotes the ring of integers, <OJ is the field of rational numbers, IR is the field of real numbers; C denotes the field of complex numbers.

An injective homomorphism (of rings, of modules or of algebras) is called a monomorphism; a surjective homomorphism is called an epimorphism. A bijective homomorphism is called an isomorphism and is denoted by the symbol ~ (if A and B are rings (or modules, or algebras) we write A ~ B if there exists an isomorphism between A and B). Furthermore, a homomorphism from an object to itself is called an endomorphism, and an endomorphism which is an isomorphism is called an automorphism. A kernel of an homomorphism 1 : A --t B (where A and B are rings or modules or algebras), i.e., the set {a E A I I(a) = O}, is denoted by Ker I; the image of 1 is denoted by Iml or by f(A). If R is a noncommutative ring then by a R-module we mean (if otherwise is not indicated) a left R-module.

If R is a commutative ring and V ~ R, then (V) denotes the ideal of R generated by the set V (i.e., the intersection of all ideals of R containing V which coincides with the set of all elements of the form 2:7=1 aiVi where a1, ... ,an E R and V1, ... , Vn E V for some positive integer n). By a finitely generated ideal of a commutative ring R we mean an ideal generated by a finite set of elements of R. If A and B are two subsets of a ring 5, then the product AB is the set of all elements of the form 2:7=1 aibi where a1, ... , an E A; b1, ... , bn E B (n E N, n 2: 1). In particular, the left ideal of 5 generated by a set V ~ 5 (i.e., the intersection of all left ideals of 5 containing V) can be written as 5V. Similarly, if M is a left module

1.2. BASIC NOTIONS OF THE THEORY OF COMMUTATIVE RINGS 5

over a ring 5 and A C 5, P C M then by the product AP we mean the set of all finite sums of the form 2::7:1 aiXi where al,.·., an E A, Xl, ... , Xn E P. The product of a subset of a right 5-module by a subset of 5 is defined similarly.

Let Ro be a sub ring of a ring R and let B be any subset of R. Then the intersection of all subrings of R containing Ro and B is the unique subring of R containing Ro and B and contained in every subring of R containing Ro and B. This intersection is denoted by Ro [B] and is called the ring extension (or the overring) of Ro generated by the set B (in this case we also say that B is the set of generators of the ring Ro[B] over Ro). Of course, Ro[B] coincides with the set of elements of the form 2::?=1 XjbjYi with Xl, ... , Xn, Yl, ... , Yn E Ro and bl , ... , bn E B for some positive integer n (if the ring R is commutative, then Ro[B] is the ring of elements of the form 2::?=1 Xibj with Xl, ... , Xn ERa; bl , ... , bn E B).

Let G be a field and F be a subfield of G (in this case we also say that G is a field extension or an overfield of F). If B is a subset of G, then the intersection of all subfields of G containing F and B is the unique subfield of G containing F and B and contained in every subfield of G containing F and B. This intersection is denoted by F(B) and is called the field extension (or the overfield) of F generated by the set B. (In this case we also say that B is the set of generators of the field F(B) over F and if B is a finite set, we say that F(B) is finitely generated field extension of F).

Clearly, F(B) coincides with the quotient field of the ring F[B]. If G is a field extension of a field F then the transcendence degree of this extension is denoted by trdegp G.

The direct sum of a family {M"hEA of modules over a ring R will be denoted by ElhEAM" and the direct product of the family {M"hEA will be denoted by

Il"EA M". If R[t] is the polynomial algebra in one indeterminate t over a ring R, then

the degree of a polynomial f(t) E R[t] is denoted by deg J(t) (if J(t) == 0; we set degJ(t) = -1). We write J = o(tm ) ifdegJ < m. If R[t l , ... ,tn ] is a polynomial algebra in n indeterminates tl, ... , tn over a ring R, then the total degree of a polynomial J E R[t 1, ... , t n ] will be denoted by deg J, and the degree of J with respect to an indeterminate tj will be denoted by degtj f.

1.2. Some Basic Notions and Results of the Theory of Commutative Rings

In this section we consider the fundamental notions and some classical results of commutative algebra. Despite the fact that an overwhelming majority of these notions and results can be found in books on commutative algebra (for example, in the wonderful book [AM69] under the influence of which the material of this section has been selected), we believe that it is reasonable to include such a section in order to make the book self-contained and to show that the roots of many conceptions of differential and difference algebra lie in the general theory of commutative rings.

Prime and maximal ideals. Radicals. Let A be a commutative ring.

1.2.1. DEFINITION. An ideal P of A is called a prime ideal of A if P f A and

6 1. PRELIMINARIES

one of the following equivalent properties holds:

(1) AlP is an integral domain (i.e., AlP has no zero divisors); (2) the set S = A \ P is closed under multiplication, that is 1 E Sand 81,82 E S

implies 8182 E S (such sets are also called multiplicatively closed or multiplicative) j

(3) for any a1, a2 E A, the inclusion a1a2 E P implies either a1 E P or a2 E Pj (4) if h, 12 are ideals of A and hI2 ~ P, then either h ~ P or h ~ P.

It follows from Definition 1.2.1 that A is an integral domain if and only if (0) is a prime ideal of A.

1.2.2. EXERCISE. Let P, P1 , ... , Pn be prime ideals of A. Prove the following statements.

(a) If h, ... ,In are ideals of A such that nf=lIi ~ P, then Ii ~ P for some i. If nf=lIi = P, then Ii = P for some i.

(b) S = A \ Uf=l Pi is a multiplicative subset of A. ( c ) If! is an ideal of A such that I ~ Uf = 1 Pi, then I ~ Pi for some i (1 :::; i :::; n). (d) If 1 : A -+ B is a ring homomorphism (A and B are commutative rings),

then, for any prime ideal Q of B, the ideal 1- 1 (Q) of A is also prime. Furthermore, there is a one-to-one correspondence between prime ideals of I(A) and prime ideals of A containing Ker I.

(e) Let T be a multiplicative subset of A such that 0 rt. T, and let M be the set of all ideals I of A such that In T = 0. Since (0) E M, the set M is not empty, and by Zorn's lemma it has maximal elements (i.e. there exists an ideal P EM such that Q n T =I 0 for any ideal Q of A such that P ~ Q). Show that every maximal element of M is a prime ideal of A.

(f) Let R = K[X1' ... , xn] be a polynomial ring in indeterminates Xl, ... , Xn over a field K and let 1 E R. Show that the principal ideal (I) = IRis prime if and only if 1 rt. K and the polynomial 1 is irreducible.

(g) Show that the ideal (n) = nZ of the ring Z is prime if and only if n is a prime number.

(h) Show that the set of prime ideals of A has minimal elements with respect to the relation of inclusion.

1.2.3. DEFINITION. Let I be an ideal of a ring A. Then the set {X E A I xn E I for some positive integer n} is an ideal of A which is called the radical of I, and is denoted by r(I). The ideal r«O)) is called the nilradical of A, it is denoted by radA (so that rad A is the set of all nilpotent elements of A: rad A = {x E A I xn = 0 for some n EN}).

1.2.4. EXERCISE. Prove the following properties of the radical.

(a) If I is an ideal of A then r(I) is equal to the intersection of all prime ideals of A containing I (in particular, rad A is equal to the intersection of all prime ideals of A). [Hint: use the result of Exercise 1.2.2 (e).]

(b) If I, J are ideals of A, then: r(r(I)) = r(I)j r(I + J) = r(r(I) + r(J))j r(I) = A if and only if I = A.

(c) If P is a prime ideal of A, then r(pn) = P for any positive integer n.


(d) Let f : A ~ B be a homomorphism of commutative rings, and let J be an ideal of B. Then r(f-l(J)) = f-l(r(J)).

(e) rad(AI rad A) = O. (f) Let D be the set of all zero divisors of A (that is D = {x E A 1 x "# 0

and xy = 0 for some 0"# YEA}). Then D = U",;tor(Annx) where Ann x denotes the annihilator of x i.e., the ideal {a E A 1 ax = O}.

1.2.5. DEFINITION. An ideal m of A is said to be maximal if m "# A and one of the following two equivalent properties holds:

(1) there is no ideal I of A such that m ~ I ~ Aj (2) Aim is a field.

1.2.6. EXERCISE. As above, let A be a commutative ring. Prove the following statements.

(a) The ring A has at least one maximal idealj if I is an ideal of A and 1"# A, then there exists a maximal ideal of A containing Ij if x is a non-invertible element of A (that is xy "# 1 for any YEA), then x is contained in a maximal ideal of A. [Hint: use the Zorn's lemma.]

(b) Every maximal ideal of A is prime. (c) If A is a principal ideal domain (that is an integral domain in which every

ideal is principal), then every prime ideal is maximal. (d) A ring with exactly one maximal ideal is called a local ring. Show that if m

is an ideal of A such that m"# A and every element x E A \ m is invertible, then A is a local ring and m is its maximal ideal.

(e) Let m be a maximal ideal of A such that every element of the form 1 + x with x E m is invertible. Show that A is a local ring.

1.2.7. DEFINITION. The intersection of all maximal ideals of a ring A is called the Jacobson radical of A and is denoted by J(A).

1.2.8. EXERCISE. Show that J(A) = {x E A 11- xy is invertible for all YEA}.

1.2.9. EXERCISE. Let A[x] be the ring of polynomials in one indeterminate x over a commutative ring A. Show that J(A[x]) = radA[x].

1. 2.10. EXERCISE. Prove the following statements (the first of which is known as Nakayama's lemma).

(1) Let A be a commutative ring, M a finitely generated A-module and I an ideal of A such that 1M = M. Then there exists an element x E I such that (1 + x)M = o. If moreover I ~ J(A), then M = O.

(2) Let A be a commutative ring, M an A-module, Nand N' submodules of M, and I an ideal of A. Suppose that M = N + IN' and that either

(a) I is nilpotent (that is In = 0 for some n E N) or (b) I ~ J(A) and N' is finitely generated.

Then M = N.

1.2.11. DEFINITION. Two ideals I, J of a ring A is said to be coprime if 1+ J = A.

8 1. PRELIMINARIES

1.2.12. EXERCISE. Let h, ... , I. be ideals of a commutative ring A such that Ii and Ij are coprime whenever if. j (1 ~ i,j ~ s). Show that h .12 . .... I. = h n 12 n··· n I. and the rings A/(h n··· n I.) and (A/h) E9 ... E9 (A/I.) are isomorphic. (This result is known as the Chinese Remainder Theorem).

Primary decomposition.

1.2.13. DEFINITION. An ideal q of a commutative ring A is called primary if q f. A and the inclusion xy E q (x, yEA) implies either x E q or yn E q for some positive integer n (in other words, q is primary if all zero divisors of the ring A/q are nilpotent elements).

The following exercise contains some properties of primary ideals.

1.2.14. EXERCISE. Let q be an ideal of a commutative ring A, q f. A. Prove the following statements.

(a) If q is primary, then r(q) = P is a prime ideal of A. (In this case we say that q is a p-primary ideal of A.) Note, that the converse is not true: let A = k[x, y, z]/(xy - z2) where k[x, y, z] is the polynomial ring in the indeterminates x, y, z over a field k. Let x, ii, z be canonical images of x, y, z, respectively, in the ring A and let P = (x, z). Show that p is a prime ideal of A, but p2 is not primary (consider the element xii = z2 E p2). However, r(p2) = p.

(b) If r( q) is a maximal ideal of A, then q is primary. (c) If there exists a maximal ideal m such that mn ~ q for some n EN, then q

is primary. (d) Let {ql}'EA be a finite set of p-primary ideals of A, where p is some prime

ideal of A. Then n'EAql is also p-primary. (e) Let q be a primary ideal of A, p = r(q) and x E A. Then the ideal (q : x) =

{a E A I ax E q} is equal to A if x E q; it is equal to q if x tI. p; and it is a p-primary ideal of A if x E P \ q.

1.2.15. DEFINITION. Let I be an ideal of a commutative ring A. A representation of I as a finite intersection of primary ideals (if such representation exists) is called a primary decomposition of I. If I = ni=1 qi is such decomposition with r(qi) f. r(qj) for if. j and qi 1J nj;tiqj (1 ~ i ~ n), then the primary decomposition 1= ni:Iqj is said to be minimal (or irredundant, or reduced).

1.2.16. EXERCISE. Show that if an ideal I has a primary decomposition, then it has also a minimal primary decomposition. [Hint: use Exercise 1.2.14(d)].

1.2.17. EXERCISE. Prove the following statement (which is known as the First Uniqueness Theorem on primary decomposition).

Let I = ni=l qj be a minimal primary decomposition of an ideal I of a commutative ring A. Let Pi = r(qi) (1 ~ i ~ n). Then PI, ... ,Pn are precisely the prime ideals which occur in the set of ideals r((I : x)) with x E A. Therefore, ideals PI, ... ,Pn are independent of the choice of minimal primary decomposition of I.

[Hint: use the results of Exercises 1.2.2(a) and 1.2.14(e)].

With the notation of the previous exercise, the prime ideals PI, ... ,Pn are said to belong to the ideal I, or to be associated with I. The minimal (with respect to


inclusion) elements of the set {.pI, ... , .pn} are called the minimal or isolated prime ideals belonging to I. The others are called embedded prime ideals. It is easy to see that if P is any prime ideal of A such that P 2 I, then P contains a minimal prime ideal belonging to I. Thus, the minimal prime ideals that belongs to I are precisely the minimal elements of the set of all prime ideals containing I.

1.2.18. EXERCISE. Let I = ni=lqi be a minimal primary decomposition of an ideal I of a commutative ring A, and let .pi = r(qi) (1 ~ i ~ n). Show that Ui=d'i = {x E A I (I : x) :j:. I}. In particular, if the zero ideal has a primary decomposition, then the set of all zero divisors of A is the union of the prime ideals belonging to (0).

Let I = ni=1 qi be a minimal primary decomposition of an ideal I of a commutative ring A. The set E of prime ideals belonging to I is said to be isolated if the following condition holds: if.pi = r(qd for some i (1 ~ i ~ n) and .pi ~ .p for some .p E E, then .pi E E.

1.2.19. EXERCISE. Let {.pill ... ,.pi n } be an isolated set of prime ideals of I. Show that C(i 1 n· . ·nqi~ is independent of the choice of minimal prime decomposition of I.

This statement is known as the Second Uniqueness Theorem on primary decomposition. It implies that the isolated components of I (i.e., the primary components qi that correspond to the minimal prime ideals .pd are uniquely determined by I.

1.2.20. EXERCISE. Let A[x] be a polynomial ring in one indeterminate x over a commutative ring A. For any ideal I of A, let l[x] denote the set of all polynomials in A[x] with coefficients in I. Show that the following statements hold:

(a) if.p is a prime ideal of A, then .p[x] is a prime ideal of A[x]; (b) if q is a .p-primary ideal of A, then q[x] is a .p[x]-primary ideal of A[x]; (c) if I = ni=lqi is a minimal primary decomposition in A, then l[x]

ni=1 q;[x] is a minimal primary decomposition in A[x]; (d) if.p is a minimal prime ideal for an ideal I of A, then .p [x] is a minimal prime

ideal for l[x] in the ring A[x];

1.2.21. DEFINITION. An ideal I of a commutative ring A is called perfect or radical if the inclusion xn E I (x E R, n is a positive integer) always implies the inclusion x E I.

Of course, any prime ideal is perfect, as well as every primary and perfect ideal IS prime.

1.2.22. EXERCISE. Let I be an ideal of a commutative ring A such that the inclusion x2 E I (x E A) implies x E I. Show that I is perfect.

1.2.23. EXERCISE. Let A[x] be a polynomial algebra in one indeterminate x over a commutative ring A, I a perfect ideal of A and l[x] the ideal of A[x] consisting of the polynomials whose coefficients belong to I. Show that l[x] is a perfect ideal of A[x].

[Hint: show that if f ft. l[x], then f2 ft. l[x].]

10 1. PRELIMINARIES

1.2.24. EXERCISE. Generalize the statement of the previous exercise to an arbitrary polynomial algebra A[(Xi)iEll over a commutative ring A.

1.2.25. EXERCISE. Let A be a commutative ring. Prove the following statements.

(a) The intersection of any family of perfect ideals of A is a perfect ideal of A. (b) If h ~ 12 ~ ... is an ascending chain of perfect ideals of A, then U~=l In is

a perfect ideal of A. (c) If I is a perfect ideal of A and R ~ A, then I : R is also a perfect ideal of

A.

Rings and modules of fractions. Localization Let A be a commutative ring and S a multiplicative subset of A. Then one can

define a relation", on the set A x S as follows: (a, s) '" (a', s') if and only if there exists S" E S such that (as' - a's) S" = O. It is easy to verify that", is an equivalence relation. Let % (a E A, s E S) denote the equivalence class of (a, s) and let S-l A denote the set of equivalence classes. We can introduce a ring structure on S-l A by defining addition and multiplication as follows: !! + ~ = ~ !!. ~ = aa' (it is

& s' $S" S $' ss' easily verified that these operations are well-defined and S-l A satisfies the axioms of a commutative ring). The zero element of S-l A is ¥ and the unit element is i. The ring S-l A is called a ring of fractions of A with respect to S. Note that S-lA = 0 if and only if 0 E S.

1.2.26. EXERCISE. Show that the canonical mapping f : A ~ S-l A from A to the ring of fractions S-l A has the following universal property: if 9 : A ~ B is a homomorphism of A to a commutative ring B such that g(s) is an invertible element of B for all s E S, then there exists a unique ring homomorphism h : S-l A ~ B such that 9 = h 0 f.

Let p be a prime ideal of A. Obviously, S = A \ P is a multiplicative subset of A so we can consider the corresponding ring of fraction S-l A which is denoted by Ap. The elements ~ E Ap with a E p form an ideal m of Ap. If ¥ E Ap \ m, then b ~ p, hence bE S, so that ¥ is invertible in Ap. Therefore, Ap is a local ring with the maximal ideal m. The process of switching from A to Ap is called localization at p.

The construction of ring of fraction can be easy adapted to modules. Let M be an A-module, and S a multiplicative subset of A. Then we can define a relation '" on the set M x S as follows: (m, s) '" (m', s') if and only if S" (sm' - s' m) = 0 for some element S" E S. As before, we have an equivalence relation. Let 7 denote the equivalence class of the pair (m, s) E M x S, and let S-l M denote the set of all such equivalence classes. Then S-l M can be considered as a S-l A-module such that !!l. + m' = s'mtsm' and !!!!l. = am where!! E S-l A !!l. !!l. E S-l M (it is

s s' sal s s ss' , s ' s' s' easily checked that these operations are well-defined). If S = A \ p for some prime ideal p of A, then we write Mp instead of S-l M.

Let f : M ~ N be an A-module homomorphism. If S is a multiplicative subset of A, then f induces the S-l A-module homomorphism S-l f : S-l M ~ S-l N such that (S-l J) (7) = J(:n) (m E M, s E S).


1.2.27. EXERCISE. Let A be a commutative ring, S a multiplicative subset of A and M, N be A-modules. Show that S-1(M + N) = S-1 M + S-1 Nand S-1(M n N) = S-1 M n S-1 N.

1.2.28. EXERCISE. Let A be a commutative ring, S a multiplicative subset of A and f : M -+ N, 9 : N -+ P homomorphisms of A-modules. Prove the following statements.

(a) S-1(g 0 f) == (S-1g) 0 (S-1 f).

(b) If the sequence 0 -t M ~ M .!4 P -+ 0 is exact, then

0-+S-1M S-I')S-1N s-19)S-1p-+0

is an exact sequence of S-1 A-modules (therefore, the correspondences M -+ S-1 M, f -+ S-1 f define an exact functor from the category of A-modules to the category of S-1 A-modules).

(c) There exists a unique isomorphism of S-1 A-modules

such that cp(% ® m) = a;, for all a E A, m E M, s E S.

1.2.29. REMARK. Recall that a module M over a commutative ring A is fiat if the functor ®AM from the category of A-modules to itself is exact (i.e., for any exact sequence 0 -+ N' -+ N -+ Nil -+ 0 of A-modules the sequence

o -+ N' ®A M -+ N ®A M -+ Nil ®A M -+ 0

of A-modules is also exact). Exercise 1.2.28(b,c) shows that S-1 A is a flat Amodule.

1.2.30. EXERCISE. Let S be a multiplicative subset of a commutative ring A and let M, N be A-modules. Show that there exists a unique isomorphism f : S-1 M ®S-IA S-1 N -+ S-1 (M ®A M) such that f (~ ® T) == mft". In particular, if p is any prime ideal of A, then Mil ®Ap N'(J ~ (M ®A Mh.

Let A be a commutative ring, S a multiplicative subset of A and S-1 A the corresponding ring of fractions. Every ideal I of A determines the ideal S-1 I = {% E S-1 A I a E I, s E S} of S-1 A that is called an extension of I. Clearly, S-1 I is generated by the set cp(I) where cp : A -+ S-1 A is the canonical mapping given by the formula cp(a) = I (a E A). Conversely, with any ideal J of S-1 A, one can associate the ideal cp-1 (J) == {a E A I T E J} of the ring A; this ideal is called a restriction of the ideal J. We summarize the basic properties of both operations in the following exercise.

1.2.31. EXERCISE. Let S be a multiplicative subset of a commutative ring A, I, It, 12 ideals of A and J, h, J2 ideals of S-1 A. Show that the following relations hold.

(a) cp-1(S-1I) 2 I, S-1(cp-1(J)) = J. (b) S-1(/1 n 12 ) = S-1It n S-1 h cp-1(J1 n J2 ) == cp-1(Jd n cp-1(J2 ).

12 1. PRELIMINARIES

(c) S-1(h + 12) = S-1 h + S-1 h rp-1(J1 + J2) = rp-1(Jt} + rp-1(J2). (d) S-1r(I) = r(S-1 I), rp-1(r(J)) = r(rp-1(J)). (e) If P is a prime ideal of A and P n S = 121, then S-1 p is a prime ideal of S-1 A

and rp-1(S-1p) = p. Therefore, the correspondence p ~ S-1p between the set of those prime ideals of A which satisfy the condition p n S = 121 and the set of all prime ideals of the ring S-1 A is bijective and preserves the relation of inclusion. The formula p' -+ rp-1 (p') defines the inverse mapping.

(f) Let 1 = n?=1qi be a minimal primary decomposition of I. Let Pi = r(q;) (1::; i ::; n) and suppose that qi are arranged in such a way that Pi nS = 121 for i = 1, ... ,m and PinS -::j:. 121 for i = m+1, ... ,n (0::; m::; n). Then the ideals S-1 1 and rp-1(S-1 I) have the following minimal primary decompositions:

1.2.32. EXERCISE. It is easy to see that if A is an integral domain and S is a multiplicative subset of A, then the ring S-1 A is also an integral domain and there is a natural isomorphism between S-1 A and a subring of the field of fractions Q(A) of A (actually, Q(A) = A(o) = T- 1 A where T = A \ (0)). Considering the localizations Ap with respect to prime ideals p of A as subrings of Q(A), show that A = npAp = nmAm where p ranges over the set of all prime ideals and m ranges over the set of all maximal ideals of A.

Noetherian and Artinian rings and modules. Let A be a commutative ring. An A-module M is said to satisfy the ascending

(respectively, descending) chain condition for submodules if any sequence of A-submodules M1 ~ M2 ~ ... (respectively, M1 ;2 M2 ;2 ... ) of M becomes stable, i.e.,~ if Mn = Mn+1 = ... for some positive integer n. A module M is said to satisfy the maximal (respectively, minimal) condition if every non-empty family of submodules of M, ordered by the inclusion relation, has a maximal (respectively, minimal) element.

1.2.33. DEFINITION. An A-module M is called Noetherian if one of the follow-ing equivalent properties holds:

(1) M satisfies the ascending chain condition for submodulesj (2) M satisfies the maximal conditionj (3) every A-submodule of M (including M itself) is finitely generated.

1.2.34. DEFINITION. An A-module M is called Artinian if one of the following equivalent properties holds:

(1) M satisfies the descending chain conditionj (2) M satisfies the minimal condition.

1.2.35. EXERCISE. Prove the equivalence of the properties (1)-(3) in Definition 1.2.33 and of the properties (1), (2) in the previous definition.


1.2.36. DEFINITION. A ring A is called Noetherian (respectively, Artinian) if it is a Noetherian (respectively, Artinian) A-module.

It is easy to see that the ring of integers ~ and the ring k[x] of polynomials in one indeterminate x over a field k are Noetherian, but they are not Artinian. The polynomial ring k[Xb ... , Xn , ... J in an infinite number of indeterminates over a field k is neither Noetherian, nor Artinian (show this).

We present below some elementary statements about Noetherian rings and modules. Their proofs can be found in any fundamental course on commutative algebra (see, for example, [ZS58], [ZS60J or [AM69J).

1.2.37. EXERCISE. Let 0 -t M' -t M -t Mil -t 0 be an exact sequence of A-modules. Prove that M is Noetherian (respectively, Artinian) if and only if both M' and Mil are Noetherian (respectively, Artinian) A-modules.

1.2.38. EXERCISE. Prove that a direct sum of a finite number of Noetherian (respectively, Artinian) modules is a Noetherian (respectively, Artinian) module.

1.2.39. EXERCISE. Prove the following statements.

(a) If A is a Noetherian (respectively, Artinian) ring and M is a finitely generated A-module, then M is Noetherian (respectively, Artinian).

(b) If A is a Noetherian (respectively, Artinian) ring and I is an ideal of A, then AI I is a Noetherian (respectively, Artinian) ring.

(c) A homomorphic image of a Noetherian ring is also a Noetherian ring. (d) If A is a Noetherian ring and S is a multiplicative subset of A, then the

ring of fractions S- l A is Noetherian. In particular, the localization of a Noetherian ring at a prime ideal is Noetherian.

1.2.40. THEOREM (HILBERT'S BASIS THEOREM). Let A be a Noetherian ring. Then the polynomial ring A[Xl, ... , xnJ in n indeterminates Xl, ... , Xn over A is Noetherian.

Sketch of the proof. We proceed by induction on n. The step of induction is trivial (since A[Xl, ... ,XnJ = A[XI, ... ,Xn-l][XnJ), so it is sufficient to prove the theorem for n = 1, i.e. to prove that the polynomial ring B = A[xJ in one indeterminate is Noetherian. Let J be an ideal of B; we will show that J is finitely generated. Let I = {O} U { leading coefficients of elements of J}. It is easy to see that I is an ideal of A, and I "# (0) if J "# (0). Since A is Noetherian, 1= (CI, ... , cr ) for some elements Cl,"" Cr EA. Let is be an element of J with the leading coefficient Ci (1 ::; i ::; r) and let s = maxl~i~r{degf;}. It is easy to prove that J = (11, ... , fr) + J n (I:~=l Axk ). Being an A-submodule of a finitely generated (hence, Noetherian) A-module I:~=l Axk, J n (I:~=l Axk) is also a finitely generated A-module: J n (I:~=l A.) = I:~l Agi for some elements gl, ... ,gm E J. Then 11, ... ,fr,gl, ... ,gm generate the ideal J. 0

1.2.41. COROLLARY. Let A be a Noetherian ring and B a finitely generated A-algebra. Then B is Noetherian.

1.2.42. EXERCISE. Prove the analogue of the Hilbert Theorem for the formal power series rings: if A is a Noetherian ring, then the formal power series ring A[[xlJ is also Noetherian. [Hint: every nonzero element of A[[xJ1 has the form

14 I. PRELIMINARIES

f = E:o Ci X ; where Cj E A for all i = 0,1, ... and there are nonzero elements among Ci. Set degf = min{i E N I Cj i= O} for f i= 0 and deg(O) = -00 and use the idea of the proof of Theorem 1.2.40].

1.2.43. EXERCISE. An ideal I of a commutative ring A is said to be irreducible if for any ideals J, K of A such that J n K = I, we have either J = lor K = I. Prove the following statements for a Noetherian ring A.

(a) Every ideal of A is a finite intersection of irreducible ideals. (b) Every irreducible ideal is primary.

It follows from Exercise 1.2.43 that every ideal of a Noetherian commutative ring has a primary decomposition.

1.2.44. EXERCISE. Let A be a Noetherian ring. Prove the following statements.

(a) Every ideal I of A contains a power of its radical (i.e. there exists n E N such that (r(1))n ~ 1). In particular, there exists mEN such that (radA)m = (0) (that is, the nilradical of a Noetherian ring is nilpotent).

(b) Let I be an ideal of A, I i= A. Then the prime ideals which belong to I are precisely the prime ideals which occur in the set of ideals {(I : x) I x E A}.

1.2.45. EXERCISE. Prove that a commutative ring A is Noetherian if and only if each prime ideal of A is finitely generated. [Hint: suppose that the family ~ consisting of all ideals of A which are not finitely generated is non-empty. By Zorn's lemma there exists a maximal element in ~, sayan ideal P. If P is not prime, there exist elements x, YEA \ P such that xy E P. Then P ~ (P, x), hence (P, x) is finitely generated ideal of A. Setting J = P : (x), we have J ;2 ~P, y) ~ Pj therefore J = (Cl' ... , cm) for som~ elements Cl, ... , Cm. E A. Show that If al + b1 x, ... , an + bnx (aj E P, b; E A, 1::; , ::; n) generate the ideal (J, x), then al, ... ,an ,CIX, ... ,Cn X generate the ideal P. The fact that the ideal P is finitely generated implies a contradictionj hence P is prime, that, in turn, contradicts to the initial assumption that all prime ideals are finitely generated].

Let M be a module over a commutative ring A. A descending chain

of sub modules of M is said to be a Jordan-Holder series (or a composition series) of M if each module Mi -1 / Mi (i = 1, ... , n) is simple (that is, it has no submodules except 0 and itself). The number n is called the length of the chain.

1.2.46. EXERCISE. (Jordan-Holder Theorem). If an A-module M has two composition series

and M = Mo' -:J MI' -:J ... -:J M' = 0

~ ~ ~ m ,

then m = n and there exists a permutation (j of {O, 1, ... , n - I} such that M;f Mi+1 ~ M~(j/ M~(j)+l. (In this case the integer n is said to be the length of the A-module Mj it is denoted by IA(M).) Furthermore, every chain Nl ~ ... ~ Nk = 0


of submodules of M can be extended to a composition series. [Hint: let I(M) denote the length of the shortest composition series of M (I(M) = +00 if M has no composition series). First of all, show that if N ~ M, and I(M) < +00, then I(N) < I(M). Then prove that any descending chain m M has length ~ I(M), that is all composition series of M have the same length I(M). The other statements of the theorem can be easily proved by induction on I(M)].

1.2.4 7. EXERCISE. Let A be a commutative ring. Prove the following statements.

(1) Let M be an A-module. Then M is of finite length if and only if M is Noetherian and Artinian A-module.

(2) Let 0 -+ M' -+ M -+ Mil -+ 0 be an exact sequence of A-modules. Then M is of finite length if and only if M' and Mil are of finite length; in this case IA(M) = IA(M') + IA(M") (thus, the function IA(-) is additive on the class of all A-modules of finite length).

1.2.48. EXERCISE. Let A be an Artinian ring. Prove the following statements:

(1) every prime ideal of A is maximal (therefore, J(A) = radA); (2) the ring A has only a finite number of maximal ideals; (3) the nilradical rad A is nilpotent.

A finite strictly increasing sequence Po C PI C ... C Pn of prime ideals is said to be a chain of prime ideals of the length n. The supremum of the lengths of all chains of prime ideals of a commutative ring A is said to be the dimension of A and is denoted by dimA (so that dim A E N or dim A = +(0). Clearly, dimZ = 1; if A is a field, then dimA = 0 and dimA[x] = 1 (A[x] is a polynomial ring in one indeterminate x over A).

1.2.49. PROPOSITION. A commutative ring A is Artinian if and only if A is Noetherian and dimA = O.

PROOF. Suppose that A is Artinian. By Exercise 1.2.48 we have dimA = 0, (rad A)n = 0 for some n EN and radA = ml n ... n mk, where ml,"" mk are all maximal ideals of A. Therefore, n:=l m? ~ (rad A)n = O. Now, it is sufficient to show that if nl ... l\p = 0 for some maximal ideals of an Artinian ring A, then A is Noetherian. To prove this fact, let us consider the chain A :J nl :J nl . n2 :J ... :J nl .. 'l\p = O. Each factor nl ... n;-dnl ... n; is a vector A/n;-space which is Artinian and, therefore, has a finite dimension. It follows, that nl ... n;-dnl ... ni is a Noetherian A-module (i = 1, ... , p). Applying the result of Exercise 1.2.37, we obtain that A is Noetherian.

Conversely, let A be Noetherian, and dimA = O. Since the zero ideal has a primary decomposition (see Exercise 1.2.43 (b)) and dimA = 0, A has only a finite number of minimal prime ideals ml, ... , mq which are also maximal ones. By Exercise 1.2.48(3), (mI' .. mq)1 = 0 for some lEN. Now, by the same reasons as in the first part of the proof, we obtain that A is an Artinian ring. This completes the proof. 0

1.2.50. COROLLARY. Any finitely generated module over an Artinian commutative ring is a module offinite length.

16 1. PRELIMINARIES

Integral dependence. Let A be a subring of a commutative ring B. An element x E B is called integral

over A ifthere exist elements al,"" am E A such that Xm+alX m- l + .. ·+am-lX+ am = O. This equality is said to be a relation of integral dependence for x over A. If every element of B is integral over A, then B is called an integral extension of A (the inclusion A ~ B is also said to be an integral extension).

1.2.51. EXERCISE. Show that the only elements of the field Q integral over IZ are the elements of IZ.

1.2.52. EXERCISE. Let A be an integral domain with the quotient field I< and let L be a field extension of I<. Show, that if an element 0' E L is algebraic over I<, then there exists a nonzero polynomial f E A[X] such that f(O') = 0, and there exists a nonzero element a E A such that aO' is integral over A.

As a consequence of this statement we obtain the following statement: if A C B is an integral ring extension and if 5 is a multiplicative subset of A, then 5- 1 A C 5- 1 B is an integral extension. -

1.2.53. EXERCISE. Let A ~ B be an integral extension and I an ideal of B. Show that B I I is an integral extension of AI A n I.

1.2.54. PROPOSITION. Let A be a subring of a commutative ring B. For every element x E B, the following conditions are equivalent:

(1) x is integral over A; (2) the ring A[x] is finitely generated as an A-module; (3) A[x] is contained in a subring of B which is finitely generated as an A

module; (4) there exists a finitely generated A-module M ~ B such that xM ~ M and

for any y E A[x], the equality yM = 0 implies y = O.

PROOF. The implications (1) --+ (2) --+ (3) are trivial, (3) --+ (4) is also easily checked (we take M to be a subring described in the condition (3)).

To prove the implication (4) -+ (1), we suppose that M = 2:7=1 Ami and xmi = 2:;=1 aijmj where aij EA. Then

n

2:(<lijX-aij)mj=O (i=I, ... ,n), j=l

where <lij is the Kronecker delta. By multiplying on the left by the adjoint of the matrix (<lijX - aij), we obtain that D = det(<lijx - aij) annihilates each mi, hence DM = O. By the condition (4) we have D = 0, so x is integral over A (in this case det(<lijX - aij) = 0 is the corresponding equation of integral dependence). This completes the proof. 0

1.2.55. REMARK. If we strengthen the first condition of (4) in the previous proposition by assuming that, for a certain ideal I of B, the inclusion xM ~ 1M holds, then there exists a relation of integral dependence for x with the coefficients from I.


1.2.56. COROLLARY. Let A be asubring ofa commutative ring B. The set of all elements of B, integral over A, forms a subring A of B. If elements Xl, ... ,Xn E B are integral over A, then the ring A[X1' ... , xn] is a finitely generated A-module.

The ring A in the previous corollary is called the integral closure of A in B. If A = A, then A is said to be integrally closed in B. A domain is said to be integrally closed if it is integrally closed in its quotient field.

1.2.57. EXERCISE. Show that, with the above notation, the ring A is integrally closed in B.

1.2.58. COROLLARY. If A ~ B ~ C are commutative rings such that the ring extensions A ~ Band B ~ C are integral, then the extension A ~ C is also integral.

1.2.59. PROPOSITION. Let A ~ B be an integral ring extension. A prime ideal P of B is maximal if and only if the ideal P n A of A is maximal.

PROOF. Clearly, it is sufficient to prove the statement for the case p = 0, i.e. to prove that an integral domain B is a field if and only if A is a field. Let A be a field and 0 i= X E B. Let xm + a1Xm-1 + ... + am = 0 (a; E A) be a relation of integral dependence for x over A of the lowest possible degree. Since B has no zero divisors, we have am i= 0, hence x-I = _a;1(xm- 1 + a1Xm-2 + ... + am-I} E B, thus B is a field. Conversely, let B be a field and 0 i:- yEA. Then y-1 E B and we can write a relation of integral dependence of y-1 over A: y-n + ai y-n+1 + ... + a~ = 0 (a~ E A). Therefore, y-1 = -(ai +a~y+·· .+a~yn-1) E A, hence, A is a field. 0

1.2.60. COROLLARY. Let A ~ B be an integral ring extension and let PI. P2 be prime ideals of B such that PI ~ P2 and PI n A = P2 n A. Then PI = P2.

1.2.61. EXERCISE. Let A ~ B be an integral ring extension and let B be an integral domain. Show that if every nonzero prime ideal of A is maximal, then every nonzero prime ideal of B is also maximal.

1.2.62. PROPOSITION. If a ring extension A ~ B is integral, and ifp is a prime ideal of A, then there exists a prime ideal p' of B such that p' n A = p.

PROOF. At first, suppose that A is a local ring with a maximal ideal p. Let p' be any maximal ideal of B. Then, by Proposition 1.2.59, p' n A is a maximal ideal of A, hence p' n A = p.

We now pass to the general case. Let S = A \ p. The extension S-l A ~ S-l B is integral (see Exercise 1.2.52), and we have a commutative diagram

A --+ B

A p =S-lA --+ S-lB

with the natural mappings a and {3. Let p" be an arbitrary maximal ideal of S-l B and let p' = {3-1 (p"). Then pIt nAp = pAp (by the first part ofthe proof), whence p' n A = {3-1(p") n A = {3-1(p" nAp) = {3-1(pAp) = p. This completes the proof. 0

As a consequence of this proposition we obtain the following well-known theorem.

18 1. PRELIMINARIES

1.2.63. THEOREM (" GOING UP"). Let A S; B be an integral ring extension, PI S; ... S; Pm S; ... S; Pn a chain of prime ideals of A, and

P~ S; ... S; P~ (1.2.1)

a chain of prime ideals of B such that P~ n A = PI, ... ' P~ n A = Pm (O:S: m < n). Then there exists an extension of (1.2.1) to a chain of prime ideals of B of the form

p~ S; ... S; P~ S; ... S; P~

such that P: n A = Pi, i = 1, ... , n.

1.2.64. LEMMA. Let A S; B be an integral ring extension and let I be an ideal of A. Then r(I B) = {x E B I there exist elements aI, ... , am E I such that xm + a1xm-1 + ... + am = O}.

PROOF. If xm + alxm - 1 + ... + am = 0 (al, ... , am E /), then xm E IB, hence x E r(I B). Conversely, let xn E I B for some n EN, n 2: 1. Then xn = 2:7=1 a:xj where a: E I, Xi E B (1 :s: i :s: k). To complete the proof it is sufficient to apply Remark 1.2.55 to the finitely generated A-module M = A[X1, ... , Xk] (it is easy to see that xn M S; 1M). 0

1.2.65. LEMMA. Let A be an integrally closed domain with a quotient field K and let f(X) = xm + aOX m- 1 + ... + am, g(X) = xn + bl X n- 1 + ... + bn be polynomials in one indeterminate X over K. Then f(X)g(X) E A[X] if and only if f(X) E A[X] and g(X) E A[X].

PROOF. Of course, we only need to proof that the inclusion f . 9 E A[X] implies f E A[X] and 9 E A[X]. Let L be a field extension of K in which f and 9 can be factorized into a product of linear factors: f(X) = n::1 (X - Oi), g(X) = n;=l (X - (3j). Obviously, Oi, {3j are integral over A. The coefficients of f and g, as symmetric functions of 01, ... , Om and {31, ... ,{3n, respectively, are integral over A and lie in K. Therefore, these coefficients belong to A, because the domain A is integrally closed. 0

1.2.66. LEMMA. Let A be an integrally closed domain and A S; B an integral ring extension such that no element of A is zero-divisor in B (that is ab t 0 if o t a E A and 0 t b E B). Let A[X] be the polynomial ring in one indeterminate X over A, u E Band '-Pu : A[X] -+ B a homomorphism defined by '-Pu (f) = f( u). Then Ker '-P" is a principal ideal generated by a monic polynomial (i.e. by a polynomial whose leading coefficient is 1).

PROOF. Let K be the quotient field of A and I = Ker '-Pu. Then the ideal I K[X] of K[X] is generated by some monic polynomial g(X) E K[X]. Furthermore, if h(X) E A[X] is a monic polynomial such that h(u) = 0, then h = 9 . hI for some monic polynomial hI E K[X], and Lemma 1.2.65 shows that 9 E A[X]. We shall prove that I = (g). Indeed, since 9 E IK[X], there exists an element 0 t a E A such that ag E I. It follows that ag(u) = 0, hence g(u) = 0, i.e. gEl. Now, if f E IS; IK[X], then there exist 0 tal E A, h E A[X] such that f = all hg, i.e. at! = fIg. Let 11 and 9 be the residue classes of hand g, respectively, in the ring (Aja1A)[X]. Then 119 = 0, hence 11 = 0 (since 9 is monic, it is not a zero-divisor). Thus, all the coefficients of h are divisible by al in A, hence a\ h E A[X]. It follows that f is divisible by gin A[X); thus, Ker '-Pu = (g). 0

1.3. GRADED AND FILTERED RINGS AND MODULES 19

1.2.67. THEOREM ("GOING DOWN"). LetAC B bean integral ring extension. If A is an integrally closed domain no element of which is zero-divisor in B, if

is a chain of prime ideals of A, and if, furthermore,

(1.2.2)

is a chain of prime ideals of B such that

qm n A = Pm, ... , qn n A = Pn (1 < m ~ n),

then there exists an extension of (1.2.2) to a chain of prime ideals of B of the form

such that qinA=Pi (i= 1, ... ,n).

PROOF. Obviously, it is sufficient to prove the theorem in the case m = n = 2. Let S denote the multiplicative subset {ax I a E A \ PI, x E B \ q2} of B. Clearly, o rt s, A \ PI ~ Sand B \ q2 ~ S. Furthermore, S n P1B = 0. Indeed, let ax E S n P1B where a E A \ PI, X E B \ q2. By Lemma 1.2.64, there exists a polynomial f(X) = X le + a1XIe-1 + ... + ale E A[X) such that ai E PI (1 ~ i ~ k) and f(aX) = O. Let II(X) = aleXIe + alale-lXIe-1 + ... + ale E A[X). Then lI(x) = 0 and, by Lemma 1.2.66, there exists a monic polynomial g E A[X) such that Ker(<p" : A[X]-+ A) = (g) (we use the notations of Lemma 1.2.66). Therefore, II = hg for some h E A[X], so for the corresponding residue classes in the ring (Ajpd[X) we have hg = II = aleXIe, hence 9 = Xl for some l, 1 ~ l ~ k, and g(X) = Xl + a~XI-1 + ... + a: where aL ... , a: E Pl. Since g(x) = 0, we have x E r(p1B) (by Lemma 1.2.64), i.e. some power of x belongs to P1B, hence to q2, contrary to the assumption. Thus, S n P1B = 0, so P1(S-1 B) is a proper ideal of S-l B which is contained in some maximal ideal m of S-l B. Let q1 = 0:- 1 (m), where 0: is a natural ring homomorphism B -+ S-l B (x -+ j). Clearly, ql nS = 0,

whence q1n(A\P1) = 0 and q1n(B\q2) = 0. It follows that qlnA ~ PI and q1 ~ q2. Since o:-lpI(S-l B)) = P1B, we have m;2 P1(S-1 B), q1 n A ;2 P1B n A ;2 PI and, therefore, q1 n A = Pl. This completes the proof. 0

1.3. Graded and Filtered Rings and Modules

A ring A is said to be graded if there is a family {A (n) I n E IE} of subgroups of additive group of A such that A = EenEZ A (n) (i.e. A is the direct sum of IE-modules A(n) (n E IE)) and A(i)A(i) ~ A(i+i) for all i,j E IE. Elements ofUnEzA(n) are called homogeneous elements of A and elements of A(i) (i E IE) are called homogeneolJs elements of degree i (if a E A(i), we shall write deg a = i). Any nonzero element a E A can be written, in a unique way, as a finite sum a1 + .. ·+am (m E N, m ~ 1), where aj E A(leo) for some integers k1, ... , km . In this case elements a1, ... , am are called homogeneolJs components of a. The abelian groups A(n) (n E IE) are called

20 1. PRELIMINARIES

homogeneous components of the graded ring A; if A(n) = 0 for all n < 0, then A is said to be a graded ring with a positive grading or a positively graded ring (in what follows, we shall mostly deal with such graded rings). It is easy to see that the homogeneous component A (0) is a subring of a positively graded ring A = EBnEl\I A(n), and the direct sum A+ = EBnEl\I,n;to A(n) is a two-sided ideal of A such that AI A+ == A (0) .

Let A = EBnEZ A(n) be a graded ring (here and below it means that A(n) (n E 2:) are the homogeneous components of the graded ring considered). A subring (respectively, left, right or two-sided ideal) B of A is called a homogeneous or graded subring (respectively, left, right or two-sided ideal) of A if B = EBnEZ(B n A(n»), i.e. for any element a = al + ... + am E B (where al,"" am are homogeneous components of a in the graded ring A) we have ai E B for all i = 1, ... , m.

1.3.1. EXAMPLE. Any ring A can be considered as a graded ring with trivial grading: A = EBnEzA(n) where A(O) = A and A(n) = 0 for n:j:. O.

1.3.2. EXAMPLE. Let K be a ring, {Xl, ... , Xn} (n EN, n 2: 1) a set of symbols and FK(Xl, ... , xn) a free left K-module whose basis is the set of symbols

{1,XillXi,Xi2,Xi,Xi2Xi" ... ,Xi," 'Xi~,"'; 1 ~ ik ~ n for k = 1,2, ... }.

If we set (klXi, ... xi~)(k2Xj. ... Xjp) = klk2Xi, .. . Xi~Xj. ... Xjp (kl' k2 E K), consider 1 as a unit and demand the validity of the distributive laws, then FK(Xl, .. . , xn) becomes a ring, which is said to be a free K-ring generated by the set of symbols {Xl"'" x n }. This ring can be considered as a positive graded

one whose sth homogeneous component Fj;) (Xl, ... ,Xn ) (s EN, s 2: 1) is the free left K-module generated by the set {Xi, ... Xi, I 1 ~ ik ~ n for k = I, ... } and

Ff) = K (we identify an element k E K with k . 1 E FiO»). Let J be the two-sided ideal of i'K(Xl, .. "Xn) generated by the set {XiXj

XjX; I 1 ~ i,j ~ n}. Then J is a homogeneous ideal of FK(Xl, ... , xn) and the ring of residue classes of FK(Xl, ... , Xn) with respect to J is called the polynomial ring over K in the indeterminates Xl, .. " xn (here we denote the residue class of an element Xi E FK(Xl, ... , xn) (1 ~ i ~ n) by the same symbol x;). This ring is denoted by K[Xl, ... , xnl and can be naturally considered as a graded ring whose homogeneous elements of degree p (p E N) are of the form kx~' ... x~n with k E K, r::;=lkj = p (clearly, in the ring K[Xl, ... ,Xn], we have XjXj = XjXj for all i,j = 1, ... ,n); such grading of K[xl, ... ,xnl will be called standard. Note, that one can also consider other gradings of R = K[Xl, ... , xn]. For example, let aI, ... ,an E 2: and let R(s) (s E 2:) be a left K -module generated by the set of all elements xi' ... x~n (11, ... , In E N) such that ~j=l ajlj = s. Obviously, R = EBsEz R(s) and R(p) R(q) S; R(p+q) for all p, q E 2:, so we have a grading of R.

In this case we say that Xi has weight aj (1 ~ i ~ n) and that the grading is obtained by assigning the weights al, ... , an, respectively, to the generators Xl,.'" Xn of the polynomial ring K[Xl' ... , xnl.

1.3.3. EXAMPLE. Let K be a field, R = K[Xl, ... , xnl (n E N, n 2: 1) a polyno-mial ring over K in a family of indeterminates {Xl, ... , xn}, and G = K (Xl, ... , xn) the field of rational fractions over K in Xl, ... ,Xn (i.e. G is the quotient field of


the integral domain R). Let F be the su bfield of G consisting of all rational fractions of the form J XI"",Xn , where f(X1, ... , Xn) E Rand g(X1' ... , xn) E Rare

g Xl "",X n

homogeneous polynomials (with respect to the standard grading of R). Then F can be considered as a graded ring with the grading induced by the following degree

function: deg C :~ ::::::: ) :::: deg f(Xl, ... ,xn) - deg g(Xl' ... ,xn). Commutative graded rings of this type are well known, they arise as homogeneous coordinate rings and function fields of projective curves in Algebraic Geometry.

l.3.4. EXAMPLE. Let A :::: EBnEZA(n) be a graded ring and J a two-sided homogeneous ideal of A. Then J :::: EBnEz(JnA(n)) and the ring ofresidue classes of A with respect to J can be considered as a graded ring: AI J:::: EBnEz(A(n)+J I J) ~ EBnEzA(n)IJ(n), where J(n):::: JnA(n).

Let A:::: EBnEZA(n) be a graded ring. A left A-module M is said to be a graded left A-module if there exists a family {M(n) I n E Z} of subgroups of additive group of M such that M :::: EBnEZ M(n) (i.e., if M is considered as a Z-module, then M is the direct sum of Z-modules M(n) (n E Z)) and A(i)MU) ~ M(i+j) for all i, j E Z. The concepts of homogeneuos elements, homogeneous elements of degree i (i E Z), homogeneous components, homogeneous (or graded) submodules of a graded module are defined in perfect analogy to the corresponding concepts for a graded ring. Note, that if N :::: EBnEZ N(n) is a homogeneous submodule of a graded A-module M :::: EBnEZ M(n) (i.e. N(n) :::: N n M(n) for all n E Z), then MIN can be also considered as a graded A-module:

l.3.5. EXAMPLE. Let A :::: EBnEZA(n) be a graded ring and F a free left A-module with a basis iI, ... , fm (m E N, m ~ 1). Then F can be naturally considered as a graded A-module F :::: EBnEZ F(n) where F(n) :::: 2:::1 A(n) k Furthermore, for any m-tuple (al, ... ,am ) E zm, one may consider the corresponding grading of F: F:::: EBnEZF(n)(a1, ... ,am) where F(n)(a1, ... ,am) :::: 2:::1 A(n-a;) J; for all n E Z. More generally, if F is a free left A-module with a basis {f>.hEA' then for any family of integers a :::: {a>'hEA we have the following grading of F: F:::: EBnEZ F(n)(a) where F(n)(a) :::: 2:>'EA A(n-a>.) f>. for all n E Z.

l.3.6. REMARK. Let A:::: EBnEZA(n) be a graded ring and M:::: EBnEZM(n) a finitely generated left A-module with homogeneous generators Xl, ... ,Xs such that Xj E M(kj) (1 :::; j :::; s) for some integers k1 , ... , k •. Then it is easy to see that M(n) :::: '"'~ A(n-kj)x' for every n E Z

~J=l J .

The following theorem (the proof of which can be found in [ZS60, Ch. VIII, § 6]) shows that if the homogeneous component A(O) of a graded ring A :::: EBnEZ A(n) is a skew field, then any grading of a free A-module is of the form described in Example l.3.5 (for some family of integers {a>. hEA).

l.3.7. THEOREM. Let A :::: EBnEZA(n) be a graded ring whose homogeneous component A(O) is a skew field, and let M be a projective graded A-module. Then

22 I. PRELIMINARIES

M is a free A-module, and every system of homogeneous generators of M over A contains some basis of M.

1.3.8. DEFINITION. Let M = 61nEZM(n) and N = 61nEZN(n) be graded modules over a graded ring A. A homomorphism C{) : M -t N is said to be a homomorphism of graded modules of degree P (p E ~) if C{)(M(n» ~ N(n+p) for all n E ~. The restriction C{)n : M(n) -t N(n+p) of C{) to M(n) (n E ~) is called the nth component of C{). By an exact sequence of graded A-modules (where A is some graded ring) we mean a sequence of the form

where Li, Li+1' Li+2' ... are graded modules over A and C{)i, C{)i+1, ... are A-homomorphisms of graded modules of degree 0 such that Ker C{)i+1 = 1m C{)i for every z.

1.3.9. DEFINITION. Let A = 61nEZ A(n) be a graded ring. A graded left Amodule F is said to be a free graded A-module if it has a basis consisting of homogeneous elements or, equivalently, if there exists a family {a~} ~EA of integers such that F ~ 61~EA A(a~) where A(a~) = 61nEZ A(n-a~) and ~ denotes an isomorphism of graded A-modules. Thus, the modules considered in Example 1.3.5 are precisely free graded A-modules.

Let M = 61nEZ M(n) be a finitely generated graded module over a graded left Noetherian ring A = 61nEZ A(n) whose homogeneous component A(O) is a skew field. Then the following algorithm for constructing a free resolution of M can be proposed.

Let {ml,"" md (k EN, k ~ 1) be a family of homogeneous generators of M over A such that mi E M(ai) for some integers a1, ... , ak. By Remark 1.3.6, if F1 is a free graded A-module with homogeneous components Fin)(al,"" ak) = 2:7=1 A(n-aj) /j, then we have the exact sequence of graded A-modules F1 ~ M -t 0 where C{)l is the epimorphism of graded modules such that C{)l (/j) = mj for all j = 1, ... , k. Clearly, Nl = Ker C{)l is a homogeneous A-submodule of Fl. Since the ring A is left Noetherian, Nl is finitely generated over A, so we can take a finite system of homogeneous generators of the graded module N 1 and (as above) consider an appropriate epimorphism ..pl : F2 -t Nl (F2 is a finitely generated free graded A-module whose grading is of the form described in Example 1.3.5). We obtain the exact sequence of graded A-modules

F2

1 o

where C{)2 = ..p1 0 or1 (orl is the injection Nl -t Fl). Considering the homogeneous A-submodule N2 = Ker..p1 of F2 and using the above reasoning, we can extend the


last exact sequence to an exact sequence of graded A-modules

(F3 is a finitely generated free graded A-module, whose grading has the form described in Example 1.3.5), etc. If the projective dimension of A-module M is finite, then the process terminates and we obtain a finite resolution of the graded Amodule M. Indeed, if pdA M = s (s EN), then N.- 1 = Ker <P.-l is a projective A-module. By Theorem 1.3.7, Ns - 1 is a free graded A-module whose grading is of the form described in Example 1.3.5 (we denote this module by Fs). Therefore, Ker <Po = 0, so the above process leads to the finite free resolution

(1.3.1)

of the graded A-module M. For example, if A = K[Xl' ... , xn] is a polynomial algebra over a field K provided with the standard grading, then a finite free resolution of the form (1.3.1) can be constructed for any graded finitely generated A-module (this fact is a direct consequence of the Hilbert syzygy theorem, see [CE56]).

Let M = ffisEz M(a) and N = ffisEz N(s) be graded left modules over a graded ring A = ffisEz A(s), and let HomA (M, N) denote the abelian group of all Ahomomorphisms from M to N (when M and N are considered not as graded Amodules, but only as left A-modules). Then the homomorphisms <P : M -+ N of graded A-modules of degree p (p E ;Z) form an additive subgroup of HomA (M, N) which is denoted by HOMA (M, N)(p). It is easy to see that their direct sum HOMA (M, N) can be considered as a subgroup of HomA (M, N), i.e. as the graded abelian group with homogeneous components HOMA (M, N)(p), p E ;Z (by a graded abelian group we mean a graded ;Z-module when ;Z is treated as a graded ring with the trivial grading, see Example 1.3.1).

1.3.10. PROPOSITION. Let M be a finitely generated graded left module over a graded ring A. Then HOMA(M, N) = HomA(M, N) for any graded left A-module N.

PROOF. Since HOMA(M, N) ~ HomA(M, N), it is sufficient to show that for an arbitrary A-homomorphism f : M -+ N there exists a finite number of homomorphisms <Pi : M -+ N (i = 1, ... , k) of graded modules of some integer degrees such that f = <PI + ... + <Pk· Let XI, ... ,Xn be a finite system of homogeneous generators of Mover A. Then each element f(xj) (1 :S j :S n) has a unique representation of

the form f(xj) =2:7~IYji where Yji (i= I, ... ,kj ; kj is a positive integer) areho-mogeneous elements of N such that deg Yjl < ... < deg Yjkj. These elements define homomorphisms of graded A-modules 9ji : M -+ N (1 :S j :S n, 1 :S i :S kj ) such

{ 0, if 1-# j,

that 9ji(Xt) = . . (it may easily be verified that the homomorphisms Yji, Ifl=J;

9ji are well-defined). Now, f = 2:';=12:7;1 9ji, so that f E HOMA(M, N). This completes the proof. 0

1. 3 .11. REMARK. If M and N are graded modules over a graded ring A = ffinEZ A(n) and M is not finitely generated over A, then it can happen that

24 I. PRELIMINARIES

HOMR(M, N) ~ HomR(M, N). For example, let A have an infinite number of nonzero components, so that there exists an element a E ITnEZACn) \ EBnEZACn). Let M = ACZ) (i.e. M is the graded A-module whose nth homogeneous component is A for any n E IZ) and let define f E HomA (M, A) by putting f(x) = LiEZ a;11";(x) for every x E M (11"; is the projection of the direct sum M = ACZ) onto its ith component, a; are components of the element a). Clearly, f f/:. HOMA(M,A).

1.3.12. EXERCISE. Let A = EBnEZ ACn) be a graded ring. Then we may consider the category A-gr of graded left A-modules where morphisms are homomorphisms of graded modules of degree o.

(a) Show that this category possesses direct sums, products, inductive and projective limits.

(b) Show that A-gr is an abelian category which satisfies Grothendieck's axiom A5 (see [Gr57, Ch. 1, Sect. 1.5]): for any family {M>'hEA of homogeneous submodules of a graded left A-module M such that for any>., J1. E A there exists v E A for which M>. ~ Mil, M JJ ~ Mil, and for any homogeneous submodule N ~ M, we have (L>'EA M>.) n N = L>'EA(M>. n N).

Let M = EBsEz MCs) be a graded left module over a graded ring A = EBsEz ACa). By the nth suspension M(n) (n E IZ) of M we shall mean the module M together with the grading with homogeneous components M(n)Cs) = MCs+n) (s E IZ). Thus, we obtain the functors Tn : A-gr~ A-gr such that Tm 0 Tn = Tm+n , Tn 0 T_ n = id (m, n E IZ).

1.3.13. EXERCISE. Show that the family {A (n) I n E IZ} is a family of generators for the category A-gr. (Recall that a family (U >'».EA of objects of a category k is said to be a family of generators of k, if for any object M E k and for any its subobject N =j:. M there exist >. E A and a morphism 'P : U >. ~ M such that 'P =j:. o'IjJ for any morphism'IjJ : U>. ~ N; 0 denotes the canonical injection N ~ M).

1.3.14. EXERCISE. Let A be a graded ring. Projective objects P of the category A-gr will be called graded projective A-modules (it means that for any epimorphism 'P : M ~ N of graded A-modules and for any homomorphism of graded A-modules f : P ~ N there exists a unique homomorphism of graded modules h : P ~ M such that 'P 0 h = f). Prove the following statements.

(a) Every free graded A-module (see Definition 1.3.9) is a graded projective A-module.

(b) A graded left A-module P is a projective object of A-gr if and only if Pis a direct summand of a free graded A-module.

(c) Let P be a graded left A-module. Then P is a projective object in A-gr if and only if P is a projective left A-module (in the usual sense). [Hint: suppose that M, Nand P are graded A-modules such that the diagram

M~N f~ /g

P

is commutative for some A-homomorphisms f,g, h, where f is a homomorphism of graded modules of degree o. Show that if 9 (respectively,


h) is a homomorphism of graded modules of degree 0, then there exists a homomorphism of graded modules hi : M --t N (respectively, g' : N --t P) of degree 0 such that f = goh' (respectively, f = g'oh). Then use statement (b) .J

(d) Let N = EBnEZ N(n) be a graded submodule of a graded left A-module M = EBnEZM(n). Show that if M = NEB K for some left A-module K, then N is a direct summand of M in the category A-gr (i.e. there exists a graded A-submodule L = EBnEZ L(n) such that M(n) = N(n) EB L(n) for all n E ~).

1.3.15. EXERCISE. Let A be a graded ring. Show that the following assertions are equivalent:

(a) E is an injective object of A-gr (the definition of an injective object is dual to that of projective one, see Exercise 1.3.14);

(b) functor HOMA(-, E) is exact; (c) for every homogeneous left ideal I of A, the morphism HOMA(a, IE)

HOMA(A, E) --t HOMA(I, E) induced by the canonical inclusion a: I --t A is surjective.

1.3.16. EXERCISE. Let a graded left module E over a graded ring A be injective (as a left A-module). Prove that E is an injective object of A-gr. Show that the converse of this statement does not hold.

1.3.17. REMARK. Let A be a graded ring. Then Exercise 1.3.14(c) shows that the projective dimension of a graded A-module M in the category A-gr coincides with pdA(M), the projective dimension of M in the category of left A-modules ..

1.3.18. PROPOSITION. Let the homogeneous component A(O) of a left Noetherian positively graded ring A = EBsEN A(s) be a skew field, and let M = EBsEN M(s)

be a finitely generated graded left A-module. Then the dimension of any vector space M(s) (s E N) over A(O) is finite.

PROOF. Let Xl, ... , xp be homogeneous generators of M over A such that Xi E M(k;) (k l , ... , kp EN). Then any element X E M(s) (s EN) can be expressed as X = I:f=l aixi with ai E A(s-ki) (1 :S i :S p); therefore, it is sufficient to show that any left A(OLmodule A(k) (k EN) is finitely generated. We shall prove this fact by induction on k. For k = 0 it is obvious. Let k > O. Since A is left Noetherian, its left ideal A+ = EBs>o A(s) is generated, as a left A-module, by a finite set {b l , ... , bq }

of homogeneous elements (b j E A(mj) for some positive integers ml, ... ,mq ). If b E A(k), then b = I:J=l cjbj for some elements Cj E A(k-mj) (1 :S j :S q). Applying the inductive hypothesis, we obtain that b can be expressed as a linear combination over A(O) of the elements Yilbj where Yjl belongs to one of the finite sets of generators of the left A(OLmodules A(k-mj) (j = 1, .. . ,q). This completes the proof. 0

1.3.19. DEFINITION. With the conditions of the previous proposition, the function 'PM : N --t N such that 'PM(S) = dimA(o) M(s) for all SEN, is called the Hilbert function of the graded A-module M.

26 1. PRELIMINARIES

1.3.20. EXERCISE. Let 0 -+ M -+ N -+ P -+ 0 be an exact sequence of finitely generated positively graded left modules over a left Noetherian positively graded ring A whose zero component is a skew field. Show that <PN(S) = <pM(S) + <pp(s) for all sEN.

Let A = K[XI, ... , xn] be a polynomial ring over a field K provided with the standard grading (so that A(O) = K), and let M = EB.eN M(') be a finitely generated graded left A-module. Then the following well-known theorem shows that for all sufficiently large n E N the Hilbert function of M is a polynomial (the proof of this theorem one can find, for example, in [ZS60]).

1.3.21. THEOREM (HILBERT-SERRE). Let M = EB.eN M(') be a finitely generated graded module over a standardly graded polynomial ring K[XI, ... , Xn] over a field K. Then there exists a polynomial <p(t) in one variable t with rational coefficients such that <p( s) = dimK M(') for all sufficiently large sEN. This polynomial is called the Hilbert polynomial of M.

1.3.22. EXERCISE. Let us consider a standardly graded polynomial ring R = K[XI' ... ' xnl over a field K as a graded module over R. Show that the Hilbert function <PR(S) of this module is the polynomial <pR(S) = ('~~~l) where the symbol

('tk) (k EN) denotes the polynomial ('+k)('+kk~l)···(·+l).

1.3.23. DEFINITION. A ring R is said to be filtered if there is an ascending chain {R. I s E ~} of subgroups of the additive group of R such that 1 E Ro and RiRj ~ Ri+j for all i, j E ~. The family of these subgroups is called a filtmtion of R and is denoted by (R.).ez; the abelian group R. (s E ~) is said to be the sth component of the filtered ring R.

If R. = 0 for all s < 0, then R is said to be positively filtered, the filtration (R.).ez is called positive and is denoted by (R.).eN. Elements of R. \R.- I (s E ~) are called elements of degree s.

Note that Definition 1.3.23 implies that Ro is a subring of R.

1.3.24. REMARK. Sometimes, a filtration of a ring R is defined as a descending chain (R~)qez of subgroups of the additive group of R such that R~R~ ~ R~+q for all p, q E Z (for example, if R is commutative and I is an ideal of R, then one may consider the descending filtration (R~)qez of R such that R~ = 0 for q < 0 and R~ = ]q for q ~ 0). However, in this book by a filtration we shall always mean an ascending filtration (i.e. a filtration in the sense of Definition 1.3.23).

1.3.25. EXAMPLE. Any ring R can be considered as a filtered ring with the trivial filtmtion (R.).ez such that R. = 0 for all n < 0, and R. = R if s ~ O.

1.3.26. EXAMPLE. Let R = K[XI, .. . , xnl be a polynomial ring in indeterminates Xl, ... , Xn over a ring K, and Rn (n EN) the set of all polynomials whose degree is less than or equal to s. Then (R.).eN is a positive filtration of R which is called standard. Similarly, one can consider the positive filtration (A.).eN of the free K-ring A = K(XI, ... ,Xn) generated by a finite set of symbols {XI, ... ,Xn} (see Example 1.3.2), such that A. (s E N) is the left K-module generated by the set {Xi t ... Xip I P ~ s, 1 ~ ik ~ n for k = 1, ... , p}.


1.3.27. EXAMPLE. Let A = ffi.EzA(·) be a graded ring. Then A can be also considered as a filtered ring with respect to the filtration (Ar )rEZ such that Ar = E.<r A(') for all r E IE.

1.3.28. DEFINITION. Let R be a filtered ring with a filtration (R')'EZ. A left R-module M is called a filtered R-module if there exists an ascending chain {M. I s E IE} of additive subgroups of M such that R.Mr ~ Mr+. for all r, s E IE. The family {M. I s E IE} is called a filtration of M, it is denoted by (M.).EZ. If M. = 0 for all n < 0, then M is said to be positively filtered and is denoted by (M')'EN; in this case the filtration itself is called positive.

The notion of a right filtered R-module is introduced in the same way. Obviously, if R is a filtered ring, then R can be considered as a left or as a right filtered Rmodule.

If the filtration of a filtered ring R is trivial, then any ascending chain of submodules of a R-module M is a filtration of M; in this case, by a trivial filtration of M we mean the chain (M')'EZ such that M. = M for all s ~ 0 and M. = 0 for all s < O.

1.3.29. DEFINITION. Let R be a filtered ring with a filtration (R.).EZ, and M a filtered left R-module with a filtration (M.).EZ. Then the filtration (M').EZ is called exhaustive if U.EzM. = M; it is called separated if n.EzM. = 0, and discrete if there exists So E IE such that M. = 0 for all s < so.

Let R be a filtered ring, M a filtered left R-module with a filtration (Ms ).EZ and L a R-submodule of M. Then Land MIL can be also considered as filtered R-modules provided with the filtrations (M. n L).EZ and (M. + LIL)'Ez, respectively. Clearly, if the filtration of M is exhaustive (respectively, discrete), then the filtrations of L and MIL are exhaustive (respectively, discrete); as for separated filtrations, the similar property holds only for L.

1.3.30. EXAMPLE. Let R be a filtered ring with a filtration (R.).EZ, and M a finitely generated left R-module with a finite system of generators {Xl, ... , xp}. Putting M. = 0 for s < 0 and M. = Ef=l R.x; for s ~ 0, we obtain a positive filtration (M.)'EZ of M which is called a filtration associated with the system of generators {Xl, ... , Xp}.

1.3.31. DEFINITION. Let R be a filtered ring and let M and N be filtered left R-modules provided with filtrations (M.)'EZ and (N')'EZ, respectively. An Rhomomorphism f E HomR(M, N) is said to be a homomorphism of filtered modules of degree p (p E IE) if f(M.) ~ N.+p for all s E IE. Homomorphisms of finite degree form a subgroup of HomR(M, N) which is denoted by HOMR(M, N); homomorphisms of degree p (p E IE) form a subgroup HOMR(M, N)p of HOMR(M, N).

1.3.32. REMARK. It is easy to see that, with the preceding notation, the fol-lowing statements hold:

(a) if p ~ q, then HOMR(M, N)p ~ HOMR(M, N)q; (b) HOMR(M,N) = UpEzHOMR(M,N)p; ( c) if f : M -+ N is of degree p and g : N -+ P is of degree q, then 9 0 f is of

degree p + q.

28 1. PRELIMINARIES

1.3.33. REMARK. Let R be a filtered ring. Then one may consider the category R-filt of filtered left R-modules where the morphisms are the homomorphisms of filtered R-modules of degree O. It is easy to see that R-filt is an additive category and, furthermore, if f E HOM(M, N)o (M, N are some filtered left R-modules with filtrations (Ms)'EZ and (Ns)sEZ, respectively), then Ker f has the induced filtration ((Ker f)s)'EZ where (Ker f). = Ker f n M. (s E &:), and Coker f is filtered by the family {1m f + N. 11m f I s E &:}). In particular, it follows that monomorphisms and epimorphisms in R-filt coincide with injective and surjective morphisms of Rfilt, respectively. Obviously, arbitrary direct sums, as well as direct products, exist in R-filt. Furthermore, for any object M E R-filt with a filtration (M')'EZ, and for any n E &:, let M(n) denote the filtered left R-module M with the filtration (M(n)s = MS+n)'EZ, Then one can consider the functors Tn : R-filt-+ R-filt (n E&:) such that Tn(M) = M(n) for all objects M E R-filt and if f E Hom(M, N)o, then Tn(f) = f. Clearly, Tn 0 Tm = Tn+m and To = Id (identity functor). In particular, every Tn is an equivalence of categories which commutes with direct sums and direct products.

1.3.34. EXERCISE. Let {M>. I ..\ E A} be a family of filtered modules over a filtered ring R. Then EB>'EA M>. and ILEA M>. can be naturally considered as filtered R-modules. Show that if the filtrations of M>. (..\ E A) are exhaustive, then the filtration of EB>'EA M>. is also exhaustive, but the filtration of the direct product ILEA M>. needs not to be exhaustive.

1.3.35. REMARK. The category R-filt (R is a filtered ring) is preabelian (see [Gr57]) but not abelian. Indeed, let M i= 0 be a left R-module with a filtration (Ms )SEZ where Ms i= 0 for some s E &:. Let M' denote the module M provided with the zero filtration (M;)sEZ (M; = 0 for all s E &:). Then the identity morphism of M is in HOMR(M', M)o but not in HOMR(M, M')o; therefore, idM is a bijective mapping which fails to be an isomorphism.

Let R be a filtered ring with a filtration (R')'EZ and let gr R denote the abelian group EBsEz R.I R.-1· Then gr R can be considered as a graded ring if we set (X+Ri-1)(y+Rj-1) = xy+Ri+j-1 (i, j E&:) for all elements X+Ri-1 E R;jRi-1, Y + Rj -1 E Rj I Rj -1 (x E Ri, Y E Rj) and demand the validity of the distributive laws. This graded ring gr R is said to be a graded ring associated with the filtered ring R (or an associated graded ring of R); the pth homogeneous component Rp I R p- 1 of gr R is denoted by (gr R)(p); the family {(gr R)(p) I p E &:} is said to be the grading associated with the filtration (R.)SEZ (or the associated grading ofthe filtered ring R). If a is an element of R of degree i (i .e. a E Ri \ Ri _ d; then the element MH(a) = a + Ri-1 E (gr R)(i) is called a head of a.

1.3.36. REMARK. Let A = EB.Ez A(') be a graded ring that can be also considered as a filtered one with the filtration (Ar = 2::.<r A(s))rEZ (see Example 1.3.27). Then each component (gr A)(p) of the associated-graded ring of this filtered ring can be canonically identified with A (p) (p E &:).

Let R be a filtered ring with a filtration (R')'EZ, gr R = EBpEz(gr R)(p) an associated graded ring, and M a filtered left R-module with a filtration (Mr )rEZ, Then the abelian group EBqEz Mq I Mq- 1 denoted by gr M can be naturally considered as a graded left gr R-module, its qth homogeneous component Mq I Mq_1 (q E &:) is


denoted by (gr M)(q). gr M is said to be a graded gr R-module associated with the filtered R-module M (or the associated graded module of M). If x E Mi \ Mi-I (i.e. deg x = i), then the element MH(X) = x + Mj-l E (gr M)(i) is called a head of x. If L ~ M, then the set UxELMH(X) is denoted by MH(L).

1.3.37. DEFINITION. Let R be a positively filtered ring with a filtration (R')'EN such that Ro is a skew field and each R. (s E N) is finitely generated as a left Romodule. Then the function WR : N -+ N such that WR(S) = dimRo R. is called the Hilbert function of R. Similarly, if M is a filtered left R-module with a filtration (MS)'EZ, such that M. is a finitely generated left Ro-module for every s E ~, then the function WM : N -+ N, such that WM(S) = dimRo Ms, is said to be Hilbert function of the filtered module M.

1.3.38. REMARK. Let R be a positively filtered ring with a filtration (R')'EN such that Ro is a skew field and each R. (s E ~) is a finitely generated left Romodule. Let gr R be the associated graded ring of R (its grading is also positive). If W Rand tpgr R are the Hilbert functions of the filtered ring Rand ofthe graded ring gr R (see Definition 1.3.19), respectively, then WR(S) = 2:j<. tpgrR(i) for all s E ~. For example, if K is a field and R = K[Xl, ... , xnl is a polynomial ring provided with the standard filtration, then gr R can be identified with R provided with the standard grading. In this case the above formula and the result of Exercise 1.3.22 show that WR(S) = 2:~=o (i~~~l) = (.~n).

1.3.39. PROPOSITION. Let R be a filtered ring with a filtration (R')SE71" M a filtered left R-module with a filtration (M')SEZ and L a submodule of M provided with the induced filtration (L. = Ln Ms )sEZ. Then MH(L) is a graded submodule of the graded gr R-module gr M.

PROOF. If x E L n Mi and a E Rj (i, j E ~), then ax E L n Mi+j. Therefore, for any elements x + Mi- 1 E MH(L;) and a = a + Rj-l E (gr R)(j) we have ax = ax + Mi+j-I E MH(Li+j) ~ (grM){i+j), so that MH(L) = EBiEZM(L;) is a graded submodule of gr M. 0

1.3.40. PROPOSITION. Let R, M and L be as in the preceding proposition, and let 7r : M -+ M / L be the canonical epimorphism of filtered modules (M / L is considered with the natural filtration {(M/ L)s = Ms + L/ L = 7r(Ms) I s E ~}).

Then there exists a canonical isomorphism gr(M/ L) :::: (gr M)/ MH(L).

PROOF. Let us consider mappings OJ : (gr M)(i) -+ (gr(M/ L))(i) (i E ~) such that Oi(X + Mi-l) = 7r(x) + (M/L)i_l for any element x + Mi- 1 E (grM)(i). These mappings (which are well-defined epimorphisms of abelian groups) give rise to the surjective mapping 0 : gr M -+ gr(M/ L) with the kernel MH(L) (so that OI(grM)(') = OJ for all i E ~). Moreover, 0 is an epimorphism of graded gr Rmodules. Indeed, if a = a + Ri-l E (gr R)(j) (a E R;) and x = x + Mj-l E (grM)(j) (x E Mj)' then O(ax) = OJ+j(ax + Mi+j-d = 7r(ax) + (M/L)i+j-l = a· (7r(x) + (M/ L)i+i-d = aO(x). Thus, we obtain the natural isomorphism of graded gr R-modules gr M/MH(L) and gr(M/ L). 0

Let R be a filtered ring with a filtration (R')'EZ and let M, N be left filtered R-modules provided with filtrations (Ms)sEZ and (N')SEZ, respectively. If

30 I. PRELIMINARIES

f E HomR(M, N)o (so that f is a morphism in the category R-filt), then f induces canonical mappings fi : (gr M)(i) -+ (gr N)(i) (i E Z) such that f;(x + Mi-d = f(x) + Ni-l for any element x E Mi. These mappings, in turn, define a homomorphism of graded gr R-modules gr f : gr M -+ gr N such that if P is another object of R-filt and 9 E HomR(N, P)o, then gr(g 0 f) = gr go gr f. Thus, we obtain a functor gr : R-filt -+ (gr R)-gr; the following proposition gives some properties of this functor.

1.3.41. PROPOSITION. Let R be a filtered ring and let M be an object of R-filt.

(i) If the filtration of M is exhaustive and separated, then M = 0 if and only ifgrM = O.

(ii) If the filtration of M is discrete, then there exists So E Z such that (gr M)(i) = 0 for all i < so.

(iii) Let 0 -+ N ~ M -4 P -+ 0 be an exact sequence of filtered left modules such that the filtrations of Nand P are induced by the filtration of M (i.e., if (Ms) s EZ is the filtration of M, then Nand P have the filtrations (Ns = Ms n N)SEZ and (Ps = (3(Ms) == Ms + N/N)'EZ, respectively). Then the

induced sequence of graded gr R-modules 0 -+ gr N ~ gr M -4 gr P -+ 0 is exact.

(iv) The functor gr commutes with direct sums and direct products.

PROOF. The first three assertions are obvious. Let us prove the forth statement. Let {M.>J>'EA be a family of objects of R-filt and M = EB>'EA M>.. If (M>'S)SEZ is a filtration of M>. (,\ E A), then (EB>'EA M>'S)SEZ is a filtration of M and we obtain that

(gr M)(s) = EB>'EA M>.s/ EB>'EA M>".-l = EB>'EA(M>.s/M>.,s-d = EB>'EA(gr M>.)(s). In a similar way we can prove that gr commutes with direct products. 0

1.3.42. DEFINITION. Let R be a filtered ring with a filtration (R.).ez, A filtered left R-module F with a filtration (F')'EZ is called a free filtered R-module if it is a free R-module and there exist a basis {X>.heA of F and a family {n>'heA of integers such that F. = L:>'EA Rs-n>.x>. for all s E Z. (Note that x>. E Fn>. \ Fn>.-l for any ,\ E A.) In this case the set of pairs {(x>., n>.)hEA is said to be a filt-basis of F.

We leave the proof of the following proposition to the reader as an exercise.

1.3.43. PROPOSITION. Let R be a filtered ring with a filtration (R,)sE7l. and M a left filtered R-module with a filtration (M. ).e7!,.

(i) M is a free filtered R-module with a filt-basis {(x>., n>.)heA if and only if M ~ EB>.eA R( -n>.) where ~ denotes an isomorphism of filtered R-modules (recall that R( -n) (n E Z) denotes the filtered ring R with the filtration

(R(-n)s = Rs-n).ez). (ii) If M is a free filtered R-module with a filt-basis {(x>., n>.)heA, then gr M

is a free graded gr R-module with the homogeneous basis {MH(X>.)heA. (iii) Let gr M be a free graded gr R-module with a homogeneous basis {X>.heA

where x>. E (gr R)(n>.) for some integers n>. (,\ E A). If the filtration of M is discrete, then M is a free filtered R-module with the filt-basis {(x>., n>.) heA.


(iv) IfF is a free graded gr R-module, then there exists a free filtered R-module F' such that gr F' == F.

(v) Let F be a free filtered R-module with a filt-basis {(x.x, n.x)hEA and let f : {X.xhEA -+ M be a mapping such that f(x.x) E Mp+n ), (oX E A). Then there exists a unique homomorphism of filtered modules 9 : F -+ M of degree p which extends f.

(vi) Let F be a free filtered R-module and let 9 : gr F -+ gr M be a homomorphism of graded gr R-modules of degree p (p E 2). Then there exists a homomorphism of filtered modules f : F -+ M of degree p such that

9 = gr f· (vii) A filtration of a free filtered R-module F is exhaustive (separated) if and

only if the filtration of R is exhaustive (respectively, separated). (viii) Let the filtration of M be exhaustive. Then there exists a free resolution of

M in R-filt:

... -+ F2 ~ FI ~ Fa ~ M -+ 0 (1.3.2)

where every Fj (j = 0,1, ... ) is a free filtered R-module, and if (Fjs)sEZ is the filtration of Fj , then cpj(Fjs ) = Imcpj n Fj-l,s (j = 1,2, ... ) and CPo (Fos) = M. for all s E Z. Moreover, if the filtration of R is discrete, then we may assume that the filtrations of Fj (j E N) are also discrete.

1.3.44. REMARK. By a finitely generated filtered R-module over a filtered ring R with a filtration (Rs)'EZ we shall mean a filtered left R-module M with a filtration (MS)SEZ satisfying the following property: there exist a finite family of integers {nl,"" nd and elements Xl, .. " Xk E M such that Xi E Mn, (1 ~ i ~ k) and Ms = L~=l Rs-niXi for all s E Z. Clearly, if M is a free filtered R-module which is finitely generated as a left R-module, then M is a finitely generated filtered R-module.

1.3.45. EXERCISE. Let a filtration of a filtered module M over a filtered ring R be exhaustive. Show that M is a finitely generated R-module if and only if there exist a finitely generated free filtered R-module F, and an epimorphism of filtered modules 7r : F -+ M such that 7r(F.) = M. for all s E Z «F')'EZ and (M')sEZ are the filtrations of F and M, respectively).

1.3.46. REMARK. Let R be a filtered ring with a positive filtration (RS)'EN

such that Ro is a skew field. Let 0 -+ M ~ N -4 P -+ 0 be an exact sequence of positively filtered left R-modules (so that a and j3 are homomorphisms of filtered modules of degree 0) and let all components of the filtrations of M, Nand P be finitely generated left Ro-modules. Then it is easy to see that WN = WM + Wp

(WM denotes the Hilbert function of a filtered R-module M, see Definition 1.3.37). However, if M is a filtered R-module whose filtration is associated with some system {Xl, ... ,xp} of its generators (see Example 1.3.30) and if 0 -+ N ~ F ~ M -+ 0 is an exact sequence of filtered R-modules such that F is a free filtered R-module with p generators /I, ... ,fp, cp(f;) = Xi (1 ~ i ~ p) and N = Kercp, then the filtration of N induced by the injection a need not be associated with a system of generators of N (even if N itself is a free R-module). Therefore, the algorithm of construction of free graded resolutions of finitely generated graded modules cannot be extended to the case of filtered modules.

32 I. PRELIMINARIES

1.4. Conservative Systems

Most of the notions and statements of this section are due to E.R. Kolchin (they are thoroughly expounded in [KoI73]).

Let R be a commutative ring, MaR-module, and E a family of R-submodules of M.

1.4.1. DEFINITION. The family E is said to be a conservative system of submodules of M if M E E and the following two conditions are satisfied:

(cl) the intersection of any family of elements of E is an element of E; (c2) if a family of elements of E is linearly ordered with respect to inclusions,

then the union of elements of the family belongs to E.

If M = R, we obtain the notion of a conservative system of ideals of the ring R. It is easy to see that the family of all submodules of M is a conservative system,

as well as the family consisting of the single element M. It is also clear that if {Ei liE I} is a family of conservative systems of submodules of M, then niEIEi is a conservative system. Therefore, if is a set of submodules of M, then there exists a smallest conservative system of submodules of M containing (it is the intersection of all conservative systems containing <I».

1.4.2. EXAMPLE. The set of all perfect ideals of a commutative ring R is a conservative system (see Exercise 1.2.25).

Let M be a module over a commutative ring R, E a conservative system of submodules of M and A eM. Then the intersection of all submodules from E containing A is called a submodule of ME-generated by A and is denoted by {Ah; (obviously, it is the smallest (with respect to the inclusion) element of E containing A). If the set A is finite and N = {Ah;, then N is said to be a finitely E-generated submodule of M and A is said to be a E-basis of N. If M = R, then the above notions are naturally applied to the ideals of R.

1.4.3. PROPOSITION. Let E be a conservative system of submodules of a Rmodule M, A ~ M and x E {Ah;. Then there exists a finite subset B ~ A such that x E {Bh;.

PROOF. If A is finite, the proposition is trivial. Let A be infinite. Then there exists a family of subsets of A such that is linearly ordered by inclusion, UCE~C = A and each element of has cardinal number strictly less than that of A. By property (c2) ofa conservative system (see Definition 1.4.1), UCE~{Ch; E E. Furthermore, A ~ UCE~{Ch; ~ {Ah;, hence, {Ah; = UCEdCh;. Now, we can complete the proof by induction on the cardinal number of A. 0

1.4.4. DEFINITION. Let M and M' be modules over rings Rand R', respectively, let E be a conservative system of R-submodules of M, and E' a conservative system of R'-submodules of M'. A mapping F : E -+ E' is called conservative (or a homomorphism from the conservative system E to the conservative system E') if the following conditions are satisfied:

(1) if A E E (i E I), then F(niEIA) = niElF(Ad; (2) if A, BEE and A ~ B, then F(A) ~ F(B);

1.4. CONSERVATIVE SYSTEMS 33

(3) if {Ai liE I} is a family of sets of the system E, linearly ordered with respect to inclusion, then F(UiEIAd = UiEIF(Ai).

Obviously, if a conservative mapping F : E -+ E' is bijective, then the inverse mapping F- l is also conservative. In this case we say that F is an isomorphism of E onto E', and that the two conservative systems are isomorphic (we write E ==' E').

Let F : E -+ E' be a homomorphism from a conservative system E of Rsubmodules of a R-module M to a conservative system E' of R'-submodules of a R'-module M'. Let A' E E' and {Ai liE I} be a family of elements of E such that F(A;) = A' for every i E 1. Then the condition (1) of Definition 1.4.4 implies that F(niElA;) = niElF(Ai) = A'. Therefore, for any A' E E', there exists a smallest (with respect to inclusion) element A E E such that F(A) = A' (A = n{B EEl F(B) = A'}). This element A is denoted by P-l(A'). If A',B' E E' and A' ~ B', then F(p-l(A') n P-l(B')) = A' n B' = A' hence P-1(A') ~ P-1(A') n P-l(B') ~ P-1(B'). Furthermore, if {A; Ii E I} is a family of sets, linearly ordered by inclusion, which are elements of the image of a conservative mapping F : E -+ E', then UiEIA: = UiEIF(P-l(AD) = F(UiElP-1(AD) also is an element of this image. Thus, we obtain the following result.

1.4.5. PROPOSITION. Let F : E -+ E' be a homomorphism from a conservative system of R-submodules of a R-module M to a conservative system of R'submodules of a R'-module M'. Then the family F(E) = {F(A) I A E E} is a conservative system of R'-submodules of M'.

1.4.6. EXAMPLES OF CONSERVATIVE MAPPINGS.

1. Let Ro be a subring of a ring Rand E a conservative system of ideals of R. Then the mapping 1 -+ 1 n Ro (I E E) is a homomorphism from the conservative system E onto a conservative system of ideals of the ring Ro which is denoted by EIRa' If Ro is an ideal of R, then the family {I n Ro I 1 E E} is a conservative system of ideals of R that is also denoted by E IRa.

2. Let f : R -+ R' be a ring epimorphism and J = Ker f. If E is a conservative system of ideals of R, then E' = {1 EEl 1 :2 J} is also a conservative system of ideals of R, and the mapping 1 -+ f(I) (1 E E') is an isomorphism of E' onto a conservative system of ideals of R' which is denoted by f(E). If f is a natural ring epimorphism R -+ Rj J (where J is an ideal of R), then the conservative system f(E) is denoted by Ej J and the isomorphism of conservative systems E' = {1 E E I 1 :2 J} and Ej J is called canonical.

3. Let 5 be a multiplicatively closed subset of a ring R, and cp : R -+ 5- 1 R a canonical ring homomorphism (cp : a -+ T for any a E R). Let E be the set of all 5-prime ideals of R, i.e. the set of ideals 1 of R such that 1 : s = 1 for every s E 5. It is easy to see that E is a conservative system and the mapping 1 -+ 5- 11 (1 E E) is an isomorphism of this conservative system onto a conservative system of ideals of the ring 5- 1 R which is denoted by 5- 1 E. This isomorphism is called canonical, the inverse conservative mapping is defined by J -+ cp-l (J).

4. Let f : M -+ M' be a homomorphism of R-modules and N = Ker f. Let E be the conservative system of all submodules of M containing N, and E' the conservative system of all submodules of M'. Then the mapping r : P -+ f- l (P) (P E E') is a homomorphism of conservative systems.

34 1. PRELIMINARIES

1.4.7. DEFINITION. Let M be a module over a ring R. A conservative system ~ of submodules of M is called divisible if A: s = {m E Mism E A} E ~ for any A E~, s E R.

1.4.8. DEFINITION. A conservative system of ideals of a ring R is called perfect if it is divisible and every element of the system is a perfect ideal of R.

In what follows, for any subsets Sand T of a ring R, the set {st I s E S, t E T} will be denoted by ST.

1.4.9. PROPOSITION. Let ~ be a divisible conservative system of ideals of a ring R. Let A and B be subsets of R. Then {A}E{Bh:; ~ {ABh:;. If~ is perfect, then {ABh:; = {Ah:; n {Bh:;.

PROOF. Since {ABh:; : A = naEA({ABh:; : a) E ~ and B ~ {ABh:; : A, we have {Bh:; ~ {ABh:; : A. Therefore, A ~ {ABh:; : {Bh:;. As above, we see that {ABh:; : {Bh:; E ~ whence {A}E{Bh:; ~ {ABh:;. This inclusion shows that ({ A h:; n {B hY ~ {AB h:;. It follows that if the system ~ is perfect, then {Ah:; n {Bh:; ~ {ABh:;, hence {Ah:; n {Bh:; = {ABh:;. 0

1. 4.10. DEFINITIO N. Let ~ be a conservative system of ideals of a ring R, J a perfect ideal of R contained in ~ and 9Jl the set of all prime ideals p of R such that p E ~ and p 2 J. Then 9Jl contains minimal (relative to inclusion) elements that are called ~-components of J. Zorn's lemma and condition (c1) of Definition 1.4.1 imply that every prime ideal P such that P E 9Jl contains a ~-component of J.

1.4 .11. PROPOSITION. Let ~ be a perfect conservative system of ideals of a ring R. Then every ideal J E ~ is the intersection of its ~-components.

PROOF. Let J E ~ and x E R \ J. Then the set = {I E ~ I I 2 J and x ct. I} is non-empty (for example, J E <I». By condition (c2) of Definition 1.4.1 and Zorn's lemma, there exists a maximal (relative to inclusion) element P of the set . If a, b E R \ P and ab E P, then x E {P, ah:;, x E {P, bh:;, whence x E {P,ah:;· {P,bh:; ~ P (see Proposition 1.4.9), and we obtain the contradiction with the inclusion P E . Thus, the inclusion ab E P implies a E P or b E P, so that the ideal P is prime and, therefore, it contains a ~-component of J. It follows that the intersection of all the ~-components of J does not contain x. Since x is an arbitrary element of R \ J, this intersection is J. 0

1.4.12. EXERCISE. A conservative system ~ of ideals of a ring R is called additive if the sum of any two ideals of ~ is an element of~. For example, if F is a set of endomorphisms of the additive group of R, then the set of ideals I of R such that f(I) ~ I for any f E F is an additive conservative system.

Show that an ideal I of a ring R belongs to an additive conservative system ~ of ideals of R if and only if for every x E R, the condition x E I implies {x h:; ~ I. [Hint: use Proposition 1.4.3].

If a conservative system ~ is not additive, the above statement is not necessarily true. For example, let R = 1Z2 [X, Y]/ I where I is the ideal of 1Z2 [X, Y] generated by X 2 ,XY,Y2 . Let x = X + I, y = Y + IE R and ~ = {O,(x), (y),(x + y)'R}. Then the ideal (x,y) satisfies the condition of Exercise 1.4.12, but (x,y) ct.~.

1.4. CONSERVATIVE SYSTEMS 35

1.4.13. EXERCISE. Let E be a conservative system of ideals of a ring R and let Spec(R, E) denote the set of all prime ideals which are elements of E. Furthermore, for any subset A of R, let V(A) denote the set of prime ideals of R containing A and D(A) denote the set of prime ideals P of R such that A i: P. Prove that Spec(R, E) = nrER{D(r) U V({rh)}.

1.4.14. EXERCISE. Let E be a divisible conservative system of ideals of a ring R and let I E E. Show that the radical r(I) of I also belongs to E. [Hint: let x E R \ r(I). Then xn rJ. r(I) for any positive integer n. Show that there exist maximal elements in the set N = {J EEl I ~ J, xn rJ. J for any n E N} and prove that such elements are prime ideals of R. It follows that for any element x E R \ I there exists a prime ideal Px E E such that I ~ Px, x rJ. Px.]

1.4.15. EXERCISE. Let E be a divisible conservative system of ideals of a ring R, I E E and J = r(I). Suppose, that J = Pl n ... n Pn where Pl, ... ,Pn are prime ideals of R such that Pi i: Pi for any different indices i,j (1 ~ i,j ~ n). Show that Pi E E for all indices i = 1, ... , n.

1.4.16. DEFINITION. Let R be a ring, M an R-module, and E a conservative system of submodules of M. Then E is called a Noetherian conservative system if it satisfies one of the following three conditions (that are evidently equivalent to each other);

(1) every element of E has a E-basis; (2) every strictly ascending chain of ideals of E is finite; (3) every non-empty set of elements of E has a maximal element.

The following result is a version of Proposition 1.4.11 for Noetherian perfect conservative systems.

1.4.17. PROPOSITION. Let E be a Noetherian perfect conservative system of ideals of a ring R. Then every ideal I E E is the intersection of a finite family of prime ideals from the system E none of which contains another. This finite family is unique, being the family of E-components of I.

PROOF. Let N = {I EEl I cannot be represented as a finite intersection of prime ideals from E}. Obviously, if h, I2 rJ. N, then h n h rJ. N. Supposing that N i= 0, we obtain that the set N has a maximal element J. Proposition 1.4.9 implies that J is a prime ideal of R (if ab E J, and a, b rJ. J, then {J, a h, {J, b h rJ. N hence J = {J, a h n {J, b h rJ. N that contradicts the choice of J). Thus, N = 0

so that the first part of the proposition is proved. Discarding superfluous prime ideals from every expression I = Pl n ... n Pn (I, Pl,··., Pn E E, and each ideal Pi is prime), we can express I as an intersection of a finite family M of prime ideals of the system E none of which contains another. If pi is a E-component of an ideal IE E, then, since npEMP = I ~ pi, we have P ~ pi for some P EM, whence P = p'. Thus, every E-component of I is an element of M. Conversely, if P EM, then P contains a E-component pi of I. As above we obtain that pi = p. This completes the proof. 0

1.4.18. REMARK. Let I be a perfect ideal of a ring R and I = Pl n ... n Pn where Pl, ... ,Pn are prime ideals of R. If E is any divisible conservative system

36 1. PRELIMINARIES

containing I, then Pi E ~ for all i = 1, ... , n (see Exercise 1.4.15), hence, P1, ... , Pn are ~-components of I. Therefore, for every perfect ideal I of R, we may refer to the components of I without specifying the conservative system.

The following result is a direct consequence of the definition of Noetherian conservative system.

1.4.19. PROPOSITION. Let ~ be a Noetherian conservative system (of submo-dules of a module or of ideals of a ring). Then:

(a) every conservative system contained in ~ is Noetherian; (b) every homomorphic image of ~ is a Noetherian conservati ve system.

1.4.20. COROLLARY. Let ~ be a conservative system of ideals of a ring R, Ro a subring of R, I an ideal of Rand S a multiplicatively closed subset of R. If ~ is Noetherian, then ~IRo, ~II, ~I I, and S-l ~ are also Noetherian.

1.4.21. COROLLARY. Let ~ be a perfect conservative system of ideals of a ring R, and I an ideal of R. Then the system ~ is Noetherian if and only if the conservative systems ~I[ and ~I I are Noetherian.

PROOF. Obviously, it is sufficient to show that if the conservative systems ~II' ~I I are Noetherian, the set ~ is also a Noetherian conservative system. Let J, J{ E ~, J ~ J{ and J n 1= J{ n I, {J + Ih; = {I< + Ih;. Then, by Proposition 1.4.9, we have J{ = J{ n {J + Ih; = {J{ J + J{ Ih; ~ {J + J n Ih; = J, whence J{ = J. Therefore, if J C J{, then either the inclusion J n I ~ J{ n I of elements of ~II is strict or the incfusion {J + Ih;1 I ~ {I< + Ih;1 I of ~I I-ideals is strict. It follows that if ~ is not Noetherian, then either ~II or ~I I is not Noetherian. 0

1.4.22. LEMMA. Let ~ be a perfect conservative system of ideals of a ring R. Suppose that ~ is not Noetherian and let N be the set of all ideals I E ~ that are not finitely ~-generated. Then there exist maximal elements of the set N, and every such maximal element is a prime ideal of R.

PROOF. The existence of maximal elements in N is a consequence of Zorn's lemma. Let P be any such maximal element. If a, bE R \ p, then {p, a }~, {p, b}~ ff. N, hence there exist finite sets A, B ~ R such that {p, a h; = {Ah;, {p, b h; = {Bh;. By Proposition 1.4.9 we have {p, ab h; = {p, ah;n{p, b h; = {Ah;n{Bh; = {ABh;; therefore, {p, ab h; ff. N, whence ab ff. p. Thus, the ideal p is prime. 0

1.4.23. PROPOSITION. Let ~ be a perfect conservative system of ideals of a ring Rand Ro a subring of R such that R is a finitely generated Ro-algebra. If the conservative system ~IRo is Noetherian, then so is ~.

PROOF. Obviously, it suffices to consider the case when R = Ro[x] for some x E R. Assume the proposition false. By Lemma 1.4.22 there is a maximal ideal p E ~ which is not finitely ~-generated, and p is prime. Since ~IRo is Noetherian, there exists a finite set A ~ Ro such that {p n Ro h; = {A h;, hence, p =F {p n Ro h:. Therefore, there exists a polynomial f = ao + a1X + ... + anxn E Ro[X] such that f(x) E p and f(x) ff. {p n Roh; (clearly, n > 0). Let f be chosen with n as small as possible; then it is easily seen that an ff. p. Since {p, anh; is finitely ~-generated, there exists a finite set B C P such that {p, anh; = {B, anh; (see Proposition 1.4.3). For any element v E p there exists a polynomial 9 E Ro[X]

1.5. DERIVATIONS AND DIFFERENTIALS 37

such that v = g(x). Dividing 9 by f we obtain an equation a~g = qf + r with q, r E Ro[X] and deg r < n, such that a~ v = q(x)f(x) + r(x) whence r(x) E p. By virtue of the minimality of n we have r( x) E {p n Ro h: = {A h. It follows that a~v E {f(x),Ah, hence anv E {f(x),Ah. Thus, anP ~ {f(x), Ah:, whence (see Proposition 1.4.9) p = pn {an,ph = pn {an,Bh = {anP,Bh: = {f(x),A,Bh, contradicting the assumption that p is not finitely ~-generated. This completes the proof. 0

1.4.24. COROLLARY. Let R[X] be a polynomial algebra in one indeterminate X over a ring R, and let the set of all perfect ideals of R be a Noetherian conservative system. Then the set of all perfect ideals of R[X] is also a Noetherian conservative system.

1.4.25. EXERCISE. Let ~ be a perfect conservative system of ideals of a ring R. Show that the following two statements are equivalent:

( 1) the system ~ is Noetherian; (2) every strictly increasing sequence of prime ideals of ~ is finite and every

ideal I E ~ has only finitely many components.

[Hint: prove the following statement. Let C be an ordered set and (Cn)nEl'I an infinite sequence of distinct finite subsets of C such that distinct elements of Cn are not comparable and each element of Cn +! is greater than an element of Cn . Then there exists an infinite strictly increasing sequence of elements of C].

1.4.26. EXERCISE. Let ~ be a Noetherian conservative system of ideals of a ring R. An ideal I E ~ is said to be ~-irreducible if the equality 1= J1 n J2 with J1 , h E ~ implies h = I or h = I.

(1) Show that every ideal I E ~ is the intersection of a finite set of ~-irreducible ideals of~.

(2) Let the system ~ be divisible and suppose that the inclusion u E {v, A h implies the inclusion u E {({u,Ah : vOO)v,Ah for all u,v E R and for any subset A ~ R. (Recall that B : VOO = {a E R I avn E B for some n E N} for any element v E R and for any subset B ~ R.) Show that every ~-irreducible ideal I E ~ is primary.

1.5. Derivations and Differentials

In this section, by a ring we shall always mean a commutative ring.

1.5.1. DEFINITION. Let A be a ring and M an A-module. An additive mapping D : A -+ M is called a derivation of A into M if D(ab) = D(a)b + aD(b) for all a,b E A.

The set of all derivations of A into M is denoted by Der(A, M).

1.5.2. EXERCISE. Show that if D1, D2 are derivations of a ring A into an Amodule M, then D1 + D2 E Der(A, M) and aD1 E Der(A, M) for any a E A; thus, Der(A, M) is an A-module. (As usual, D1 + D2 and aD1 are mappings from A to M such that (D1 + D2)(x) = D1(X) + D2(x) and (aD!)(x) = aD1(x) for any x E A).

The A-module Der(A, M) is called the module of derivations of A into M.

38 I. PRELIMINARIES

1.5.3. EXERCISE. Let D be a derivation of a ring A into an A-module M, and B = Ker B = {a E A I D(a) = O}. Show that B is a subring of A (in particular, D(l) = 0), and if A is a field, then B is a subfield of A.

1.5.4. DEFINITION. Let k be a ring, A a commutative k-algebra, and M an A-module. A derivation D : A -t M is called a derivation over k or a k-derivation if D(A) = 0 for any A E k.

The set of all k-derivations of A into M will be denoted by Derle (A, M). Note, that if a E A, A E k and D E Derle(A, M), then D(Aa) = AD(a). It is also evident that Derle (A, M) is an A-submodule of Der(A, M).

In what follows we shall mainly deal with derivations of fields, so we formulate here some basic results on such derivations.

1.5.5. PROPOSITION. (see [Bou70b, Chap. V, § 9, Proposition 4]). Let n be a field, E a subfield ofn and F a purely transcendental field extension of E contained in n. Let (Xi)iEI be a pure transcendence basis of F over E. Then for any derivation D of E into n and for any family (UdiEI of elements of n there exists a unique derivation D of F into n such that DIE = D and D(Xi) = Ui for all i E I.

1.5.6. PROPOSITION. (see [Bou70b, Chap. V, § 9, Proposition 5]). Let n be a field, E a subfield ofn, and F a separable algebraic field extension of E contained in n. Then every derivation D of E into n can be uniquely extended to a derivation D of F into n.

1.5.7. PROPOSITION. (see [Bou70b, Chap. V, § 9, Theorem 2]). Let n be a field, K a subfield ofn and E = K(Xl, ... , xn) a finitely generated separable field extension of K contained in n. Let trdegK E = r. Then dimn DerK(E, n) = rand there exists a subset B = {Xi" ... , Xi.} of the set {Xl, ... , xn} such that B is a transcendence basis of E over K and E is a separable algebraic field extension of K (B). (In this case the set B is called the sepamting basis of the field E over K).

Let G be a field, F a subfield of G and DerF G the vector G-space of all Fderivations of G into itself (so that DerF G = Derp(G, G)). Let (DerF G)· = HomG(DerF G, G) be the dual vector G-space for DerF G. Then for every element 'TJ E G we can consider the element d'TJ E (DerF Gt such that (d'TJ)(D) = D(1]) for any D E DerF G. It is easy to check that the mapping 1] -t d'TJ is F-linear and d(1]1'TJ2) = 'TJl d1]2 + 'TJ2 d'TJl for any elements 'TJ1, 'TJ2 E G. (Indeed, d('TJ1'TJ2)(D) = D('TJ1'TJ2) = 'TJID(1]2) + 'TJ2D('TJd = ('TJld'TJ2 + 'TJ2d1]d(D) for every D E DerF G). Let nF(G) denote the vector G-subspace of (DerF Gt generated by the set of all elements d'TJ with 'TJ E G. nF(G) is called a module of differentials associated with the field extension G 2 F. Later on, we will use the following statement about the module of differentials associated with a finitely generated field extension.

1.5.8. PROPOSITION. Let G = F('TJ1, ... , 'TJ1e) be a finitely generated field extension of a field F. Then nF(G) is a finitely-dimensional vector G-space with the generators d'TJ1, ... , d'TJle.

PROOF. Obviously, it is sufficient to show that every generator d( (( E G) of the vector G-space nF(G) can be written as a linear combination of the elements d'TJ1, ... , d'TJk with coefficients in G. But this fact is almost evident: if ( E G, then


we can express ( in the form ( = f 1/" ... ,1/, where f and 9 are polynomials in k 9 T}l,···,'1k

indeterminates over F. Therefore,

1 d( = 2 ( ) [g (TJ1, ... , TJk) df (TJ1 , ... , TJk) - f( TJ1, ... , TJn) dg (TJ1 , ... , TJk)],

9 TJ1,'" , TJk

hence, d( is a linear combination of dTJi (1 ::; i ::; k) with coefficients from G (since df(TJ1,' .. , TJk) and dg(TJ1, ... , TJk) can be represented as such combinations). This completes the proof. 0

1.5.9. PROPOSITION. Let F be a field of zero characteristic, G a field extension of F and (TJ"')"'EA a family of elements of the field G. Then the family (TJ",)aEA is algebraically independent over F if and only if the family (dTJ"')"'EA of elements of the vector G-space OF (G) is linearly independent over G.

PROOF. Let the family (TJ"')"'EA be algebraically independent over F. Then we can extend this family to a transcendence basis B of Gover F. Let B' = B \ {(TJa)"'EA}. By Proposition 1.5.5, for any element (3 E A there exists a Fderivation D{J of the field F(B) into G such that

D{J(ry",) = { I,

0,

if 0: = (3,

if 0: =! (3

and D{J(b) = 0 for all bE B'. Since G is an algebraic extension of F(B) and all fields considered are of zero characteristic, for any (3 E A there exists a unique derivation D{J of G into itself which extends D{J (see Proposition 1.5.6). Now we can prove that the family (dTJ"')"'EA is linearly independent over G. Indeed, if 2.:;;'=1 CkdTJi. = 0 for some indices i1, ... ,im E A and for some elements C1, ... ,Cm E G, then

m m

Cp = 2: CkDip(TJi.) = (2: Ckd%)(Di p) = 0 k=1 k=1

for all p = 1, ... , m. Conversely, suppose that the family (dTJ"')"'EA is linearly independent over the

field G, but the family (ry"')"'EA is algebraically dependent over F, so that there exist a positive integer m and a polynomial f(X 1, ... ,Xm ) over F such that f( TJi,. ... , TJi m ) = 0 for some indices i 1, ... , im E A. Since Char F = 0, we may assume that It (TJi, , ... , TJi m) =! 0 for some index k (1 ::; k ::; m). Differentiating

the equality f(TJi" ... , TJi m ) = 0, we obtain that 0 = df(ryi" ... , TJi m ) = 2.:;;'=1 Akd% where Ak = It (ryi, , ... , TJi.), so that not all of A 1, ... , Ak are equal to zero. Therefore, the elements (dTJ"')"'EA are linearly dependent over G that contradicts our assumption. This completes the proof. 0

Now, let A be a commutative ring, B an A-algebra and p : B0A B -+ B a natural homomorphism of A-algebras (p : b 0 b' -+ bb' for all b, b' E B). Let [ = Ker p. Then one can naturally consider [/[2 as a B 0A B-module whose annihilator is [. Therefore, [/[2 can be treated as a module over (B 0A B)/ [ and also as a B-module (since the A-algebras (B 0A B)/ [ and B are canonically isomorphic). Such B-module [/[2 is called a module of Kohler differentials of B over A and is denoted by OBIA. Since b 01 - 10 b E [ for any b E B, one can consider the well-defined mapping dBIA : B -+ [/[2 such that dBIA (b) = (b 01 - 10 b) + [2 for all b E B. It is easy to see that dB1A is a derivation of B into OBIA'

40 1. PRELIMINARIES

1.5.10. PROPOSITION. Let B be an algebra over a ring A. Then the pair ([lBIA, dBIA) has the following universal property: if M is a B-module and D is an A-derivation of B into M, then there exists a unique B-module homomorphism !.p : [lBIA -t M such that !.p' dBIA = D.

PROOF. For any element x 129 Y of B Q9A B (x, Y E B) we have

x 129 Y = xy 1291 + (x 129 1)(1129 Y - Y 1291) = p(x 129 y) 1291 + (x 1291)(1 129 Y - Y 1291)

(p is the natural homomorphism BQ9A B -t B; as above, its kernel is denoted by I). Therefore, ifLi XiQ9Yi E [(Xi, Yi E B), then Li XiQ9Yi = Li(xiQ91)(IQ9Yi-YiQ91). Since (1129 Yi - Yi 1291) + [2 = dBIAYi, any element of [lBIA = [/[2 has the form Li XidBIAYi (we identify the canonical image of Xi 1291 in (B Q9A B)/[ with x;), so that the set {dBIAY lyE B} generates [lBIA as a B-module. This proves the uniqueness of !.p.

In order to prove the existence of !.p, let us consider a trivial extension B * M of the algebra B by the B-module M, that is the additive group B EEl M with the ring structure via the multiplication by the formula (a, x)(b, y) = (ab, ay+ bx) (a, bE B; x, Y E M). It is easily checked that we really obtain a ring with the unit (1,0) and this ring B * M is an A-algebra. Let 1 : B Q9A B -t B * M be a homomorphism of B-algebras such that I(x 129 y) = (xy, xD(y)) for all x, Y E B. Since I(I) ~ M (we identify M with the B-submodule {(O, z) I z E M} of B * M) and M2 = 0, we have 1(I2) = 0, so that 1 induces a homomorphism j of B-algebras BQ9A B/ [2 -t B * M which maps dBIAY E [lBIA to 1(1129 Y - Y 129 1) = (0, Dy). The restriction of j to [lBIA gives a B-linear mapping !.p : [lBIA -t M with !.p . dBIA = D. This completes the proof. 0

1.5.11. COROLLARY. Let B be an algebra over a ring A and M a B-module. Then there exists a canonical isomorphism of B-modules

PROOF. Obviously, the mapping D -t !.p(D) whose existence is proved in Proposition 1.5.10 is a B-homomorphism and its kernel is equal to 0. It is also surjective: if !.p' E HomB([lB/A, M) then !.p' is the image of the derivation D E DerA(B, M) defined as follows: if TJ E B, then D(TJ) = !.p(dB/ATJ). 0

1.5.12. EXAMPLE. Let A be a ring and B an A-algebra generated (as an Aalgebra) by a family (Xi)iEl' Then the B-module [lBIA is generated by the family (dBIAXi)iEI. If the family (X;)iEl is algebraically independent over A (so that B is a polynomial A-algebra in the family of indeterminates (Xi)iEl), then [lBIA is a free Bmodule with the set offree generators (dBIAXi)iEl' Indeed, let L;;'=1 AkdBIAxi. = ° for some elements AI, ... , Am E B (i l , ... , ik E I) and let -f) f) denote the partial x,. derivation of the ring B with respect to Xi •. Then -f)f) E DerA(B, B), hence, (see x,. Corollary 1.5.11) there exists a homomorphism of B-modules /;. : [lBIA -t B such that

if j = i k

if j -=j:. i k


for any j E I. Now, for every I = 1, ... , m, we have A/ = /;, (2:;;'=1 AkdBIAXik) = fdO) = 0; so that the family (dBIAXi)iE/ is linearly independent over B, i.e., this family is a basis of the free B-module nBIA.

1.5.13. EXERCISE. Let k be a field and K a separable algebraic field extension of k. Show that nKlk = O. [Hint: if 0 E K, then there exists a polynomial f(X) E k[X] such that f(o) = 0, J'(o)::f 0 (by J'(X) we mean the usual derivative of f(X)). Since 0 = dKlk(f(O)) = J'(o)dKlkO, we obtain that dKlko = 0].

1.5.14. EXERCISE. Let B ----+ B'

I I A ----+ A'

be a commutative diagram of rings and ring homomorphisms.

(a) Show that there exists a natural homomorphism of B-modules nBIA -+ nB'lA' that induces a natural homomorphism of B-modules

(1.5.1)

(b) Show that if B' = BQ9A A', then (1.5.1) is an isomorphism (so that nB'lA' ==' nBIA Q9A A' ==' nBIA Q9B B').

(c) Show that if 5 is a multiplicatively closed subset of the A-algebra Band B' = 5-1 B, then nB'lA ==' nBIA Q9B B' ==' 5- 1n BIA'

1.5.15. THEOREM. Let k,A,B be commutative rings, <p: k -+ A and1/;: A -+ B ring homomorphisms (whose actions allow to consider B as an A-algebra and as a k-algebra). Then:

(1) there exists an exact sequence of B-modules

(1.5.2)

where v : (dAlka) Q9 b -+ bdBlk1/;(a) and u : bdBlkb' -+ bdBIAb' for all a E A; b, b' E B;

(2) the mapping v is invertible from the left (i.e. v is injective and 1m v is a direct summand of the B-module nBIA) if and only if every k-derivation of A into a B-module T can be extended to a derivation of B into T.

PROOF.

(1) Obviously, u is surjective. Since dBIA1/;(a) = 0 for any a E A, we have u· v = O. It remains to show that Keru = Imv (i.e., that Keru ~ Imv). Let T be a B-module. Then we have an exact sequence

(where 0 is defined in a natural way, and {3 : D -+ D . 1/; for any D E Derk (B, T)), and the canonical isomorphisms

(1.5.3)

42 1. PRELIMINARIES

induce the exact sequence of B-modules

Setting T = Coker v, we obtain that Ker u = 1m v, so that the sequence (1.5.2) is exact.

(2) Note that a homomorphism of B-modules 'Y : M' ---* M is invertible from the left if and only if the induced mapping 'Y. : HomB(M,T) ---* HomB(M',T) (cp ---* cP .'Y) is surjective for every B-module T. Applying this remark to the homomorphism v and using the isomorphisms (1.5.3) we obtain that v is invertible from the left if and only if the mapping Derk(B, T) ---* Derk(A, T) (which is induced by v) is surjective for every B-module T . This completes the proof. 0

1.5.16. COROLLARY. With the notation of Theorem 1.5.15, the mapping

is an isomorphism of B-modules if and only if every k-derivation of A into a Bmodule T can be uniquely extended to a derivation of B into T.

1.5.17. THEOREM. Let k be a commutative ring, A an k-algebra, m an ideal of A and B = A/m. Suppose, that 0: : m ---* QAlk ®A B is the homomorphism of A-modules such that o:(x) = dAlkX ® 1 for every x E m, and d is the B-linear mapping m/m2 ---* QAlk ®A B induced by 0: (obviously, 0:(m2) = 0). Then the following assertions hold.

(i) The sequence of B-modules

m/m2 ~ QAlk ®A B ..;. QBlk ---* 0 (1.5.4)

is exact (v is the same as in sequence (1.5.2)). (ii) Let Al = A/m2. Then QAlk ®A B ~ QAdk ®A, B.

(iii) The homomorphism d is invertible from the left if and only if the extension o ---* m/m2 ---* Al ---* B ---* 0 of the k-algebra B by m/m2 is trivial over k.

PROOF.

(i). First of all, note that the natural epimorphism A ---* B implies the surjectivity of v. Obviously, v . d = 0, so that it is sufficient to prove that the sequence

(which is induced by the sequence (1.5.4)) is exact for every B-module T. This fact is, in turn, equivalent to the exactness of the sequence

(see isomorphisms (1.5.3)) where f is defined naturally, and 9 : D -* Dim for every D E Derk(A, T). Since the last sequence is evidently exact, we obtain that the sequence (1.5.4) is also exact.


(ii). Note that a homomorphism of B-modules N' ~ N is an isomorphism if and only if the induced mapping Homn(N, T) ~ Homn(N', T) is isomorphism for any B-module T. Applying this remark to our case we obtain (see isomorphisms (1.5.3)) that the second statement of the theorem is equivalent to the evident fact that the natural mapping Derk(Alm2, M) ~ Derk(A, M) is an isomorphism for any Aim-module M.

(iii). By (ii), we can replace A by Al in the exact sequence (1.5.4), so we can suppose that m2 = O. Let w . c5 = 1 for some B-linear mapping w : nAlk ®A B ~ m/m2.

Setting D(a) = w(dAlka ® 1) for every a E A we obtain the k-derivation D : A ~ m such that D(x) = x for every x Em. Let f : A ~ A be a mapping defined as follows: f(a) = a - D(a) for every a EA. Obviously, f is a homomorphism of k-algebras and f(m) = 0, hence f induces a homomorphism of k-algebras j : B = Aim ~ A. Since f(a) - a E m for all a E A, j is a section of the ring extension o ~ m ~ A ~ B ~ O. The converse implication of (iii) can be proved by the permutation of the above arguments. 0

Skew polynomial rings. In conclusion of this section we consider a special type of rings that are widely

used in the ring theory as a source of examples and counter-examples. At the same time, these rings are closely connected with differential and difference algebraic structures (actually, rings of differential and difference operators are rings of this type).

1.5.18. DEFINITION. Let R be a ring (not necessarily commutative) and let {3 be an injective endomorphism of R. An additive mapping c5 : R ~ R is called a {3-derivation of the ring R (or a skew derivation of R associated with the endomorphism (3) if c5(ab) = (3(a)c5(b) + c5(a)b for any a, bE R.

Let R be a ring (not necessary commutative), let 0"1 = {al, ... , an}, 0"2 = {{31, ... , {3m} be two sets of injective endomorph isms of R, and let d1 , ... ,dm be skew derivations of the ring R associated with the endomorphisms {31 , ... , {3m, re-spectively. Furthermore, suppose that any two elements of the union of the sets 0"1, 0"2 and Ll = {d1 , ... , dm } commute with each other (i.e. A(J.L(a)) = J.L(A(a)) for any A, J.L E 0"1 U 0"2 U Ll, a E R). Let n be a free commutative semigroup with free generators Zl," . ,Z2m+n so that each element wEn has a unique representation of the form w = zt, ... z;~..t~ (il,"" i2m+n EN). Then the set S of all formal finite sums L:wEO aww (coefficients aw belong to R and only a finite number of these coefficients are not equal to zero) can be naturally considered as a left R-module. Moreover, S becomes a ring if we define the multiplication of elements of S in such a way that for every a E R we set Zja = f3;(a)zj + dj(a) for i = 1, ... , m, Zja = f3;(a)zj for i = m + 1, ... , 2m, z;a = a;(a)zj for i = 2m+ 1, ... , 2m+n, and extend this definition by the distributive law. The ring obtained is called a skew polynomial ring over R associated with the set 0"1 U 0"2 U Ll. This ring is denoted by R[Zl, ... ,Z2m+n;dl, ... ,dm;{31, ... ,{3m;al, ... ,an] or by R[ZI' ... , Z2m+n; d1 (f31), ... ,dm(f3m); f31, ... ,f3m; aI, ... ,an]' Of course, the usual polynomial ring in indeterminates ZI, ... ,Z2m+n over R is a particular case of the ring of skew polynomials constructed (if all {3j and aj are identical automorphisms

44 1. PRELIMINARIES

of Rand d1 , ... , dm are zero derivations of R, we obtain the usual polynomial ring R[Zl"'" Z2m+n]).

If d1, ... , dm are usual derivations, then R[ Zl, ... , Zm+n ; d1, ... , dm; aI, ... , an] denotes the skew polynomial ring in m + n indeterminates Zj such that Zj a = aZi + di(a) (1 :S i :S m) and zja = aj(a)zj (m + 1 :S j :S m + n) for any a E R (in this case we consider 0'2 as an empty set). Furthermore, if A = 0, then we can naturally consider the skew polynomial ring R[ Zl, ... , Zn; aI, ... ,an] in n indeterminates Zl,"" Zn (zja = aj(a)z; for i = 1, ... , n).

The following theorem can be proved by the same method as is used in the standard proof of Hilbert basis theorem (see the proof of Theorem 1.2.40). We believe, it would be a good exercise for a reader to give this proof.

1.5.19. THEOREM. Let R be a left Noetherian ring, and let

be the skew polynomial ring, where aI, ... , an; f31, ... ,f3m are automorphisms of R and di (f3;) is a f3i-derivation of the ring R (1 :S i :S m). Then the ring S is left Noetherian.

CHAPTER II

NUMERICAL POLYNOMIALS

In this chapter we consider properties of polynomials with rational coefficients which have integer values for any sufficiently large integer arguments (such polynomials are called numerical). It is shown that for any given subset E of ilm one may associate with E some finite family of numerical polynomials (these polynomials are called dimension polynomials of E; to a certain degree, they characterize the set E itself). The main attention is attracted to the univariate dimension polynomials associated with subsets of Nm , because the problem of determination of such polynomials is a part of the problem of computing Kolchin's differential dimension polynomial of finitely generated differential field extensions.

2.1. Definition and some Properties of Numerical Polynomials

2.1.1. DEFINITION. A polynomial l(t1, ... ,tn) in n indeterminates t1, ... , tn with rational coefficients is called numerical iff l(t1,"" tn) E il for all sufficiently large t 1 , ... , tn E il, i.e. there exists an-tuple (T1,"" rn) E il such that I( 81, ... ,8n ) E il for any vector (81, ... , 8n ) E il with Ti S 8i (i = 1, ... , n).

It is clear that any polynomial with integer coefficients is numerical. As an example of an univariate numerical polynomial with non-integer coefficients one may consider the polynomial

( t)=t(t-1) ... (t-m+1) (mEil,m~l), m m!

(2.1.1)

which, for any integer t ~ m, gives the number of combinations by t things of m. Sometimes we shall consider the expression (!) for nonpositive values of m by

assuming

We shall also set

G) = 1, (:) = 0

(:) + = { ~!)

for m < O.

ift > 0

if t < O.

(2.1.2)

Many relationships that are valid for regular binomial coefficients are also valid for polynomials of the form (2.1.1). In particular, it can be directly tested that

(2.1.3)

The following proposition gives us some relationships between "binomial" numerical polynomials which will be used later.

45


46 II. NUMERICAL POLYNOMIALS

2.1.2. PROPOSITION. The following identities hold for any n,p, r E N:

(2.1.4)

(2.1.5)

(2.1.6)

(2.1.7)

(2.1.8)

PROOF. The validity of (2.1.4) and (2.1.5) can be easily derived from (2.1.3) by induction on n.

Before proving (2.1.6)-(2.1.8), let us note that if the values of numerical polynomials f(t) and g(t) coincide for all sufficiently large integer values of t, then f(t) == g(t). Therefore, in the proof of (2.1.6)-(2.1.8) we can (and we shall) suppose that t E ~,t ~ n.

Comparing the coefficients of xn in the identity (1 + x)t+k = (1 + x)t(1 + x)k, we obtain (2.1.6). In order to obtain (2.1.7), first we shall prove that

t (~) ( i ) = 2k (n) ;=0 l n - k k

(2.1.9)

for any k, n EN (as usual, we suppose (~) = 0 for k > n). Indeed, by the directly tested identity

we have

t(7)(n~k) =t(~)(i+:-n) = (~).t C-(~-k)) .=0 .=0 .=n-k

= G) t, G) = (~)(1 + I)k = 2k (~). Now we can complete the proof of (2.1.7) using (2.1.6) and (2.1.9):

~ G) C : i) = ~ (7) ~ G) (n ~ k)

= ~ G) ~ (7) (n~k) = ~2k(~) G)'

2.1. DEFINITION AND PROPERTIES OF NUMERICAL POLYNOMIALS 47

Now we can apply (2.1.6) and the obvious identity

to obtain (2.1.8):

~(-lt-i2i (:) C: i) = ~(-lt-i2i (7) 1; G) C ~ k)

= ~(-lt-i2i(:) ~ G) (1) = t G) t(-1t-i2i (:) (1)

k=O .=0

= ~ G) (~) ~(-1t-i2i (: ~:)

= 1; G) G) 2k ~(_1t-i2i-k (: =~)

= 1;2kG) G) ~(-1)(n-k)-j2j(n~~:J = 1;2k (~) G)(2 -It-k = 1;2k G) G)'

This completes the proof of the proposition. 0

Note that if f(t) is a numerical polynomial in one variable t, then its first difference 6.f(t) = f(t + 1) - f(t) and the next differences 6.2 f(t) = 6.(6.f(t)), 6.3 f(t) = 6.(6. 2 f(t)), etc. are also numerical polynomials. In particular, (2.1.3) shows that

(2.1.10)

2.1.3. PROPOSITION. Let f(t} be a numerical polynomial of degree m in one variable t. Then f(t) can be represented in the form

m (t + i) f(t) = ~ ai i ' (2.1.11)

where aD, al, ... , am are integers uniquely defined by f(t).

PROOF. Dividing the polynomial f(t) by c-;;,m) in the ring Q[t], we obtain f(t) = am c-;;,m) + r(t) where am E Q[t) and deg r(t) ::; m - 1. Dividing r(t) by e-;;'~11) (in


Q[t]), we obtain f(t) = ame~m) + am_le~~~l) + rdt) where degrl(t) ~ m - 2. Continuing this process, we arrive to the expression

m (t+i) f(t) = ~ai i ' aj E Q (i = 0,1, ... , m), (2.1.12)

where the rational numbers ao, al, ... , am are uniquely defined by f(t). We have to show now that aj E Z (i = 0,1, ... , m). We shall do it by induction on m = deg f(t).

If m = 0, then the numeriqueness of f(t) implies f(t) = ao E Z. Suppose that m > ° and the existence and uniqueness of representation (2.1.11) (with integer coefficients ao, aI, ... , am) is proved for all numerical polynomials of degree less than m. Considering finite differences of the both parts of (2.1.12) and using (2.1.10) we obtain

flkf(t) = };ai+ke+~+k) (k=I,2, ... ,m).

The polynomial flk f(t) is numerical, hence flm f(t) = am E Z. Applying the inductive hypothesis to the polynomial f(t) - am e~m) = L:~~l aj eti) (whose degree does not exceed m - 1), we obtain that ao, al, ... , am-l E Z. 0

Proposition 2.1.3 implies, in particular, that the leading coefficient of any numerical polynomial f(t) of degree m equals L!>:!!(t) , hence f(t) can be represented in the form

(2.1.13)

(As usual, o(tm ) denotes a univariate polynomial with rational coefficients whose degree does not exceed m - 1.)

Furthermore, since eti) E Z for any t E Z, and i E N, Proposition 2.1.3 implies the following result.

2.1.4. COROLLARY. If f(t) is a numerical polynomial in a variable t, then f(t) E Z for anyt E Z.

2.1.5. COROLLARY. Let f(tl, ... , tn) be a numerical polynomial of degree m in n variables tl, ... , tn. Then the polynomial f(tl, ... , tn) can be represented in the form

where ai, ... i" are integers uniquely defined by f(tl, ... , tn).

PROOF. We use induction on n. The case n = 1 is considered in Proposition 2.1.3.

Let n > 1. Then there exists a vector (81, ... , 8n) E Nn such that if ri ~ 8i,

for all i = 1, ... , n - 1, then f(rl, ... , rn-l, tn) is a numerical polynomial in one variable tn which, by Proposition 2.1.3, can be represented in the form


where dn = deg t ,. f and ai,. (tl' ... , tn-l) (0 ~ in ~ dn) are uniquely defined numerical polynomials of degree less than or equal to m - in (since deg e"i~i,.) = in for all in = 0, 1, ... , dn). By the inductive hypothesis,

where ail, ... ,i,. E &: for all indices i l , ... in which appear in the sum. Therefore,

f(tl, ... ,tn )

~Si,,~g.n J [(;"';!?'eN'-~;';'-';' (" :;,) ... ('.-:.~:.-')] en ~ in)

i l +···+i,,_l Sm-i,.

n (t. + i.) ai! ... i" II 3 i. 3

j=l 3

(in the latter sum we assume that ail ... i,. = 0 for all multiindices (i l , .. . ,in ) such that in > dn). 0

2.1.6. PROPOSITION. Let f(t) = amtm+am_ltm-l+ .. '+alt+aO be a numerical polynomial of degree m in one variablet such that f(s) E &: for any s E &:, s > So (so is an integer). Then there exists a numerical polynomial g(t) with the following properties:

(1) g(s) = f(so + 1) + f(so + 2) + ... + f(s) for any s E &:, s > so; (2) degg(t) = m + 1; (3) the leading coefficient of g(t) is equal to m~l am.

PROOF. By Proposition 2.1.3, f(t) can be represented in the form f(t) = I:~o hi eti) , where bo, bl, ... , bm E &:, and it is easy to see that bm = am ·m!. Hence

m 0-00-1 (so + 1 + i + k)

f(so + 1) + f(so + 2) + ... + f(s) = L: bi L: i i=O k=O

for any s E &:, s > so. Applying relation (2.1.4), we can replace the internal sum in the right-hand side of the latter equation by Cot!!1) - ('°titl) hence

.~o f(s .) = ~ b. [(s + i + 1) _ (so + i + 1)] = ~ b. (s + i + 1) _ A ~ o+J ~. i+l i+l ~. i+l ' 3=1 .=0 .=0


h A ",m b (so+i+1) '1l Th th . I I . I were LJi=O i i+1 E u..... us, e numenca po ynomta

g(t) = 2::::0 bi (tt!t1) - A satisfies all the conditions 1) - 3) (the degree of this polynomial is equal to m + 1, and the coefficient of tm +1 is equal to the coefficient of tm+1 in the polynomial bm C~~t1), i.e. to the number (:+1)! = m~1 am). The proposition is proved. 0

In conclusion of this section, we shall give solutions of some combinatorial problems closely connected with the problem of computation of differential and difference dimension polynomials treated below.

For any integers m and r (m > 0, r ~ 0), let Jl+(m, r) denote the number of solutions of the equation

Xl + X2 + ... + Xm = r (2.1.14)

in positive integers Xi. Let Jl(m, r) denote the number of solutions of equation (2.1.14) in non-negative integers Xi, and [1.( m, r) the number of solutions in integers Xi of the equation

(2.1.15)

2.l.7. PROPOSITION. With the preceding notation,

+ (r-I) Jl (m, r) = m _ 1 ' (2.l.l6)

( m+ r -1) Jl(m,r) = m-I ' (2.l.l7)

and

(2.1.18)

PROOF. First of all, we shall prove (2.1.17). For this purpose, we associate with every solution (Xl, ... ,Xm ) E Nm of equation (2.1.14) the ordered set ofr zeros and (m - 1) ones which is constructed in the following manner: we put Xl zeros, then one 1, then we put X2 zeros, then one 1 and so on. After the last 1 we put Xm

zeros. It is easy to see that the correspondence is one-to-one and Jl( m, r) equals the number of the sets described above. On the other hand, this number is equal to the number of all (m - 1) element subsets of the set {I, 2, ... , m + r - I}: a subset {il, ... ,im-d (1 ~ il, ... ,im- l ~ m+r-I) corresponds to the ordered set of zeros and ones, where the ones are in the positions i l , ... , i m - l . Hence,

( ) (m+r-l) Jlm,r = m-l . Any solution (Xl, ... , xm) of equation (2.1.14) in positive integers corresponds

to the solution (x~, ... , x:n) E Nm of the equation Xl + ... + Xm = r - m, where xi = Xi - 1 (1 ~ i ~ m). Conversely, each solution (x~, ... , x~) E Nm of the last equation corresponds to the solution in positive integers (x~ + 1, ... , x~ + 1) of equation (2.1.14). Hence

Jl+(m, r) = Jl(m, r - m) = (r - m + m - 1) = (r -1 ) . m-I m-I


(Notice that (2.1.16) also holds for r < m, because m = 0 for k,1 EN, k < I.) To prove (2.1.18)' note that the number of m-tuples (Xl, ... , Xm) E Nm , for which

Xl + ... + Xm = r and all coordinates, except of xk" ... , Xki (1 :S i:S m), vanish, is equal to p+ (i, r) = G= D. It follows that the number of elements (Xl, ... , Xm) E Nm

for which IXII + ... + IXm I = r and all coordinates, except of Xk" ... , Xki' vanish, equals 2i G=D. Thus, there exist (7)2 i (:=D elements (Xl, ... ,Xm ) E ~m, such that IXII + IX21 + ... + IXml = r and just i coordinates of the vector (Xl"", Xm)

are different from zero (i = 0,1, .. . ,m). Hence, jl(m,r) = 2::;:12 i (7)(:=i). The proposition is proved. 0

2.1.8. PROPOSITION. Let il = (Ul, ... ,Um ), V = (Vl, ... ,Vm ) E Nm , Ui:S Vi

(i = 1, ... , m). Let Cmr(il, v) (m, r E N, m 2: 1) denote the number of solutions (Xl, ... , Xm) E Nm of equation (2.1.14) such that Ui :S Xi :S Vi (i = 1, ... , m), and let R = Ul + ... + U m ! r = VI + .,. + Vm , di = Vi - Ui + 1 (1 :S i:S m). Then

Cmr (il, v) = (m + : ~ ~ - 1)

k=l l<j,<···<ik<m dj;+ ... +djk ~r-R

( m + r - R - d· - ... - d· - 1) 3' 3k m-l .

(2.1.19)

PROOF. It follows from the definition of Cmr (il, v) that this number equals the coefficient of t r in the polynomial

P(t) = tR (1 + t + ... + t d ,-l)(1 + t + ... + td2 - l ) ... (1 + t + ... + t dm - l ).

Indeed, every solution (Xl, ... ,Xm ) E Nm (Ui:S Xi:S Vi for i = 1, ... ,m) of equation (2.1.14) is in one-to-one correspondence with a monomial t r (with coefficient 1), which is obtained by expanding the polynomial P(t), ifin ith brackets we take the factor e'i-Ui (i = 1, ... , m). It follows that the number of such monomials is equal to Cmr(il, v).

Since 1 + t + ... + t di - l = (1 - t)-1(1 - t di ) (1 :S i :S m), we have P(t) = tR (1 - t)-m ITT=1 (1 - t dj ). Furthermore, since (1 - t)-l = 2:::0 t i in the formal power series ring lQl[[t]] (this identity is a direct consequence of the obvious equality (l-t)2:::o ti =1),

00

(1 - t)-m = (1 + t + t 2 + ... )(1 + t + t2 + ... ) ... (1 + t + t2 + ... ) = LOtti, 1=0

where (in accordance with the above considerations) a coefficient CI is equal to the number of solutions (Xl, ... , Xm) E Nm ofthe equation Xl + ... + Xm = l. Therefore (see Proposition 2.1.7), CI = (m~:~l), so that (1 - t)-m = 2::~0 (m~:~l)tl. This relationship shows that the coefficient of t r in the polynomial

P(t) =tR{~ (m:~~ l)t1} Q(1-tdj )

= {f (m + ~ ~ ~ -1)t1 } . {I + I)-1)k L tdi,+ ... +di k }

I=R k=l l~j,<···<jk~m


is equal to

( m + r - R - d· _ ... - d· - 1) 31 3k

m-l

o

Let p(m, r) and p(m, r) denote the number of solutions (Xl, ... , xm) E Nm of the inequality

Xl + ... + Xm :::; r (2.1.20)

and the number of solutions (Xl, ... , Xm) E ~m of the inequality

(2.1.21)

respectively, (m, r E Nj m> 0).

2.1.9. PROPOSITION. With the above notation,

(2.1.22)

(2.1.23)

PROOF. Since p(m,r) = L:~=oJ.l(m,k) = L:~=o (m;~ll) = L:~=o (m+;-l), we

obtain (applying (2.1.5) for t = m -1, n = r) that p(m, r) = (m-l:r+l) = (r~m). In order to prove (2.1.23), we shall use formula (2.1.18):

Since (tD = 0 for k < i,

t;e=D =~e=D =~(q~~~I) = ~ C - ~ + q) = C -1 ~ ~ ~ i + 1) = G)

(see (2.1.5)). Thus, p(m,r) = L:~o2;(7)(:), and the other equalities in (2.1.23) immediately follow from (2.1.7) and (2.1.8). The proposition is proved. 0

2.2. DIMENSION POLYNOMIAL OF A MATRIX

2.2. Subsets of Nm and their Dimension Polynomials. Dimension Polynomial of a Matrix

53

Let (WI ~I), ... ' (wm ~m) be ordered sets. Then their direct product P = n::1 Wj can be ordered in different ways. We shall mainly consider two order relations on the set P: the product order ~, when the inequality (al, . .. , am) ~ (b l , ... , bm ) holds iff aj ~i bi for all i = 1, ... , m, and the lexicographic order --< which is defined by the following condition: (al, ... ,am) --< (bl, ... ,bm ) in P iff there exists an index k such that al = bl , ... , ak-I = bk- I , but ak =f bk, ak ~k bk. It is easy to see that if each set Wi (i = 1, ... , m) is well-ordered with respect to the order ~i, then the set P = n:':::1 Wi is well-ordered with respect to the lexicographic order (in general, the set P is not even linearly ordered relative to the product order).

In accordance with the above considerations, we shall consider the product order ~ and the lexicographic order --< on the set Nm of all m-tuples with non-negative integer entries, and also on the sets of the form Nm x Nkl X ... X Nk p (m EN, ki E Z, kj 2: 1 for i = 1, ... , p), where N k = {I, 2, ... , k} for any k E Z, k 2: 1. The sets N and Nk are always considered with the natural order, with respect to which they are well-ordered, as it is easy to see. This natural order we shall denote by the same symbol ~ that we use for the product order, there will be no ambiguities in this case.

2.2.1. LEMMA.

(1) Any infinite subset of Nm x Nk (m, kEN, k 2: 1) contains an infinite sequence strictly ascending with respect to the product order and such that the projections of all its entries onto Nk are equal.

(2) There exists some order ~o on the set Nm x Nk with respect to which this set is well-ordered and the two following conditions hold:

(a) (i1, ... ,im,j) ~o (it+el, ... ,im+em,j)forallil, ... ,im,el, ... ,em E N,jENk;

(b) if(il, ... ,im,j) ~o (i~, ... ,i~,j') then (il + el,.·.,im + em,j) ~o (., + ., + .') fc all· ..,., E IN zi el,···,Zm em,} or ZI,· .. ,zm,zl,·.·,Zm,el,.·.,em 1'l,

j,j' E Nk; (c) the set Nm x Nk is well-ordered with respect to any linear order sat

isfying the condition (a).

PROOF. It is easy to see that if E is an infinite subset of Nm x N k , then there exists an infinite subset EI ~ E such that the projections of all elements e E EI onto Nk are the same. Thus, in order to prove the first statement of the lemma it is sufficient to show that any infinite subset of Nm contains an infinite subsequence strictly ascending relative to the product order. Let F be an infinite subset of Nm . If the set of the first coordinates of elements of F is infinite, then there exists an infinite subset G ~ F such that the first coordinates of any two its different elements do not coincide. Therefore, there exists an infinite sequence G I ~ G such that the first coordinates of elements of G I form strictly ascending sequence in N. If the set of the first coordinates of elements of F is finite, then there exists an infinite subset F' of F such that all its elements have the same first coordinate. In both cases there exists an infinite subsequence FI ~ F consisting of different elements of


F such that their first coordinates form non decreasing sequence in N. Similarly, we can choose from the sequence Fl an infinite subsequence F2 of different elements whose second coordinates form nondecreasing sequence in N and so on. As a result, we obtain an infinite sequence Fm of elements of Nm strictly increasing relative to the product order. Thus, the first statement of the lemma is proved.

Let us consider the order ~o on the set Nm x Nk such that (i l , ... , im, j) <0 (i~, ... , i~,j') if and only if(L;=o ill, j, ii, ... , im) -< (L;=o i~,j', i~, ... , i~) ("-<" denotes the lexicographic order on N x Nk X Nm) for any elements (ii, ... , im , j), (i~, ... , i~,j') E Nm x N k . The set Nm x Nk is well-ordered relative to this order (since the set N x Nk X Nm is well-ordered relative to the lexicographic order) and the conditions (a) and (b), obviously, hold.

We prove now the last statement of the lemma. Let ~o be a linear order on the set Nm x Nk satisfying the condition (a) and let F by an infinite subset of Nm x Nk. By the first statement of the lemma, there exists an infinite subsequence Fl ~ F strictly increasing relative to the product order and such that all its elements have the same projection on Nk. Let us show that Fl is also strictly increasing relative to the order ~o. Indeed, if I' = (ii, ... , i m , j), f" = (i~, ... , i~, j) E Fl and f' -::j:. f", il ~ i~, ... , im ~ i~, then f' ~o f" = (il + (i~ - iJ), ... , im + (i~ - im),j) since the order ~o satisfies the condition (a). Thus, any strictly decreasing (relative to the order ~o) sequence of elements of Nm x Nk is finite (otherwise, as we have seen, it contains a strictly increasing subsequence that is impossible for a decreasing sequence), so that the set Nm x Nk is a well-ordered one. The lemma is proved. 0

By a partition of the set Nm = {I, ... , m} we shall mean its representation as a union of p disjoint non-empty subsets (for some positive integer p):

(2.2.1)

where (1"] n Uk = 0 for j -::j:. k (1 ~ j, k ~ p). Let m E Z;:, m ~ 1 and let U be a subset of N m . For any p-dimensional vector

(51, ... , 5p ) E NP, let U (51, ... , 5m ) denote the set of elements u = (Ul, .. . , up) of U, for which the inequalities LiEu 1 Ui ~ 51, ... , LiEUp Ui ~ 5p hold. Furthermore, if E ~ Nm then VE will denote the set of all elements v E Nm which do not exceed any element of E relative to the product order ~ on Nm . (In what follows, unless otherwise specified, all comparisons of elements of Nm are considered relative to the product order). Thus, the inclusion v E VE is equivalent to the condition that the inequality e ~ v does not hold for any element e E E.

In what follows hu denotes the function Z;:P -t N such that hu (51, ... , 5p ) = Card U (51, ... , 5p ) for any (51, ... , 5p ) E NP and 1(51, ... , 5p ) = 0 if at least one of the arguments is negative.

2.2.2. LEMMA. Let U ~ Nm and let 0 be the result of the parallel translation of the set U by a vector a = (al, ... ,am ) E z;:m, i.e. 0 = {a + iilii E U} ~ z;:m. Suppose also that 0 ~ Nm . Then for all (51, ... , 5p ) E Z;:P we have hu (51, ... , 5p ) = ha(51 + aU, ,···, 5p + aup ), where aUj = LiEUj ai (j = 1, ... , p).

PROOF. Obviously, the parallel translation by the vector a that maps a point u E U(51' ... ' sp) to a point U E 0(51 + aU" •.• , 5p + aup ) is a bijective mapping

2.2. DIMENSION POLYNOMIAL OF A MATRIX

hu (Sl, ... , Sp) = Card U (Sl , ... , sp)

= Card tJ(Sl + aa" ... , sp + aap)

=hO(Sl+ aa" ... ,Sp+aap).D

55

2.2.3. LEMMA. Let J{ ~ zm and let L = {x E Nm Ix exceeds no point of J{

relative to the product order on zm}. Then there exists a subset H ~ Nm such that hYH (Sl, ... , sp) = hL(Sl, ... , sp) for all Sl, ... , sp E Zp.

PROOF. For each point a = (aI, ... , am) E J{ we define f( a) = (iI, ... , jm) as follows: ji = max(O,ai), i = 1, .. . ,m. Let H = UaEKf(a). Then H ~ Nm is the desired subset. Indeed, x E Nm \ L iff x E N m and x exceeds some point a E J{,

that is equivalent to the inequality x 2: f(a), a E K. 0

For any given subset E ~ Nm and for any element e = (e1, ... , em) E Nm, let E1 = EUe, and let V(Sl' ... 'Sp) = hyE (Sl, ... ,Sp) - hYE ,(Sl, ... ,Sp) for all Sl, ... , sp E zP. It is clear that v( Sl , ... , sp) = hu (Sl , ... , sp) where U is the set of elements v E VE such that v 2: e. Applying Lemma 2.2.2 to U and (-e) we see that V(Sl, . .. , sp) = hO(Sl - ea" ... , sp - eap ). Furthermore, denoting by J{ ~ zm the result of the parallel translation of E by the vector (-e) and applying Lemma 2.2.2, we obtain that L coincides with tJ. Therefore, hO(Sl' ... 'Sp) = hYH(Sl, ... ,Sp) where the set H is defined by the following condition: x = (Xl, ... , Xm) E H iff there exists an element r = (r1, ... , rm) E E such that Xj = max(O, rj -ej), j = 1, ... , m.

Thus, we have proved the following formula:

(2.2.2)

for all Sl, ... , sp E Zp.

2.2.4. LEMMA. Let E ~ Nm (m > 1), let ik E Uk, (1 ~ k ~ p), and let E contain an element whose ik th coordinate equals 1 and all other coordinates equal zero. Let E denote the set of all elements e = (e1, ... , em-I) E Nm-1 such that (e1, ... ,eik_"O,eik+" ... ,em-d E E. Consider the following partition of the set Nm - 1 :

Nm - 1 = U Ui U ih where Uk = Uk \ {ik} i=l, ... ,p

i#

(if Uk = {ik} then Nm - 1 = Ui=l, ... ,pUi). Then hYe(Sl' ... 'Sp) = hYt;(Sl, ... ,Sp) i#

for all (Sl, . .. , sp) E Zp.

PROOF. The map cp VE;(Sl' ... 'Sp) --t VE(Sl, ... ,Sp) such that cp( VI, ... , Vm-1) = (VI, . .. , Vik_,' 0, ... , Vm -1) is an one-to-one mapping of the set VE;(Sl' ... 'Sp) onto the set of all elements v = (V1, ... ,Vm ) E VE(Sl, ... ,Sp) with zero ik th coordinate, i.e. onto the set VE (Sl, ... , sp). 0


2.2.5. THEOREM. Let E be a subset of Nm (m ~ 1). Then the following statements hold:

(1) there exists a numerical polynomialwE(t1, ... , tp) in p independent variables t1, ... , tp such that WE(Sl, ... , Sp) = Card VE(Sl, ... , sp) for all sufficiently large (S1, ... , Sp) E NP;

(2) if no element of the set E exceeds another element of E, then the set E is finite;

(3) the total degree degwE of the polynomial WE does not exceed m, degti WE ::; kj (j = 1, ... , p), and the equality deg WE = m is equivalent to E = 0; in the last case WE(t 1, ... , tp) = rr~=l Ci:jki), where kj = Card O"j (j = 1, ... , p);

(4) WE(t1, ... ,tp)~Oiff(0, ... ,0)EE.

PROOF. (1) It is clear that if F is the set of all minimal elements of E then VF = VE so we can (and we will) suppose that E is finite and its elements are pairwise incomparable. Let E = {c1, ... ,cr}, where Ci = (eil, ... ,eim) (i = 1, ... ,r) and let lEI = L:~=1 L:j=l eij· We shall prove the statement by induction on lEI. If lEI = 0, then either E = 0, or E consists of the only element (0, ... ,0). In the first case VE = Nm and Proposition 2.l.9 implies that

so the numerical polynomial WE(t1, ... , tp) = rr~=l Citkj) satisfies the desired

condition. In the second case (when (0, ... ,0) E E), VE = ° hence hVE (Sl, ... , Sp) = ° for all (Sl, ... , Sp) E NP, so that we can take WE(t 1, ... , tp) = 0. Thus, statement (1) is proved for lEI = 0. Furthermore, if m = 1, then E contains only one point e, so that WE(tt) ~ e is the desired polynomial.

Now, let lEI> ° and m > l. Then there exists an element e = (e1, ... , em) E E that is different from (0, ... ,0). Let ei. > 0, where ik E O"k (1 ::; k ::; p), and let r be an element of Mm whose ik th coordinate is 1 and all other coordinates equal zero. Applying relationship (2.2.2) to E and r, we obtain

for some H ~ Mm such that IHI < lEI. - 1-

By Lemma 2.2.4, hV(EUr) (Sl, ... , Sp) = hvi'; (Sl, ... , sp) where E C Mm- , lEI < lEI. Using inductive hypothesis we can suppose that there exist numerical polynomialswt(t1, ... ,tp) andw2(t1, ... ,tp) such that hV(Eur)(Sl, ... ,Sp) =Wt(Sl, ... ,Sp) and hVH(Sl' ... 'Sp) W2(Sl, ... ,Sp) for all sufficiently large (Sl, ... , sp) E MP. Therefore, the numerical polynomial

satisfies the requirement of the first statement of the theorem. (2) The second statement of the theorem is a direct consequence of Lemma

2.2.1(1).

2.2. DIMENSION POLYNOMIAL OF A MATRIX 57

(3) As we have already seen, if E = 0, then WE(t1, ... ,tp) 1l~=1 Cjtjkj ), degwE = m. Therefore, in order to prove the third statement of the theorem, it is sufficient to prove that degwE < m if E i= 0. Let ej (1 ~ j ~ m be the smallest value of the jth coordinate of elements of E, and let e = (e1, ... , em). Applying relationship (2.2.2) to E and e we obtain WE(S1, ... , sp) = WEue(Sl, ... , sp) +WH(S1 -eU1 , ... , sp - eup ). Since each element of E exceeds e, we have WEUe = We, whence WE(S1, ... , sp) ~ W(el, ... ,em )(S1, ... , sp) for all sufficiently large S1, ... , sp EN. Applying (2.2.2) once again, we obtain the inequality

WE(S1, ... ,Sp) ~W0"(S1, ... ,Sp)-W0"(s1-eUl,· .. ,sp-eup)

= IT (Sj :. kj ) _ IT (Sj + kj ~ ~iEUj ej )

1=1 1 1=1 1

for all sufficiently large (S1, ... , Sp) E NP, that implies the inequalities degwE < m and degtj wE(t1, ... ,tp) ~ kj , (j = 1, ... ,p).

(4) As we have seen above, WE == 0 if (0, ... ,0) E E. On the other hand, if wE == 0, then VE (S1, ... , sp) = 0 for all sufficiently large (S1, ... , sp) E NP whence VE = 0. 0

2.2.6. DEFINITION. The polynomial WE(t1, ... ,tp) whose existence is proved in Theorem 2.2.5 is called a dimension polynomial of the set E associated with partition (2.2.1) of the set Nm.

As a corollary of Theorem 2.2.5 we obtain the following result due to E. Kolchin.

2.2.7. THEOREM. For every set E ~ Nm (m ~ 1) the following statements hold:

(1) there exists a numerical polynomial WE(t) in one variable t such that WE(S) = Card VE(S) for all sufficiently large sEN;

(2) degwE :S m, and degwE = m iff E = 0 (in this case WE(t) = c~m»); (3) WE(t) == 0 iff (0, ... ,0) E E.

2.2.8. DEFINITION. The polynomial WE(t) whose existence is stated by Theorem 2.2.7 is called a Kolchin polynomial of the subset E ~ Nm.

2.2.9. REMARK. It follows from the arguments in the beginning of the proof of Theorem 2.2.5 that the dimension polynomial of a set E ~ Nm is equal to the dimension polynomial of the finite set F consisting of all minimal elements of E (a partition of Nm is supposed to be fixed). Therefore, in order to find dimension polynomial of a subset of Nm, it is sufficient to have a method for computing dimension polynomials of finite subsets F ~ Nm whose elements are mutually incomparable.

By this reason, in what follows we shall always work with finite sets E = {e1, ... , en} ~ Nm and write elements of such set as a matrix of dimension n x m, with the rows e1, ... , en. We shall denote this matrix by the same letter E. By a dimension polynomial of a n x m-matrix E we shall mean the dimension polynomial of the set of rows of E (that is considered as a subset of Nm). In the following theorem we formulate some properties of dimension polynomials of subsets of Nm that have already been proved as properties of dimension polynomials of matrices. All entries of matrices and vectors considered below belong to N.


2.2.10. THEOREM. Let E = (eij) be a n x m-matrix, e = (ell ... , em) a vector, and Nm = 0"1 U ... U O"p a partition of the set Nm. Then

(1) (2.2.3)

where E U e is the matrix that is obtained by adjoining the row e to the matrix E, H = (hij ) is nxm-matrix with the elements h ij = max(eij -ej, 0), i = 1, ... , n, j = 1, ... , m, and eOj = EkEoj ek;

(2) if n ~ 1, then

(2.2.4)

where r = (enl, ... , enm ), and H = (hij ) is (n - 1) x m-matrix such that hij = max(eij - enj, 0);

(3) any permutation of rows of E does not change the dimension polynomial of the matrix E;

(4) if i, j E O"k, and a matrix E is obtained from E by interchange of i th and jth columns, then WE = WE;

(5) if epj ~ eqj for j = 1, ... , m, then WE = WE" where matrix El is obtained from E by omitting the pth row (such row we shall call superfluous);

(6) WE(Sl, ... , sp) = 0 iff E has a zero row (in this case we have degwE = -1); (7) if E is non-empty, i.e. it has at least one row, then degwE < m; the

dimension polynomial of the "empty" matrix equals n~=l cjtkj), where

kj = CardO"j, (j = 1, ... ,p); (8) if E contains the row (1,0, ... ,0), then WE = wE" where El ~ Nm- l is

the matrix obtained from E by omitting firstly the rows in which the first coordinate is greater then 0, and then the first (zero) column. In particular, if E contains the row (1,0, ... ,0) and there is 0 somewhere in the first column, then degwE < m - 1 (we consider here the following partition of Nm- 1: Nm- 1 = U i# O"i U (O"k \ {I}), where k is the element ofNp such

i=l, ... ,p

that 1 E O"k);

(9) ifn> 1 and r= h, ... , rm), where rj = mini=l eij (1 ~ j ~ m), then

where H is a n x m-matrix obtained by subtracting (rl' ... ,rm) from every row of E (in particular, every column of H contains 0).

Let us fix n x m-matrix E with the rows {el, ... , en} and a partition Nm = 0"1 U· . ·U O"p. To compute WE(t 1 , ... , tp), we can apply relation (2.2.4) "at random" to the rows of matrix E; this procedure leads to the combinatorial formula (2.2.5) below that gives explicit expression for WE(t 1 , ... , tp).

To be more precise, let us introduce some notation. For any l, n E N such that n ~ 1, 0 ~ 1 ~ n, we shall denote by A(l, n) the set of alll-element subsets of the set Nn = {I, ... , n}, and for every ~ E A(l, n) we set E{ = {ejlj EO. Furthermore,


let ee denote a minimal element of the set Nm , which is greater than or equal to every element of the set E€ (as before, elements of Nm are compared relative to the product order" S;", if the contrary is not said explicitly). If e = 0, then eE = (0, ... ,0); ife t= 0 then, obviously, eE = (eO, .. ·,eEm), where eEi = maxjEdejd (i = 1, ... , m). Let lei = 2:hEUj eEh (j = 1, ... ,p).

2.2.11. PROPOSITION. With the preceding notation, the following relationship holds:

~ 1 '" rrP (t. + k· - h') WE(t l , ... ,tp)=L.,..(-I) L.,.. J t. J.

1=0 EEA(I,n)j=l J

(2.2.5)

PROOF. We proceed by induction on n. The case n = 0 follows from Theorem 2.2.10.

It n > 0, then by formula (2.2.4) we have

WE(t l , ... , tp) = WE, (tl,"" tp) - WH(tl - rl,"" tp - rp),

where El is a (n - 1) X m-matrix obtained from E by omitting the last row, rj = 2:kEUj enk, and H = (h;j)' where hij = max(e;j - enj, 0), i = 1, ... , n - 1, j = 1, ... ,m. By the inductive hypothesis,

and

where

Hence

WE,(tl, ... ,tp) = ~(-I)I E IT Cj+kr- hj ) 1=0 EEA(',n-l)j=l J

gej= E~axhik+ri= E (~axmax(eik-enk,O)+enk) .EE 'EE kEUj kEoj

= E rrE~x( eik, enk) = hUm,j . kEUj

WE(tl, ... , tp) = ~(-I)1 E IT Ci + kt.- IEj) 1=0 EEA(I,n-l)j=l J

+ i)-I)I E IT Cj + kj ~ IEUm,j)

1=0 EEA(I-l,n-l)J=l

= i) -1)' E IT Cj + kr-IEj). 0 1=0 €EA(I,n)j=l J

Proposition 2.2.11, in particular, implies, that the Kolchin polynomial of the set E can be represented in the form

(2.2.6)


2.2.12. EXAMPLE. Let m = 2, l~h = 0"1 U 0"2, where 0"1 = {I}, 0"2 = {2}, and E = {el = (1,2), e2 = (3, I)}. Then

V'E = {(0,0),(1,0),(2,0), ... ;(0, 1), (0,2), ... ;(1, 1),(2,1)},

and the direct computations show that Card V'E(Sl, S2) = Sl + S2 + 3. The same result we can obtain using relationship (2.2.5) where lel = le2 = °

for e = 0; lel = 1, le2 = 2 for e = {I}; lel = 3, le2 = 1 for e = {2}; and lel = 3, Ip = 2 for e = {I, 2}:

WE(tl, t2) = el 71) (t2 71) - [e l +: -1) e2 +: -2)]

+ Cl + : - 3) e2 + : - 1) + Cl + : - 3) e2 + : - 2) = tl + t2 + 3.

Let us consider now some properties of the coefficients of dimension polynomials.

2.2.13. DEFINITION. Let W(tl, ... , tp) be a numerical polynomial

with ah, ... ,ip(W) E ;?C;. (According to Corollary 2.1.5, every numerical polynomial in p indeterminates can be written in such form.) A coefficient ail, ... ,ip(W) is called a leading coefficient of w, if ail +jl, ... ,ip+jp (w) = ° for all nonzero vectors (il, ... , jp) E NP. Note that a leading coefficient can be equal to zero.

Let E be a nxm-matrix, Nm = O"lU" ·UO"p a partition ofthe set Nm , kj = Card O"j

(1 ~ j ~ p) and WE the associated dimension polynomial of E. By Theorem 2.2.5, akl, ... ,kp is a leading coefficient of WE, and if E f:. 0, then all akl , ... ,1'i- l , ... ,kp(WE) (i = 1, ... , p) are leading coefficients of WE.

2.2.14. LEMMA. Let W(tl, ... ,tp) be the sum of numerical polynomials Wl(tl, ... , t p) and W2(tl, ... , tp), let all leading coefficients ofpolynomialswl and W2 be non-negative, and let a coefficient ail, ... ,ip (w) be a leading one for w. Then the coefficient ail, ... ,ip(wI) is a leading one for Wl, ail, ... ,ip (W2) is a leading coefficient of W2, and ail, ... ,ip(W) = ah, ... ,ip(Wl) + ail, ... ,ip(W2)'

PROOF. Suppose that ah, ... ,ip(wd is not a leading coefficient for Wl. Then ail+rl, ... ,ip+rp (Wl) f:. ° for some nonzero vector (rl, ... , rp) E NP. Clearly, we may assume that coefficient ail +rl , ... ,ip+rp (wI) is a leading coefficient of Wl. The coefficient ail, ... ,ip(W) is a leading one, hence ail+rl, ... ,ip+rp (w) 0, that is ail+rl, ... ,ip+rp (Wl) + ail+rl, ... ,ip+rp (W2) = 0. Therefore ail+rl, ... ,ip+rp (W2) is a lead-ing coefficient of W2 (since ail +rl , ... ,ip+rp (wI) > ° and ail +rl , ... ,ip+rp (W2) ~ 0, i.e. their sum can not be equal to zero). 0


2.2.15. LEMMA. Let ai" ... ,ip(W) be a leading coefficient OfW(tl, ... ,tp). Then ai" ... ,ip(Wt} is a leading coefficient of

Wl(tl, ... , tp) = w(it, ... , ti-l, ti + 1, ti+l, ... , tp) (1:S i :s p)

and ai" ... ,ip (w) = ai, , ... ,ip (WI).

PROOF. Let

By formula (2.l.5) we have e+~+l) = ejj ) + E{~~ (ltk), hence

Wl(tl, ... ,tp)= L aj" ... ,jp(w)CI-:it)···Cp-:jp) (j" .. ,jp)ENP Jl Jp

+ L L aj" ... ,j'_l,k,j,+" .. ,jp(w) CI -: it) ... Cp -: jp). (j" ... ,jp)ENpk<j, Jl Jp

Since aj" ... ,jp(w) = 0 for all UI, ... ,jp) > (i l , ... , ip), then aj".,jp(Wl) = 0 for all UI, ... , jp) > (iI, ... , ip) whence ai".,ip (w) = ail, .. ,ip (wt). 0

2.2.16. LEMMA. Let E ~ Nm, Nm = Uf=lO"i be a partition ofNm, W(tl, . .. , tp) the dimension polynomial of E associated with this partition and ai" ... ,ip(W) a leading coefficient ofw. Then ai" .. ,ip(W) 2: o.

PROOF. We shall prove this statement by induction on lEI = Ei,j ei,j' If lEI = 0, then either w(t l , ... , t p) = 0, or w(t l , ... , tp) = n~=l ejtkj ) hence each leading

coefficient is either zero or one. Let lEI> 0 and let ail, ... ,ip(W) be a negative leading coefficient. Since the matrix E contains a nonzero column, we ca apply formula (2.2.3) to E and (0 ... 010 ... 0) and get W(tl, ... ,tp) = WEl(tl, ... ,tp) + WH(tl, ... , tj -1, ... , tp), where lEd < lEI and IHI < lEI· By Lemma 2.2.15, every leading coefficient of the polynomial W H (t 1, ... , t j - 1, ... , tp ) is a leading coefficient of W H (tl , ... , tp), and by inductive hypothesis all leading coefficients of polynomials WE and WH are non-negative. It remains to apply Lemma 2.2.14 to complete the proof. 0

2.2.17. THEOREM. Let E = (eij) be a nxm-matrix, Nm = O"IU· . ·UO"p a partition ofNm , withCardO"i = ki' (1:S i:S p), andak,_j" ... ,kp-jp aleadingcoefIicientofthe dimension polynomialwE(t1 , ... , tp) associated with this partition. Let us represent every element c E AU;, ki ) (1 :s i :s p) as a subset c = (.AI, ... , .Aj') of a suitable set O"i, (1 :s i :s p), and let E. denote the matrix consisting of .AI, ... , .Aj, th columns of E. Then the following statements are valid.

(1) For any choice of combinations Ci E AUi, ki ) (i = 1, ... ,p) the dimension polynomial of the n x Ul + ... + jp)-matrix E.,E. 2 ••• E. p is constant (E" ... E. p is the matrix consisting of the column-matrices E." i = 1, ... , p).

(2) ak,-j" .. ,kp-jp(WE) = WE., ... E.p· " EA(j, ,kl), ... ,' pEA(jp,kp)


PROOF. We shall prove both statements by induction on lEI = 2:7=1 2:;:1 eij.

If lEI = 0, then either E = 0 or wE = 0, so that both statements are evident. The case m = 1 is also evident. Now let lEI> 0, m > 1, and suppose that ith column of E (for some i E U r, 1 :'S r :'S p) contains a nonzero element. Let us apply formula (2.2.3) and the eighth statement of Theorem 2.2.10 to E and to the vector whose ith coordinate equals 1 and all other coordinates equal zero. Then

where E S; Nm - 1 and W E(t 1 , ... , tp ) is the dimension polynomial of E associated with the partition U1 U··· U Ur -1 U ifr U Ur+l U··· U up of Nm - 1 (ifr = Ur \ {i}). Of course, lEI < lEI and IHI < lEI. By Lemmas 2.2.14-2.2.16, ak.-j" ... ,kp-jp(wE )

and ak.-j., ... ,kp-jp(WH) are leading coefficients of wE and WH, respectively, and

where kr = kr - 1, )r = jr - l. Applying the inductive hypothesis to matrices E and H we obtain

evEA(jv,kv) /.Itr

frEA(jr,kr-1)

wE ••.. Eir ... E. p + L WH ••.. H. p ek~ak

L wE ••.. Eir .. E. p + L WH ••.. Hir···H. p evEA(jv,kv)

/.Itr frEA(jr,kr-1)

+ evEA(jv,kv)

/.Itr frEA(jr,kr-1)

where Hi is the ith column of H. Note that if 10k S; Uk (k = 1, ... , p) and i r/:. lOr, then Ee., ... ,e p = He., ... ,e p (in particular, WE ••.....• p is constant). Let 10k S; Uk (k = 1, ... ,p, k f. r), and 6r S; ifr · Applying formula (2.2.3) to the matrix Ee • ... E fr ... EepEi

and the vector (0, ... ,0,1) we obtain

(we see that it is a constant) and

ak1-ill ... ,kp-jp(WE) = L wE611 ... ,Eir,,,·,E6p,Ei + L WEW1)···,Ecp

ekCak ekCak k~r if'r ir~or

= L WE •• , ... ,E.p ·

e:k~O'k

This completes the proof. 0


2.2.18. COROLLARY. With the conditions of Theorem 2.2.17, the leading coefficient ak,-il, ... ,kp-jp is equal to zero iff for any E1 E A (it , kt}, ... , Ep E A(jp, kp), there is a rowe of E such that all its coordinates that are "marked" by the combinations Ei (i = 1, ... ,p) equal O. (As before, we assume Ei S; (J'i and Card Ei = ji for i = 1, ... ,p. The last condition of the corollary means that if e = (e1' ... , em), then ek = 0 for all k E Ei, i = 1, ... , p.)

2.2.19. DEFINITION. The total degree of a numerical polynomial W(t1"'" tp) in the indeterminates t1, ... , tp, is called a degree of w(t1, ... , t p) and is denoted by degw. Furthermore, the sum Li,+ .. +ip =>' ai" .. ,jp(w) is called a total coefficient of w associated with the given A and is denoted by a>.(w). If A = degw, then a>. = a>.(w) is called a leading total coefficient of w.

2.2.20. COROLLARY. Let E = (eij) be a non-empty nxm-matrix, Nm = Uf=l (J'j a partition ofNm , WE(t1, ... , tp) the corresponding dimension polynomial of E, and T = degwE. Then

(1) T < m; (2) ifmini=l eij = 0 for j = 1, ... , m, then T < m - 1; (3) am-I(WE) = L~l mini=l eij, and ak" ... ,kr-1, ... ,kp = LjEu r mini=l eij,

(1::; r ::; p); (4) deg WE < m - k, where 0 < k ::; m, iff for any natural numbers i1 , ... , ik

such that 1 ::; i 1 < ... < ik ::; m, there exists a row of E whose i 1 , ... , ik th coordinates equal zero;

(5) let degwE ::; m - k, where 0 < k ::; m. Then am-k(wE) = LEEA(k,m)WE~' where EE (e = (it, .. ·,jk) E A(k,m)) is the matrix consisting of j1, ... , jk th columns of E.

PROOF. (1) The inequality T < m follows from Theorem 2.2.5. (2) Since E is non-empty, ak" ... ,ki-1, ... ,kp is a leading coefficient of WE (t1, ... , tp).

By Theorem 2.2.17, it is equal to the sum of minimal elements of columns with indices from (J'j. Since mini=l ejj = 0 for all j, we have ak" ... ,ki-1, ... ,kp = 0 for all i, hence deg WE < k1 + ... + kp - 1 = m - 1.

m n

= "" rpin eij. ~'=1 j=l

p p

Lak" .. ,ki-1, ... ,kp(WE) = L L WEj i=l i=l jEui

p n

"" "" min e .. ~ ~ i=l 'J i=l jEUi

(4) Assuming deg WE < m - k, we have that if i1 + ... + ip ~ m - k, then ai, , ... ,ip is a leading coefficient of WE and it is equal to O. In this case, by Corollary 2.2.18, for any sets E1 E A(k1 - i1 , k1), ... , Ep E A(kp - ip, kp) there exists a row of E which has zero ith coordinate for each i E U~=l E". Let 1 ::; it < ... < jk ::; m, and let Er = {jtljl E (J'r, I = 1, ... , k}, (r = 1, ... , p). It is evident that 0 ::; Card Er ::; kr whence ir = kr - CardEr E N. Thus, Er E A(kr - ir,kr) (r = 1, .. . ,p) and i1 + ... + ip = k1 + ... + kp - Card(E1 U··· U Ep) = m - k, so that there exists a row of E whose it, ... ,ik th coordinates equal O.

Conversely, if degwE ~ m - k, then aj" ... ,ip(WE) ::f. 0 for a family of indices i1 , ... , ip such that i 1 + ... + ip ~ m - k. By Corollary 2.2.18, there exist elements


101 E A(k1 - i 1, k1)' ... , lOp E A(kp - ip, kp) such that for any i E Ut=lE"" E contains no row with zero ith coordinate. If we set (= 101 U·· ,UEp, then Card ( = E~=l kj -

Ej=l ij :S k, hence there exist indices 1 :S i1 < i2 < ... < i k :S m such that the matrix E contains no row in which i1, ... , ik th coordinates are equal to zero.

(5) Since am-k(wE) = Ei,+".+ip=m-k ai"".,ip(WE) and degwE :S m - k, all coefficients in this sum are leading ones. By Theorem 2.2.17,

am-k(WE) = L ak,-j" .. ,kp-jp(WE) j'+"+jp=k

L L WE., ,.",E.p = L WEE' j,+,,·+jp=k e,EA(j"k,) {EA(k,m)

<pEA'(jp,kp)

where (= 101 U·· ·UEp. 0

In the case p = 1, i.e. for Kolchin dimension polynomials, Corollary 2.2.20 can be also received from results of [Si75] (see also [CF87]). Note, that the degree and the total leading coefficient of dimension polynomial of a n x m-matrix do not depend on the partition of Nm .

2.2.21. EXAMPLE. Let us calculate the degree and the total leading coefficient of the dimension polynomial of the m x 2m-matrix

o 1

o

o 1 o 0

1 0

o 1

o

It is easily seen that if 1 :S i1 < i2 < ... < im - 1 :S 2m, then E contains a row whose i1 , ... , im - 1 th coordinates are equal to zero. Therefore (see Corollary 2.2.20(4)), degwE :S m. We compute the coefficient am(wE) by the formula that was obtained in Corollary 2.2.20(5). Nonzero summands correspond to the combinations ( of the form {I + mAl, 2 + mA2,"" m + mAm }, where Ai E {O, I} for i = 1, ... , m. The number of such combinations is equal to 2m and WEE = 1. Thus, am(wE) = 2m. 0

2.3. Algorithms for Computation of the Dimension Polynomials

In the previous section we noted (see Remark 2.2.9) that with the fixed partition (2.2.1) of Nm (m E N, m 2:: 1), the dimension polynomial of any set F ~ Nm is equal to the dimension polynomial associated with the set of all minimal elements of F. Therefore, in order to be able to calculate the dimension polynomial of any subset of Nm , it suffices to be able to calculate such a polynomial for every finite set F ~ Nm (moreover, one may suppose that elements of F are pairwise incomparable with respect to the product order).

Let E = (eij) l<i<n be a n x m-matrix over N, that is a matrix with n rows l<'j<m

and m columns whose entries are non-negative integers. Regarding the rows of E

2.3. ALGORITHMS FOR COMPUTATION OF THE DIMENSION POLYNOMIALS 65

as elements ofI~m and denoting the ith row (en, . .. , eim) by ei (1 ::; i ::; n) we receive a subset E = {e1' ... , en} ~ Nm associated with E. Recall, that with fixed partition (2.2.1) ofNm, the dimension polynomial of the nxm-matrix E is precisely the dimension polynomial of the set E. If in partition (2.2.1) we have p = 1, then the corresponding dimension polynomial of E is called a Kolchin polynomial of E.

Let WE(t1, ... , t p ) be the dimension polynomial of a n x m-matrix E over N. By Definition 2.2.6, we have WE(81, ... ,8p ) = CardVE(81, ... ,8p ) for all sufficiently large (81, ... , 8 p ) E NP, where VE denotes the set of all elements of Nm which are not greater than or equal to any element of E with respect to the product order, so that v E VE iff the inequality ej ::; v does not hold for every ei (1 ::; i ::; n). (Recall, that if A ~ Nm and partition (2.2.1) is fixed, then A(81, ... , 8p ) denotes the set {(a1' ... ,am) E AI LiEu, aj ::; 81, ... , LiEUp ai ::; 8p }).

It has been already noted, that in order to calculate the dimension polynomial of any subset of Nm, it suffices to know how to calculate the dimension polynomial of any n x m-matrix E over N. One of the methods of such calculation is based on formula (2.2.5).

Let E = (eij) 1<i<n be a nxm-matrix over N. We will use the following notation 1~j<m

that was introduceain Section 2.2:

for every subset ~ ~ Nn, and fcj = LhEUj ejh (1 ::; j ::; p), where 0"1, ... , O"p are the elements of the fixed partition (2.2.1) of Nm . By Proposition 2.2.6 (see formula (2.2.5)) we can write

P( ) n P( ) t·+k· t·+ko-fco wE(t1, ... ,tp )=II J k. J +~)-I)1 L II J { J,

j=l J 1=1 {EA(I,n) j=l J

where kj = Card O"j (1::; j ::; p), and A(l, n) (1 ::; I::; n) denotes the family of all I element subsets ofNn = {I, ... , n}.

Using the last formula, we can suggest the following Algorithm 2.3.1 for computing the dimension polynomial WE(t1, ... , tp ) associated with the given nxm-matrix E and with the fixed partition (2.2.1) ofNm such that 0"1 = {ill, ... , i 1k , }, 0 •• , O"p = {ip1' ... , ipkp }.


2.3.1. Algorithm (E,n,m,p,ul, ... ,up,w) Input: n EN; mE N;p EN; Ul, .. " up is a partition ofNm ;

E is a n x m-matrix. Output: w = WE(tl, ... , tp ) is the dimension polynomial of E associated with

the given partition Variables: IB is a vector of the type true/false with indices 1, ... , n.

Begin

V is a vector of the type N with indices 1, ... , m. S is a variable ofthe type ± 1 F is a vector of the type N with indices 1, ... ,p.

do for /I = 1, ... , P kv := Carduv

WE(tl, ... ,tp ) :=0. do for every vector I B

V:= (0, ... ,0) S:= 1 do for j = 1, ... , n

if IBU) then

End

do for /I = 1, ... ,p

lti :=max(lti,eji), i= l, ... ,m S:=-S

Fv := LiEu" lti W(tl' ... , tp ) := W(tl' ... , tp ) + S . I1~=l e,,+~:-F,,)

2.3.2. Algorithm (E, n, m,w) Input: n EN; mE N;p E N; E is nxm-matrix. Output: WE(t) is the Kolchin dimension polynomial of E Variables: IBis a vector of the type true/false with indices 1, ... , n.

Begin

V is a vector of the type N with indices 1, ... , m. Sl is a variable of the type ±1 S2 is a variable of the type N.

w(t) := e'!;.m) do for every vector I B

V := (0, ... ,0) Sl := 1 do for j = 1, ... , n

if I B(j) then

End

lti := max(lti, eji), i = 1, ... , m Sl := -Sl

S2 := Vl + ... + Vm w(t) := w(t) + Sl (t+r:- S2)


Using (2.2.6), we obtain a modification of Algorithm 2.3.1 for the computation of the Kolchin dimension polynomial of a matrix E (see Algorithm 2.3.2).

It is easy to see that the asymptotic complexity of Algorithms 2.3.1 and 2.3.2 has the order n x 2n where n is the number of rows of the matrix E (by Theorem 2.2.10(5), we may assume that these rows are pairwise incomparable with respect to the product order on Nm).

Another algorithm for the computation of Kolchin dimension polynomials has been suggested by F.Mora and H.M.Moller [MM83]. Their algorithm is based on the following considerations. Clearly, in (2.2.6) we can have the equality e~ = ee for two different subsets ~ and e of Nm such that Card ~ and Card e are even and odd, respectively (we use the notation of Proposition 2.2.11). Then the suitable summands in (2.2.6) can be canceled and we can group all terms of the sum that correspond to the same element r E Nm.

Let T = T(E) be the set of all elements r E Nm which are equal to at least one of the elements e( where ~ ~ Nn . Then formula (2.2.5) implies that

WE(t) = Lf)-l)k L C+:-lrl)=LJL,.C+:-lrl), rET k=O {~EA(k,n): e(=r} "ET

(2.3.1) where Irl is the sum of all coordinates of r, and

n

JL,. = L (2.3.2)

Let 11 = (111, ... ,JIm), .. . ,fq = (lq1, ... , !qm) be elements of Nm . Then the element! = (It, ... ,fm) E Nm where fj = max1<i<q{fij} (1 ~ j ~ m) is called the least common multiple of 11, ... , lq and is den;;t~d by lcm{/t, ... , lq}.

Obviously, if the matrix E1 is obtained by attaching the rowe = (e1' ... , em) to the matrix E, T1 = T(Et} , and V.l~lr E Ttl is the set of coefficients (2.3.2) in the relationship (2.3.1) for the polynomial WE, (t), so that

for every r E T1 , then

I { JLr - L{uET: lcm(u,e)=,.} JLu JLr =

- L{UET: lcm(u,e)=,.} J.lu

if rET

if r E T1 \ T (2.3.3)

Thus, the calculation of the polynomial WE(t), i.e. the calculation of the coefficients J.l,. (r E T) in (2.3.1)' can be based on (2.3.3)' if we start from the empty matrix (whose number of rows is equal to zero and whose Kolchin dimension polynomial is equal to e;;,m)) and consecutively attach the rows of E finding the set T and the coefficients JLr (r E T) on each step (see Algorithm 2.3.3).


2.3.3. Algorithm (E,n,m,w) Input: n E N; mEN; E is n xm-matrix. Output: w is the Kolchin dimension polynomial of E Variables: T, Tl are sets of the type {vector of the type N with indices

1, ... ,m};

Begin w:=O

f-t, f-tl are vectors of the type Z with the indices from T, T1 .

T:= {(O, ... ,O)} f-t(0, ... , 0) := 1 do for i = 1, ... , n

Tl :=T do for every u E Tl

f-tl(U) := f-t(u) do for every u E Tl

r := lcm( u, e;), ei is the ith row of the matrix E ifr E T, then f-t(r) := f-t(r) - f-tl(U) else T := T U r; f-t(r) := -f-tl(r)

do for every u E T w(t) := w(t) + f-t(u)(t+m,;ltil)

End

Since at each kth step (1 ~ k ~ n) of the algorithm every element U E Tl is the least common multiple of some subset of {el' ... , ek _ d (i.e. U = ee for some ~ <; Nk-l), there are at most (k - 1) different possibilities for choice of each coordinate of u, hence at kth step (1 ~ k ~ n) the set Tl contains at most (k - l)m elements. The calculation of all elements r = lcm(u, ek) requires fewer than or equal to m(k - l)m comparisons, and we may assume (under a sufficiently economical method of sorting) that the number of tests for the inclusions rET does not exceed km log k for all sufficiently large kEN. Thus, the asymptotic complexity (with respect to n) of Algorithm 2.3.3 does not exceed m L~=2 [(k - l)m + km logk). Since m L~=2 [(k - 1)m + km log k] < 2m L~=2 km log k, the asymptotic complexity has the order nm +l log n.

By analogy with Algorithm 2.3.3, we may consider the algorithm of computing the dimension polynomial WE (tl' ... , tp ) of a matrix E associated with an arbitrary partition (2.2.1) ofNm where 0"1 = {ill, ... , ilk,}' ... ' O"p = {ipl , ... , i pkp }. In order to construct such an algorithm, let us write the relation (2.2.5) as

(2.3.4)

where the set T = T(E) has the same sense as above, the elements f-tr are given by (2.3.2), kj = CardO"j, and rOj are elements that were defined in Theorem 2.2.10(1). If matrix El is obtained from the matrix E by attaching of a rowe = (el, ... , em) (for definiteness, assume that e is (n + l)th row of Ed, Tl = T(Ed and {f-t~: r E T1 } is the set of coefficients in the relationship of the form (2.3.4) written for the


2.3.4. Algorithm (E, n, m,p, 0'1, ... , up,w) Input: n EN; m E N;p E N; 0'1, ... , Up is a partition of Nm ;

E is n x m-matrix. Output: w = WE(t 1, . .. , tp ) is the dimension polynomial of E associated with the

given partition Variables: T, T1 are sets of the type {vector of the type N with indices

1, ... , m}.

Begin

/1, /11 are vectors of the type Z with the indices from T, T1 .

F is a vector of the type N with indices 1, ... , p.

do for !I = 1, ... ,p

k" := Carduv

w:=O T:= {(O, ... ,O)} /1(0, ... ,0) := 1 do for i = 1, ... , n

T1 :=T do for every U E T1

/11(U) :=/1(u) do for every U E T1

r := icm(u, ei), ei is the ith row of the matrix E if rET, then /1(r) := /1(r) - /1du) else T:= TU r; /1(r) := -/11(r)

do for every u E T do for !I = 1, ... , P

Fv := L:jEOv Uj w(t) := w(t) + /1(u) n~=l ev+~:-Fv)

End

matrix E 1 , then the relationship between /11" (r E Td and /11" (r E T) is given by (2.3.3). Thus, the dimension polynomial WE(t1, ... , tp ) of any n x m-matrix E may be computed by Algorithm 2.3.4, whose asymptotic complexity has the order nm +1 log n (as the same of Algorithm 2.3.3).

The following algorithms for computation of dimension polynomials of arbitrary nxm-matrix E reduce the problem to the computation of dimension polynomial of a matrix with a fewer number of rows than E. By one of these algorithms (see Algorithm 2.3.14 below) we can compute the coefficients /1T in (2.3.4) for the dimension polynomial WE (t1' ... , tp ) that give us the expression for the dimension polynomial. For the justification of the algorithm we need some properties of the coefficients /11", which are stated in Lemmas 2.3.5-2.3.9,2.3.11 and 2.3.13 below. The last of these lemmas establishes a relationship on which the algorithm of computation of /1T is based.

In order to indicate the dependence of the coefficients /1T on the matrix E let us denote these coefficients by /11" (E) and extend this notation to the case of arbitrary


vector T E Nm by setting

J.lT(E) = rT, {Ii ifTET

0, if T E Nm \ T.

(We recall that T = T(E) is the set of all elements T E Nm such that each T is equal either to (0, ... ,0) or to the least common multiple of some rows of E; elements of T will be called admissible elements or admissible vectors for E.)

2.3.5. LEMMA. Let E = (e;j) 1<;<n be a n x m-matrix and let an element l<j<m

T = (Tl, ... ,Tm ) E Nm majorize ail rows of this matrix, i.e. T is greater than or equal to each row of E (with respect to the product order on Nm). Then J.lT(E) = J.l(l, ... ,1)(H) where H = (h;j) l<i<n is the matrix with elements hij =

l~j~m

{ 1, if eij = Tj, (. ) . 1 = 1, .. . ,n .

0, If eij ::j:. Tj,

PROOF. Let e = {il"'" ik} E A(k, n) (1 ::; k ::; n), ee = lcm{e;lJ"" eik}' and he = lcm{hi1 , ... , hik} (ei and hi denote the ith rows of the matrices E and H, respectively). Let us show that the equality ee = T is equivalent to the equality he = (1, ... ,1). Indeed, if ee = T, then Tj = max{e;.j, ... , eiki} (1 ::; j ::; m), so that for each j = 1, ... , m there exists an index A(j) E Nk such that hi>.(j) = 1. Thus, the jth element ofthe row hi>.(j) is equal to 1, hence he = lcm{hilJ ···, h ik } = (1, ... ,1). Conversely, if he = (1, ... ,1), then for each j = 1, ... , m there exists a number v = v(j) E Nk such that hi"j = maXiEdhij} = 1, i.e. ei"j = Tj. Therefore, ee 2: T whence ee = T (since the element T is greater than or equal to each row of E). Thus,

n n

J.lT = E E (_I)k = E E (_I)k = J.l(1, ... ,l)(H). 0 k=O {eEA(k,n): eE=T} k=O HEA(k,n): hE=(l, ... ,l)}

Now let us consider some properties of dimension polynomials of matrices, whose elements are equal either to 0 or to 1 (the matrix H in the condition of Lemma 2.3.5 is a matrix of such type). For shortness, we shall write 1'1 (E) instead of 1'(1, ... ,1) (E), where E is a n x m-matrix and (1, ... ,1) E Nm .

2.3.6. LEMMA. Let E be a nxm-matrix whose elements are equal either to 0 or to 1. Let WE(tl, ... , tp ) be the dimension polynomial of E associated with partition (2.2.1) ofNm • Then J.l1(E) = (-I)mWE(-I, ... , -1).

PROOF. By (2.3.4) we have

and, obviously, each coordinate of any vector T = (Tl, ... , T m) E T = T( E) is equal either to 0 or to 1. If T ::j:. (1, ... ,1), then 0 ::; ITI = L~1 Ti < m, therefore there exists a number v E Np such that Taj = LhEa" Th < k". In this case the polynomial


C+k~~Taj) vanishes at t = -1, hence the polynomial rr~=l Cj+k:j-Ta j ) vanishes at

tl = -1, ... , tp = -1. Thus,

p (t + k· - k) wE(-I, ... ,-I)=J.ldE )II J :. J

j=l J (t" ... ,tp )=( -1, ... ,-1)

= (E) IIP tj(tj - 1) ... (tj - kj + 1) III k. 1

j=l J. (t" .. ,tp)=(-l, ... ,-l) P

= 1l1(E) II(-I)k j = (-I)mlldE). 0 j=l

It follows from Lemma 2.3.6 that

(2.3.5)

where WE(t) is the Kolchin dimension polynomial of E. If WE(t) = 2:7:::0 a; (tti) (ao, a1, ... ,am E ::2::) then WE( -1) = ao, so that III (E) is equal to the free term of the Kolchin polynomial WE(t).

2.3.7. LEMMA. Let E be a nxm-matrix, each element of which is equal either to 0 or to 1.Then:

(1) if E has a zero row, then 1l1(E) = 0; (2) if E has a zero column, then IldE) = 0; (3) the value J.ll(E) is invariant with respect to permutations of rows (or

columns) of E; (4) if E consists of the single row (1, ... , 1), then III (E) = -1; (5) assume that the first row of E is (1,0, ... ,0) and the first elements of the

other rows of E are equal to O. Then Ill(E) = -lldH), where the matrix H is obtained by omitting the first row and the first column of E.

PROOF. All assertions of the lemma follow from (2.3.5) and from the properties of dimension polynomials proved above (these properties are considered with respect to the Kolchin dimension polynomial of a matrix E).

(1) If E has a zero row, then WE(t) == 0 (see Theorem 2.2.10(6)). Applying Lemma 2.3.6, we obtain that Ilt{E) = O.

(2) If each element of vth column of E is equal to zero (1 ~ v ~ m), then formula (2.3.2) implies that III (E) = 0 (indeed, with the notation of (2.3.2), the vth coordinate of any vector e{ (~ ~ N"m) is equal to zero, so that e{ * (1, ... ,1) for every subset ~ ~ N"m}.

(3) Obviously, a permutation of rows (or columns) of E does not change the value WE(t) and, therefore, the value Ill(E) = (-I)mWE(-I) (see statements (3) and (4) of Theorem 2.2.10).


(4) Let E consist of the single rowe = (1, ... ,1). Since Ve(s) = CardNm(s) -Card{(l + U1, ... , 1 + um) l(u1, ... , um) E Nm(s - m)} = (';;.,m) - (~) for all sufficiently large sEN, we have

WE(t) = (t + m) _ (t ) = (t + 1) ... (t + m) _ t(t - 1) ... (t - m + 1) . m m m! m!

Therefore, J-l1(E) = (-l)mwE(-l) = (_1)m(_1)m+1 =-l. (5) By Theorem 2.2.10(8), WE(t) == WH(t), hence, J-l1(E) = (-l)mwE(O) =

_(_1)m-1wH(0) = -J-l1(H). 0

Let E = (eij) 1<i<n be a n x m-matrix over N, jj; = {e1, ... , en} be the set of l<j<m

all rows of E and- e-= (e1' ... ' em) be an element of Nm. Let E U e denote the (n+ 1) x m-matrix obtained by attaching of the rowe to the matrix E (without loss of generality, we may assume that e is the (n + l)th row of E U e). The following lemma establishes the connection between the dimension polynomials of matrices E and E U e (for the same partition (2.2.1) of Nm). As above (see the proof of Theorem 2.2.5), lEI denotes the sum L7=1 L7=1 eij of all elements of E and IEI", denotes the sum L7=1 LjE", eij, 1 :s I :s p (in particular, lei denotes the sum of all coordinates of an element e E Nm and lela/ = LjE", ej for every component (T/

(1 :s I :s p) of the partition (2.2.1))

2.3.8. LEMMA. Let E be a n x m-matrix (n, mEN; m > 1, n ~ 1), whose elements are equal either to 0 or to 1. 1f the first column of E consists only of zeros and the matrix E1 is obtained from E by omitting of this zero column, then wE(-2) = -wE,(-I).

PROOF. By formula (2.2.3) written for E and e = (1,0, ... ,0) in case p = 1, we have WE(t) = WEue(t) + WH(t - 1), where H = (h ij ) l<i<n is a matrix with

l~jSm the entries hij = max{ eij - ej, O} (e1 :::: 1, e2 = 0, ... , em :::: 0 are the coordinates of e). Obviously, H = E, and WEue(t) = WE,(t) (see Theorem 2.2.10(8)), so that WE(t) :::: WE, (t) + WE(t -1) and, in particular, WE, (-1) +WE( -2) = WE( -1). Since E contains a zero column, Lemma 2.3.7(2) implies that WE( -1) = (_l)m J-ldE) = 0, hence wE(-2) = -wE,(-l). 0

2.3.9. LEMMA. Let E = (eij) l<i<n (m, n E N, m > 1) be a n x m-matrix 1<j<m

whose entries are equal either to 0 or -to 1. Suppose that ej1 = 1 for j = 1, ... , r and eij = 0 for j = r+1, ... ,n (1:S r:S n). Then J-l1(E) = J.l1(EI}-J.l1(E2), where the matrix E1 is obtained by omitting the first column of E, and E2 is obtained by omitting the first r rows of E 1 .

PROOF. Applying (2.2.3) to E and e = (1,0, ... ,0) in case p = 1 (so that the dimension polynomial is the Kolchin polynomial in one variable t), we get WE(t) = WEue(t) + WH(t - 1), where H :::: (hij) l<i<n is n x m-matrix with the

l<j<m . { 0, if j = 1, - -

entnes hij :::: max{e;j - ej, O} :::: .. By Theorem 2.2.10(8), we have eij, 1f J =j:. l.

WEue(t) :::: WE2(t), whence WE(t) = WE2(t) +WH(t -1). Furthermore, using Lemma


2.3.8, we may write wH(-2) = -WE. (-1), hence wE(-I) = wE,(-I) - WE, (-1). By Lemma 2.3.6, we obtain now that

o 2.3.10. COROLLARY. Let E = (eij) l<i<n (m, n E~, m> 1) be a nxm-matrix

l<j<m whose entries are equal either to 0 or to -1.-Assume that there exist p, q E ~m such that eip 2: eiq for all i = 1, ... , n. Then J-ll (E) = J-ll (El ), where El is obtained by omitting the pth column of E.

PROOF. Since the Kolchin dimension polynomial of E is invariant under each permutation ofrows (or columns) of E, the value J-ll(E) = (-I)mwE(-I) has the same property. Therefore, without loss of generality, we may suppose that p = 1 and there exists l' E ~n-l such that ejl = 1 for j = 1, ... ,1', and ejl = 0 for j = l' + 1, ... , n. By Lemma 2.3.9, we have J-ldE) = J-ll(El ) - J-ll(E2), where E2 is obtained by omitting the first column and the first l' rows of E (since 0 ::; eiq ::;

eil = 0 for i = l' + 1, ... , n, each element of qth column of E2 is equal to zero). Hence, J-ldE2) = 0 (see Lemma 2.3.7(2)), so that J-ll(E) = J-ldEd. 0

2.3.1l. LEMMA. Let E = (eij)l<i<n (m,n E~, m> 1) be a nxm-matrix l<j<m

whose entries are equal either to 0 or to 1. Assume that E has a row e = (el' ... , em) such that ej = 1 for 1 ::; i ::; l' and ei = 0 for l' < i ::; m (1' E ~m-l). Then jJl(E) = jJl(E \ e) - jJl(Ed, where the matrix E \ e is obtained by omitting the rowe of E, and (n - 1) x (m - 1')-matrix El is obtained by omitting the first l'

columns of E \ e.

PROOF. Applying formula (2.2.4) to E and e = (1,0, ... ,0) (in case p = 1), we get WE(t) = WE\e(t) - wE, (t - 1'), where the matrix El is obtained from E by attaching l' zero columns from the left. Now, (2.2.3) shows that wE, (t) = wE,ue(t) + wE, (t - 1'), hence

Since El contains a zero column, Lemma 2.3.7(2) implies that jJl (Ed = 0, hence

(see Lemma 2.3.6) and

jJl(E) = (-I)mwE(-l) = (-I)mWE\e(-I) + (-I)mwE,ue(-I)

= jJl(E \ e) + jJl(El U e).

Since each of the first (1' - 1) columns of El U e majorizes the 1'th column of this matrix, Corollary 2.3.10 implies that jJdEl U e) = jJl(OEl U (1,0, ... ,0)), where OEl is the matrix obtained by attaching the zero column to El from the left. Now,


applying Lemma 2.3.7(5), we obtain PI (OE1 U (1,0, ... , 0)) = -PI (E1), that implies the desired relationship P1(E) = P1(E \ e) - pdEt}. 0

Let E = (eij) l<i<n be a n x m-matrix. By Theorem 2.2.10(5), the omitting a l<j<:m

superfluous row ofE-does not change the dimension polynomial of E and, therefore, does not change the value PI (E). Furthermore, if each entry of E is equal either to 0, or to 1, then Corollary 2.3.10 implies that the omitting superfluous columns of E does not change the value P1(E) (the pth column of E = (eij) l<i<n (1 ~ P ~ m)

l<i<m is called superfluous, if there exists a number q E Nm such that i=j p and eip 2: eiq for all i = 1, ... , n) .

Thus, in the process of calculation of PI (E) (where E is a n x m-matrix whose entries are equal either to 0 or to 1) we may first of all omit the superfluous rows and columns (according to Lemma 2.3.7(5), these cancellations are performed with the suitable change of sign of P1(E)) and then omit all rows and columns using the relationship of Lemma 2.3.9. After that we may choose one of the following two alternatives: to use Lemma 2.3.9 for the calculation of pdEt} (where E1 is the matrix obtained from E by the process of cancellations described above) or to calculate P1(Et} with the help of Lemma 2.3.11, i.e. to "decompose" E1 by rows or by columns, respectively. Obviously, if the number of rows of E1 is greater than the number of its columns, then the "moving by columns" with the help of Lemma 2.3.9 is preferable, otherwise it would be a good plan to use Lemma 2.3.11 for the calculation of P1(E1).

2.3.12. EXAMPLE. Find the value P1(E) for the matrix

(0011) o 1 0 1 01 1 0

E= 1001 . 1 0 1 0 1 1 0 0

( m)IY:n:~'~m(~ ~';)9' ~: ::::: t:.~, ~::t=t:~ (:~ ~::E~~~:~~~'a: 010 110 1 00

superfluous, therefore, pdEl) = PI ( G! D) = -1 (see Lemma 2.3.7(5)). Ap

plying Lemma 2.3.9 once again, we obtain (by virtue of the statements of Lemma

2.3.7) that pdE2) = PI (( ~ D) -PI ((1, 1)) = 1 + 1 = 2. Hence, pdE) =

pdE1 ) - pdE2 ) = -3. Another method of calculation of pdE) is based on Lemma 2.3.11: Pl(Et} =

", ( m m) -", (G m -1 = ", (G : : m -", (G m -2 =

PI (( ~ ~ ~ ~ )) - PI (( ~ ~)) - 2 = -3.


As a consequence of Lemma 2.3.11 we have the following statement, on which the algorithm for calculation of the dimension polynomial of a matrix may be based (see Algorithm 2.3.14 below).

2.3.13. LEMMA. Let E = (eij) l<i<n be a n x m-matrix over Nand T = l<j<m

(TI, ... ,Tm ) E T = T(E) (as before, r(£) denotes the set of all admissible vec-tors of E, i.e. the set of all elements T E Nm that are equal either to (0, ... ,0), or to the least common multiple of some rows of E). Let K be the matrix consisting of all those rows of E, which are majorized by the vector T (for definiteness we shall suppose that the rows of K are disposed in the same order as they are disposed in the matrix E). Furthermore, let k = (kl , ... , km ) be one of the rows of K and let K \ k be the matrix that is obtained by omitting the row k in K. Then, for any subset J = {i l , ... , it} ofNm with ii < ... < il (1 ~ I ~ m), we have

Ih(K, m) = J.Lr(I{ \ k, m) - J.Lr' (K \ k, J), (2.3.6)

where J.Lr(K, m), J.Lr(J{ \k, m) are the coefficients J.Lr that are defined by (2.3.2) for the matrices J{ and J{ \k, respectively, and J.Lr' (K \ k, J) are similar coefficients for the vector T' = (Ti l' ... , Ti,) E N1 (instead of T) and for the (n - 1) x (m -I)-matrix which is obtained from E by omitting the columns with the indices j E Nm \ J.

PROOF. Without loss of generality we may assume that k is the first row of K. It follows from Lemma 2.3.5 that J.Lr(K, m) = J.LI(H), where H = (hij) I<i<n is

I<j<m

{I, if eij = Tj - -

the nxm-matrix with the entries hij = . , (i = 1, ... , n). Similarly, 0, If eij :I Tj

J.Lr(I<\k, m) = J.LI(Ht} and J.Lr,(K\k, J) = J.LI(H2 ), where the matrix HI is obtained by omitting the first row of H, and H2 is obtained by omitting the columns with the indices iI, ... , i l in HI. Now, applying Lemma 2.3.11 we obtain the relationship J.LI(H) = J.LI(Ht} - J.Ll(H2) which implies the desired statement. 0

Under the fixed partition (2.2.1) of Nm , the calculation of coefficients J.Lr (T E T = T(E)) in (2.3.4) for the dimension polynomial of an arbitrary nxm-matrix E (and, therefore, the calculation of the dimension polynomial itself) may be carried out by the following scheme: first, we apply (2.3.6) to the matrix E (constructing the matrix K of those rows of E that are majorized by the vector T. Clearly, the coefficients J.Lr for K are the same as for E). After that we compute the values J.Lr (I< \ k, m) and J.Lr' (K \ k, J) by applying (2.3.6) again, and so on, until the "empty" matrices (i.e. the matrices whose numbers of rows are equal to zero) appear.

To determine the asymptotic complexity of Algorithm 2.3.14 for sufficiently large n EN, note, first, that if the vector T = (TI,"" Tm) E T is fixed, then the construction of J{ = J{(T) requires at most mn comparisons of numbers (at this step we store all pairs (k, j) E Nn x Nm , for which ekj = Tj). Furthermore, the execution of elementary operations for all calls of algorithm NEXTINDEX (for the fixed T) requires at most hIh2 ... hm comparisons, where h" = h,,(T) (1 ~ 1/ ~

m) denotes the number of rows k = (kI , ... , km) of J{(T) such that k" = T" for all 1/ = 1, ... , m. It is easy to see that the total number of operations for all calls of algorithm NEXTINDEX (up to a constant factor, this number is equal to


2.3.14. Algorithm (E, n, m, T, J.L) Input: mE f:I, n E f:I; E is n x m-matrix Output: T is the set of admissible vectors of E;

J.L is a vector of the type Z with indices in T. Variables: J is a set of the type 1, ... , m;

Begin

K is a set of the type {vector of the type f:I with indices in1, ... ,m}.

to form the set T of admissible vectors do for every T E T

End

J.Lr := 0 J:={l, ... ,m} K := {ei : ei ~ T} where ei is a row of E Vr := 1 NEXTINDEX (J, K, Vr , J.Lr)

Algorithm NEXTINDEX (J, K, Vr , J.Lr) Input: J is a set of the type 1, ... , m

K is a set of the type {vector of the type f:I with indices 1, ... , m}. Vr is an element of the type ± 1

Output: J.Lr is a vector of the type f:I with indices in T. Global variables: mE f:I;

T is vector of the type f:I with indices in 1, ... , m.

Begin do for every row k = {kl"'" k m } of K such that kil = Ti, where i 1 is the first

element of J

End

K := K \ k

J' := {j E J: kj = Tj}

J = J\ J' cases

K = 0&J = 0 => J.Lr := J.LT + Vr K =F 0&J =F 0 => NEXTINDEX(J,K,vr,J.Lr)

J:= JU J'

LrET h1(T) ... hm(T)) does not exceed Card{(al, .. " am) E f:lml for every 11 E f:l m there exists i = i(lI) E f:l n such that av = eiv} which is less than or equal to nm. Therefore for sufficiently large n E f:I the asymptotic complexity of Algorithm 2.3.14 has the order nm+l.

Another scheme of calculation of the dimension polynomial WE (t 1, ... , tp ) of n x m-matrix E (under the fixed partition (2.2.1) of f:lm) is the following one. For n < m one may calculate the polynomial we{t1 , . .. , tp ) using Algorithm 2.3.1. Let n 2: m. In this case we apply to E relation (2.2.4) in which e is the row with the maximal entry in the first nonzero column of E. (Trivial cases: if E = (0)


we have WE = 0; if n = 1, we apply Algorithm 2.3.1.) It is easy to see that the number of zero columns of H (see (2.2.4)) is greater than the one of E, and each of the matrices E \ e, K has less rows than E has. After that, we apply the procedure described to the matrix E \ e, and so on, until we receive a matrix, whose dimension polynomial can be calculated by Algorithm 2.3.1. At the end of the process we obtain the representation of the desired polynomial WE(t 1 , .. . , t p ) as a linear combination of the numerical polynomials We, WH 1 , .•• , WH,,_l (with shifted arguments) such that any matrix Hi has precisely i rows and the number of its zero columns is greater than the same of E by one. The polynomial We and some of the polynomials WHi are calculated by Algorithm 2.3.1 (in those cases in which, according to the above considerations, one has to apply this algorithm). In order to calculate the other polynomials WHi we apply relation (2.2.4) once again and continue in the same manner. Remark, that if the first column of E is a zero one and m > 1, then the number of operations in the calculation of WE by the suggested scheme coincides with the same for the calculation of the dimension polynomial of a n x (m - I)-matrix. Furthermore, if E is a n x I-matrix, then all its rows, except of the one, where the element mini<i<n {eid occurs, are superfluous, so that the calculation of the dimension polynon"ii~l by the formula WE = mini<i<n{eid requires (n - 1) operations. Thus, if f(n, m) denotes the number of e~~entary operations (such as addition, comparison or multiplication) that are required for the calculation of the dimension polynomial W d t 1, ... , tp ) of a n x m-matrix E, then f(n, m) :S (n - 1) + f(n - 1, m) + f(n - 1, m - 1). Since f(n, 1) = n - 1, we have

f(n,2) :S 2(n - 1) + f(n - 1,2) :S 2«n - 1) + (n - 2)) + f(n - 2,2) :S ...

n(n - 1) 2 :S2«n-l)+(n-2)+···+I)= 2 ·2:Sn;

f(n,3) :S n2 + (n - 1)2 + ... + 1 :S n3 ;

f(n, k) :S nk - 1 + (n - I)k-l + ... + 1 :S nk

etc. Therefore, the algorithm for computation of the dimension polynomial based on the above scheme (see Algorithm 2.3.15 below) has the asymptotic complexity O(nm).

The symbol":" that is used in Algorithm 2.3.15 denotes the following operation on vectors:

(al, ... ,am)":"(b1, ... ,bm) = (Cl, ... ,Cm ),

where Ci = max(a; - bi , 0) for all i = 1, ... , m. Furthermore, if m > 2, then ei (1 :S i :S m) denotes the ith coordinate of an element e E Nm.

In the final part of the section we shall consider the algorithm for computation of dimension polynomials, whose asymptotic complexity is less than one of Algorithms 2.3.1,2.3.4,2.3.14, and 2.3.15. We shall also give some algorithms for the computation of the leading coefficients of dimension polynomials.

Let Q[tl, ... , tp ] be the ring of polynomials in p indeterminates t1, ... , tp over the field of rational numbers. For every sEN, j E Np , let Ll. (j) and Ll- 1 (j) be


2.3.15. Algorithm (E,n,m,p,ul, ... ,up ,w) Input: n EN; mE N;p E N;Ul, ... , up is a partition ofNm ;

E is n x m-matrix. Output: W = WE(t l , ... , tp) is the dimension polynomial of E associated with the

given partition Variables: hEN

Begin

J is a set of the type {vector of the type N with indices 1, ... , h}; e is a vector of the type N with indices 1, ... , m; F is a matrix of the type N, its number of columns is equal to m and the number of rows does not exceed n.

do for II = 1, ... , p k" := Card u"

h:= n F:=E if h < m

then Algorithm 2.3.1. (E, n, m,p, Ul, ... , up,w) else J := {nonzero columns of F}

cases

End

J = 0 =>W:= 0 Card J = 1 => c:= minimal element of j E J

W := g C" :" k") -g C" + ~: - c" )

h { c, if j E u" were c" = 0 th .

, 0 erWlse else j := {the first element of J}

e := { the first row of F with the minimal element of jth column}

c" = LiE17v ei (II = 1, ... ,p); F:= F\e Algorithm 2.3.15 (F, h - 1, m, p, Ulo ... , up, w) v:=w F:= {f":"'e: f a row of F, f"l- e} Algorithm 2.3.15 (F,h-l,m,p,ul, ... ,Up ,w) W := V(tl' . .. , tp) - W(tl - Cl, ... , tp - cp)

operators that act on Q[tl' ... ,tp] as follows:

and if


(ai, ... ip E Q) for all (i1,"" ip) E NP), then

A -1( ')f( ) _ ~ .. (t1 + i1) (tj-1 + ij - 1) (tj + ij + 1) u J t1, ... , tp - L..J a. 1 ...• p • • • '. •

. . %1 %j-1 %j+l (.l ...... p)ENP

(2.3.7')

For p = 1 we shall omit the index j and write simply tJ.. and tJ. -1 instead of tJ..(I) and tJ. -1(1), respectively.

Remark, that operators tJ..(j) and tJ.-1(j) (8 E N,j E Np) satisfy the following identity:

tJ..(j)tJ. -1 (j)f(t1, ... , tp) =f(t1, ... , tp) + f(t1, ... , tj-1, tj - 1, tj+l, ... , tp) + ... + f(t1,"" tj-1, tj - 8 + 1, tj+1,"" tp) (2.3.8)

In particular,

Indeed, let

(8 EN, ail ... ip E Q for all (i1 ... ip) E NP, and ai1 ... ip = 0 for almost all (i1 .. , ip) E W). By (2.1.4) we have

.-1

= E f(t 1, ... , tj-1, tj - r, tj+1,"" tp). r=O

2.3.16. LEMMA. Let E = (ei;) 1<i<n be a nxm-matrix over N and a partition 1<j<:m

Nm = uf=lO'i of the set Nm be fixed-: Let k E 0'/ (1 ~ I ~ p), a = min{eik leik #- O},


and let E1 be the matrix obtained by omitting the kth column and all rows of E with nonzero kth elements. Furthermore, let H = (h ij )l<i<n be the nxm-matrix

l<j<m

{ max(eik - a, 0), if j = k - -

with the entries hij = " (1 ~ i ~ n, 1 ~ j ~ m). Then eij, lf J :/: k

WE(t1, ... , ip) = ~a(l)~ -l(l)WE, (i 1, ... , ip)

+ wH(i1, ... , il_1, il - a, il+1,"" ip) (2.3.9)

where wE(i 1, ... ,ip), wH(i1, ... ,ip) are the dimension polynomials of E and H, respectively, associated with the partition Nm = Uf=lO"i, and WE,(t 1, ... ,tp) is the dimension polynomial of E1 associated with the partition N m - 1 = 0"1 U ... U 0"1-1 U iTl U 0"1+1 U ... U 0" p, where iTl = 0"1 \ {k} (if 0"1 = {k}, then the latter partition of Nm - 1 consists of the p - 1 components 0"1, .. . ,0"1-1,0"1+1, ... ,O"p).

PROOF. Applying (2.2.3) to E and (0, ... ,0, a, 0, ... ,0) (a is the kth coordinate of this vector), we obtain

Now, let us apply (2.2.3) to E U (0, ... ,0, a, 0, ... ,0) and (0, ... ,0,1,0, ... ,0) (where 1 is on the kth place). By Theorem 2.2.10(8), we get

WEU(O, ... ,O,a,O, ... ,O) (i1' ... ,ip) = WE, (t1' ... , t p)

+ W EU(O, ... ,O,a-1,O, ... ,O)(t1, ... , il- 1, tl - 1, tl+ 1, ... , ip)

Repeating the same procedure a times, we obtain the equality

WEU(O, ... ,O,a,O, ... ,O)(t1,' .. , tp) =WE, (t1," ., ip) + WE, (t 1, . .. , tl- 1, il - 1, tl+ 1, . .. , ip)

+ ",+wE,(i 1, ... ,tl- 1,tl- (a-l),il+1, ... ,ip),

that implies (2.3.9) (see (2.3.8)). 0

Now we can suggest the following scheme of computation of the dimension polynomial WE(t1, ... , ip) of a matrix E = (eij) l<i<n based on the formula (2.3.9).

l<j<m First, we choose the vector (a, 0, ... ,0) E Nm , where a = min1 <i<n {ei1 : ei1 :/: O}, and apply Lemma 2.3.16, reducing our problem to the computation of the dimension polynomial of a matrix E1 with (m - 1) columns and of the dimension polynomial of a matrix H = (hij) l<i<n such that ° ~ hil < ei1 (1 ~ i ~ n). To

l<j<m determine WH(t 1, ... , ip) we apply [2.3.9) (with H instead of E) and continue the procedure until we obtain the representation of WE (t1' ... ,ip) as a sum of some dimension polynomials of matrices with (m - 1) columns and the dimension polynomial wH,(i1, ... ,ip), where H1 is a nxm-matrix with the zero first column. After the calculation of WH, (i1,"" tp) we apply the above procedure to the second column and so on.


2.3.17. EXAMPLE. Let us calculate the Kolchin polynomial WE (t) of the matrix

E=(~ ~ ~). First, we find a = 2. Then we apply (2.3.9) and obtain

WE(t) = ~2~ -lWEI (t) + WH(t - 2),

where El = (2,0) and H = (~ ~ ~). Clearly, WH(t)

2.2.10(6)) and WEI (t) = ('t2) - (t+;-2) , whence (see (2.3.8))

o (see Theorem

WE(t) = ~2~ -lWEI (t) = WEI (t) + WEI (t - 1)

= e~2) - G) + e~1) - e;1) =4t.

Note that the computation of the Kolchin polynomialwE(t) by one of Algorithms 2.3.2, 2.3.3, 2.3.14, or 2.3.15 leads to the representation of this polynomial in the form WE(t) = et3) - 2e+;-2) + (,+;-4). However, if we apply Lemma 2.3.16, then

the polynomial WE(t) is obtained as a sum of polynomials of the form ak ('ti) (i E Z, kEN, ak E Z). The maximal degree of these polynomials is less than the same one for the polynomials, which arise in the similar representation of WE(tl, when WE(t) is calculated by one of Algorithms 2.3.2, 2.3.3, 2.3.14, or 2.3.15.

A matrix E over N will be called normalized, if every column of E contains a zero. We shall see below that if E is a normalized nxm-matrix then the algorithm of computation of the dimension polynomial WE(tl, ... , tp ), which is based on Lemma 2.3.16, requires fewer operations than the same for arbitrary n x m-matrix. At the same time, one can use Theorem 2.2.10(9) to reduce the problem of computation of the dimension polynomial of an arbitrary nxm-matrix over N to the same problem for a normalized n x m-matrix.

2.3.18. LEMMA. Let E = (eij) l<i<n be a nxm-matrixoverN and let degwE = l<j~m

o (recall that by Corollary 2.2.20,-degwE does not depend on the choice of the partition ofNm ).

(1) If elj = 0 for j = 2, ... , m and eil = 0 for i = 2, ... , n then WE = eUwEl where (n - 1) x (m - 1 )-matrix El is obtained by omitting the first row and the first column of E.

(2) Let H be the matrix that is obtained from E by replacing its first column by the zero one. Then WH = O.

(3) lfa = minl9~n{eilleil -::j:. O}, then

(2.3.10)

where El is the matrix obtained from E by omitting the first column and all rows with nonzero first elements, and E2 = (e:j) l<i<n where

l~gm

I _ { eij, if 2::; j ::; m, (. _ ) eij- .. l-1, ... ,n.

max{eil - a,O}, If J = 1.


PROOF. (1) If ell = 0, then E has a zero row, hence WE = 0 (see Theorem 2.2.10(6)). If ell > 0, then applying (2.2.3) to E and e = (1,0, ... ,0) E Nm , we obtain

Thus, the induction on ell gives us the desired result. (2) The fourth statement of Corollary 2.2.20 (with k = m-l, il = 2, ... , im - 1 =

m) shows that E contains a row, in which only the first coordinate may be distinct from zero. Therefore, H contains a zero row, hence WH = O.

(3) Relationship (2.3.10) IS a consequence of (2.2.3), written for E and (a,O, .. . ,0) E Nm . 0

Using Lemma 2.3.18, we can propose the following method of calculation of the Kolchin polynomial WE of a matrix E in the case when degwE = 0: apply the relationship (2.3.10) to WE (where a is the minimal nonzero entry of the first column of E), then write the similar representation for E2 and so on. After a finite number of such steps we obtain a representation of WE as a sum of the Kolchin polynomials of matrices Fl, ... , Fr (r E N, r 2: 1) with (m - 1) columns and of a Kolchin polynomial WH, where the matrix H = (hij ) l<i<n has the elements

l<j~m

{ eij, if 1 < i < n, 2 < j < m - -

hij = . -. - . - - so that WH = 0 (see Lemma 2.3.18(2)). 0, If 1 < l < n, J = 1

Applying the described-procedure to each of the matrices Fl' ... ' Fr we reduce the computation of WE to the computation of Kolchin polynomials of some matrices with (m - 2) columns and so on, until we obtain matrices, everyone of which has a single column. As we know, the Kolchin polynomial of such matrix coincides with its minimal element.

In general case (without the condition deg WE = 0) the computation of the dimension polynomial WE(t 1 , ... , tp ) of a n x m-matrix E = (eij) l<i<n by the

l<j~m

above scheme (using (2.3.9) instead of (2.3.10)) may be reduced to the -computation of dimension polynomials of matrices with fewer than m number of columns and of the dimension polynomial of some n x m-matrix with the zero first column. More precisely, if the first column of E contains nonzero elements, then we set a = minl<i<n{eilleil =j:. O} and apply (2.3.9). Then we apply the same relation to H (see Lemma 2.3.16) and so no. As a result, we obtain a decomposition of the polynomialwE(tl, ... , tp ) into a sum of polynomials of the form WE; (t 1, ... , ti-l, ti

ai, ti+l, ... , t p ) (1::; i ::; p), where ai EN and Ei is either a matrix with fewer than m number of columns, or a nxm-matrix with the zero first column. To compute the dimension polynomials of matrices of the latter type we apply the above method to the second column and so on, until we obtain a representation of WE(t 1, ... , t p ) as a sum of dimension polynomials of matrices with fewer than m number of columns and dimension polynomials of matrices with at most two nonzero columns. The dimension polynomials of matrices of the latter type can be found with the help of the following statements.


2.3.19. LEMMA. Let E = (eij) l<i<n be arbitrary nxm-matrix, Nm = Uf=l (1'i 1<j<:m

a partition of the set Nm , ki = Card(1'i, (i = 1, ... ,p), and WE(tl, ... ,tp) the dimension polynomial of E, associated with the partition. Let eij = 0 for all j E (1'1 and i = 1, ... , n. Then

where El is a n x (m - kt}-matrix that is obtained by omitting all columns whose indices belong to (1'1.

PROOF. The desired relationship can be obtained by formula (2.2.5) as follows:

( tl + kl) = kl WE,(t2, ... ,tp).

(Since e€i = 0 for all i E (1'1, we have to = 0.) 0

2.3.20. LEMMA. Let

('" e12 0 ...

e2l e22 0 ... E=

enl en2 0 ... !) be a normalized n x m-matrix over N (m 2: 2). Let 0 = ell < e2l < ... < en 1 and e12 > e22 > ... > en 2 = O. Then

n-1

WE(t) = L ~(ei+,.,-eil)~ -lwei (t - eil)' ;=1

where ei = (ei2, 0, ... , 0) E Nm - l (i = 1, ... , n - 1).

(2.3.11)

PROOF. We proceed by induction on n. The case n = 1 is trivial. Let n > 1 and assume that the statement of the lemma is valid for any matrix with fewer than n number of rows that satisfies the conditions of the lemma. To prove relationship (2.3.11) for n x m-matrix E = (eij) l<i<n with eij = ° (1 ::; i ::;

1 <j<:m n, 3 ::; j ::; m) we note first that if a = minl-<i<n{eit!eil i=- o} = e21 then WE(t) = ~a~-lWE,(t)+wH(t-a), where El -;-(e12,O, ... ,O) E Nm - 1 and

H ('" ~ '" ::: ~ ... ~) (see Lemma 2.3.16). The first row of H is

enl - e2l en 2 ° ... °


superfluous, hence WH = WH, where HI = ('" ~ c"

enl - e2l en 2 0 inductive hypothesis, we have

whence

n-l

WH(t) = WH, = L ~(ei+",-ei')~ -lwei(t - e;l),

;=2

n-l

WE(t) = ~e2'~ -lWE, (t) + L ~(e.+",-ei,)~ -lWei(t - eiI) ;=2

n-l

= L ~(ei+",_eil)~-lWe.(t - ed). 0 ;=1

2.3.21. COROLLARY. Let E = (::; :::) be a nx2-matrix over N such that

enl en 2

0= ell < e2l < ... < enl, e12 > e22 > ... > en2 = O. Then

n-l

WE(t) = L(e;+l,l - eil)ei2.

;=1

2.3.22. LEMMA. Let E = (::; ::: :

enl en 2 0

(2.3.12)

matrix over N (m 2: 2) such that 0 = ell < e2l < ... < enl, e12 > e22 > ... > en2 = O. Let Nm = 0'1 U 0'2 be a partition of Nm such that 0'1 = {I, 3, 4, ... , k}, 0'2 = {2, k + 1, ... , m}, 2::; k ::; m. Then

n-l WE(t l ,t2) = LWei(t l - eidwhi(t2),

i=l

where WE(tl, t2) is the dimension polynomial of E, associated with the given par-.. fIN . - ( . . 0 0) "Tk-l d h· - (. 0 0) "Tm-k+l tltlon 0 P'rn, e. - e.+l,l - ell, , ... , En, an I - e,2, , ... , E n

(1::; i ::; n - 1).

PROOF. We proceed by induction on n. The statement of the lemma is evident for n = 1. Suppose that the statement has been established for any matrix that


satisfies the condition of the lemma and whose number of rows is fewer than n. By Lemma 2.3.16, we obtain that if a = min1:5;:5n{eilleil f:. O} = e21, then

( e31 ~ e21 :~~ ~

where El = (e12' 0, ... ,0) E Nm - 1, H = :

eni - e21 en2 0

... 0) ... 0

. . o

is (n-1)x

m-matrix, and WE, (tI' t 2), WH(t1, t 2) are the dimension polynomials associated with the partition Nm - 1 = iTl UiT2 such that iTl = {2, 3, ... , k-1}, iT2 = {I, k, ... , m-1}. By the inductive hypothesis, we have WH(tl,t2) = "£';;;21Wei(t1 - eit +e21)whi(t2), hence

n-1

= W(e'2,O, ... ,O)(t2)W(e2"O, ... ,O)(t1) + LWei(tl - ei1)whi(t2) ;=2

n-l = LWei(tl - e;1)wh,(t2). 0

;=1

The following Algorithm 2.3.23 of computation of the Kolchin polynomial WE (t) of nxm-matrix E is based on the above scheme. In accordance with this scheme, we use (2.3.8) to represent the polynomial WE(t) as a sum of the Kolchin polynomials of matrices with fewer than m columns and of a Kolchin polynomial of a n x mmatrix E' with at most two nonzero columns (without loss of generality one may assume, that the first two columns of E' are nonzero ones). The polynomial WE' (t) is computed with the help of relationship (2.3.10). At the beginning, we rearrange the rows in order to dispose the entries of the first column according to the conditions of Lemma 2.3.20 (such rearrangement requires'" n log n elementary operations). After that we see that if the second nonzero column is not arranged in the reverse order, then the matrix E' has superfluous rows (precisely those rows ei (1 ::; i ::; n), for which e12 2: ej2 for some j EN, 1 ::; j < i). Thus, we obtain the following estimation of the number f(n, m) of elementary operations, which are required for the computation of a Kolchin polynomial WE(t) of a nxm-matrix E with the help of Algorithm 2.3.23:

k

f(n, m) ::; n logn + f(n, m - 1) + L f(b;, m - 1) ;=1

::; nlogn+nf(n,m-1)

where 1::; k < n; b1 , ... ,bk EN; 1::; b;::; n (i = 1, . .. ,k). Therefore, Algorithm 2.3.23 has the asymptotic complexity", nm - 1 logn for m 2: 2 (if m = 1, then the asymptotic complexity", n).


2.3.23. Algorithm (E, n, m,w(t)) Input: mEN, n E N, E is a nxm-matrix Output: w(i) is the Kolchin polynomial of E Variables: Po, PI are polynomials,

Begin w(t) := 0

Ns is the current value of the first coordinate, NR is the next value of the first coordinate, Eo is a sequence of (m - I)-tuples

if n = 0 then w(t) := c;;.m)

End

else v := (VI, ... , vm ), where Vj = minl<i<n{eij}(j = 1, ... , m) E := (eij - Vj) l<i<n - -

l<j~m

K := {indices of nonzero columns of E} cases Card K = 2, K = {j, p} =? sort row in ascending order of jth column

omit superfluous rows w := w(t) + 2.::7:/ ~(e'+l.j-eij)~-lwei(t - eij), where ei = (eip, 0, ... ,0) E Nm - 1

Card K > 2 =? take k E K interchange the kth column with the first one Ns :=0 Eo:= 0 do for every nonzero eil in ascending order

NR:=eil Eo: add sequence of rows {(ej2, ... ,ejm)lejl = Ns} No := the number of vectors in Eo Algorithm 2.3.23 (Eo, No, m - 1, Po(t)) Po(t) := ~(NR-Ns)~ -1 Po(i) w(t) := w(t) + Po(t - Ns) Ns:= NR

E : set all elements of the first column equal to zero Algorithm 2.3.23 (E, n, m, P1(t)) w(t) := w(t) + Pdt - Ns)

w(t) := w(t - Ivl) + c;;.m) - C+m,:lvl)

2.3.24. EXAMPLE. Let us compute the Kolchin polynomial of the matrix E = 1 a a 1

r-2 a 1 a r-3 a 2 a

a r-i-l a (1' E N, l' 2': 3) with the help of Algorithm 2.3.23. In process

1 a r-2 a a 1 a r-2

of computations we subsequently apply (2.3.8) starting from the last column of E.


First of all, let us write

1 0 0 0 r-20 0

where El = (~~; ~ ~ ), H =

1 0 r-2

r-30 2 0

o r-i-1 0 . By Theorem 2.2.10(8), we

1 0 r-2 0 o 1 0 r-3

have

Applying (2.2(.~~;~ ~l )and (1,0,1)' we obtain WE, (t) = W(l,O,l)(t) + WH, (t - 2),

r-40 1

where H1 = ::: ' so that formula (2.3.10) implies that 1 0 r-4 o 0 r-3

r-3 r-3

WH, (t - 2) = L: ~1~ -lW(r_2_i,O)(t - 2 - (i - 1)) = L: W(r-2-i,O)(t - i-I) i=l ;=1

= ~ [e -i ~ 1 + 2) _ C -i-I + 22- (r - 2 - i))]

= ~ [e-~+I) - e+~-r)] (r - 3)(r - 2) (r - 2)(r - 3)(2r - 5)

= 2 t- 6 .

Therefore,

WE (t) = (t + 3) _ (t + 3 -2) + (r - 3)(r - 2) t _ (r - 2)(r - 3)(2r - 5) , 3 3 2 -'-----'--'--6---'--'-----'-

( 1)2 (r - 3)(r - 2) (r - 2)(r - 3)(2r - 5) = t+ + 2 t- 6 '

hence

() ( t + 2) (t + 4 - r) ( 1)2 WE t = 3 - 3 + t +

+ (r - 3)( r - 2) t _ ..:....( r_-_2~)('_r_-_3-'-')(_2r_-_5.....:..) 2 6

r 2 r + 2 r(r2 - 6r + 11) =2"t + -2-t - 6 .


2.3.25. Algorithm (E, n, m, p, 0'1, ... , O'p, w) Input: the same as in Algorithm 2.3.15 Output: the same as in Algorithm 2.3.15

Begin kll := Card 0'11 (v = 1, .. . ,p) R:=0 do for j from 1 to p

if O'j # 0 then R:= Ru {j} if R = {j} then Algorithm 2.3.23(E, n, m, wo(tj))

W(t1, ... , tp) := wo(tj) else

if n = 0 then W(t1' ... , tp) := n~=l e;t;k;) else v := (V1, ... , vrn) where Vj = mill;~l {eij}

E := (eij - Vj) l<i<n l<j<rn

s := (Sl, ... , sp) -where Sj = L:iEU; Vi

(t t ) - n~ (t;+k;) _ (t;+kj-a;) w 1,···, P - 3=1 k; kj

K := indices of nonzero columns of E do for j from 1 to p

if (I( n O'j = 0 and O'j # 0) then E: omit columns with indicies from O'j O'j := 0; kj:= kj; kj:= 0; K:= 0

Algorithm 2.3.25 (E, n, m - kj , 0'1, ... , O'p, wo)

( t +k") WO(t1, ... ,tp):= JkJ WO(t1 ... ,tp) J

cases Card K = 2, K = {j, k}, j E 0'1, k E 0' r, I # r =>

sort the rows in ascending order of kth column omit superfluous rows WO(t1, ... , tp) := L:7:11 Wei (tl - eij )Whi (tT) where ei = (ei+1,j - eij,O, ... ,0) E Wk" hi = (eik,O, .. . ,0) E Wkr

Card K > 2 => take nontrivial k E O'j n K Wo := 0; Ns:= 0 EO:= 0 do for every nonzero eik in ascending order

NR := eik Eo : add sequence of rows {(en, ... , flk, ... , elrn)lelk = Ns} No := the number of vectors in EO Algorithm 2.3.25 (Eo, No, m - 1,0'1, ... , O'j \ {k}, ... , O'p, Po) Wo (t1, ... , tp) := wo(tl, ... , tp) + Ll(NR_Ns)(j)Ll-1 (j)PO(t 1, ... , tp) Ns:= NR

E : set all elements of the kth column equal to zero Algorithm 2.3.25 (E, n, m, 0'1, ... , O'p, P1) Wo(t1, ... , tp) := Wo(tl, ... , tp) + P1(t 1, ... , tj - Ns, .. ·, t p)

W(t1' ... ' t p) := WO(t1 - Sl, ... , t1 - Sp) + W(t1, ... , tp) End


Now, we can give a recursive algorithm for computation of the dimension polynomial WE(tl, . .. , tp), (p 2: 1) that is based on Lemma 2.3.16 (see Algorithm 2.3.25 below). For the sake of convenience, we suppose that ()j ::: 0 (i.e. kj ::: Card ()j ::: 0) for some j (1 :::; j :::; p) in partition (2.2.1) of Nm (if ()j ::: 0, then the polynomial WE(t l , ... ,tp) does not depend on tj).

Let us consider the problem of computation of the leading coefficient of the Kolchin polynomial. Let E ::: (eij) 1 <i<n be a n x m-matrix over N, and let

lSj~m

m (t + .) we(t) ::: ~ ai(E) i Z (2.3.13)

be its Kolchin polynomial. Then, by Theorem 2.2.10(7) and Corollary 2.2.20(3), we have

{I, if E is the empty matrix (n ::: 0)

am(E) ::: . 0, otherwise.

{ 0, if E is the empty matrix,

am-dE)::: m . . Lj=lmml:Si:Sn{eij}, otherwise.

The computation of the coefficient am _2(E) (if m 2: 2) can be based on the following statement.

2.3.26. LEMMA. Let E ::: (eij) l<i<n be a nxm-matrix over N such that m > 1 1 <?:j <m

and the first column of E is the z~r;; one. Let WE(t) ::: L~=o ai(E) Cti) be the Kolchin polynomial of E and let 0 < degwE :::; T (T E Nm ). Furthermore, let El be the n x (m - I)-matrix obtained by omitting the first (zero) column of E. Then degwE, :::; T -1 and aT(E) ::: aT-l(Ed.

PROOF. Applying (2.2.3) to E and (1,0, ... ,0) E Nm, we obtain that WE(t) ::: WEl(t) +WE(t -1). Thus, ifwE(t)::: L~=o ai(E)eti), then

WE, (t) ::: ~ ai(Ed C ~ i) ::: ~ ai(E) [ C ~ i) -C + ~ - 1) ]

hence degwEl :::; T-1 and ai(El ) ::: ai+l (E) for all i ::: 0,1, ... , T-l. In particular, aT(E) ::: aT-dE!). 0

Note, that if the degree of the Kolchin polynomial WE(t) ::: L~o ai(E) Cti) of a nxm-matrix E::: (eij) i<i<n is less than or equal to T (T EN, 0:::; T:::; m), then

lSj~m

(2.3.14)

where a::: minl<i<n{eilleil =j:. O}, the matrix El is obtained from E by omitting the first column-a~d all rows with zero first entries, and E2 ::: (ei j ) l<i<n is the

1 <?:j<m n x m-matrix with the entries - -

I { eij, if j =j:. 1 . . eij::: { . }.. _ (1:::; 1 :::; n, 1:::; J :::; m).

max e.l - a, 0, lf J - 1.


(Relationship (2.3.14) can be easily established, by applying (2.2.3) to E and ( a, 0, ... , 0) E Nm).

Before computing the coefficient am -2(E) of Kolchin polynomial (2.3.13) we remark, that, without loss of generality, one may assume that deg WE ~ m - 2. Indeed, applying (2.2.3) to E and e = (minl<i<n{eil}, ... ,minl<i<n{e;m})' we

obtain that WE(t) = c;;..m) - C+:-Iel) + wE,(i ..::: lei), where E' ~ (e~j) l<i<n is l~j<m

the matrix with entries e~j = eij - minl~i~n{eij} (1 ~ i ~ n, 1 ~ j ~ mf Using (2.1.4), we may rewrite the latter representation ofwE(t) as follows:

Since degwE' ~ m-2 (see Corollary 2.2.20(4)), we have am _2(E) = am -2(E')lelCl~l-l), hence the computation of am -2(E) can be replaced by the computation of am -2 (E'), so that in the following considerations we suppose that deg W E' ~ m - 2 and WE (t) = 2:;:~2 a;(E) Cti), where ai(E) E ;Z (i = 0,1, ... , m - 2). Moreover, we suppose that E has at most two nonzero columns (it should be noted, that if E has a single nonzero column, then the Kolchin polynomial WE(t) coincides with the minimal element of this column). Assuming that the first column of E is nonzero one we order its elements and apply (2.3.14) for T = m - 2. Since El (see (2.3.14)) has (m-1) columns, the computation of aT(Et) may be reduced to the choice of the minimal elements in the columns of E1 (see Corollary 2.2.20(3)). It is easy to see that such a choice requires (m-1)b 1 elementary operations, where b1 is the number of rows of E1 . Applying (2.3.14) to E2 we reduce the computation of aT (E2) (in the right-hand side of (2.3.14)) to the computation of the coefficient a2(En ) of a Kolchin polynomial of a matrix E2 with (m - 1) columns (this matrix is obtained by adjoining some additional rows to E1). We need at most (m - 1)b2 elementary operations in order to calculate aT (E21 ) (here b2 denotes the number of rows of En). Then we subsequently apply (2.3.14) until we obtain a matrix with the zero first column. By Lemma 2.3.26, one may omit such a column, then apply (2.3.14) to the new matrix, and so on.

The asymptotic complexity g(n, m) of the algorithm described does not exceed

k

nlogn+ L bim+ g(n,m -1) ~ 2nlogn + n(m +m -1) + g(n,m - 2) ~ ... ;=1

biEN b1+···+b.=n


~ (m - l)n logn + n(m + (m - 1) + ... + 2) + g(n, 1)

( m+ 1) =(m-1)nlogn+n 2 -mnlogn.

2.3.27. Algorithm (E, n, m, a m -2) Input: mE N,n E N,E is nxm-matrix with degwE ~ m - 2 Output: am _2(E)

Begin a m -2 := 0 r := number of zero columns of E m :=m-r E : omit zero columns of E sort the rows in ascending order of the entries of the first column Ns :=0 i:= 1 Eo :=0 do while i < n

do while eil = Ns and i ~ n

NR:= eil

Eo := Eo U (ei2' ... , eim) i := i + 1

a := (al, ... , am-d where ai is the minimal element of ith column of the matrix consisting of vectors from Eo

Eo :=a am -2 := am -2 + lal(NR - Ns) Ns:= NR

E : omit the first column Algorithm 2.3.27(E, n, m - 1, b) am -2 := am -2 + b end

Now, using Algorithm 2.3.27 and formula (2.3.14)' we can find the leading coefficient of the Kolchin polynomial of any matrix. The complexity fk(n, m) of computation of this coefficient for a matrix E with degwE = m - k (1 ~ k ~ m) does not exceed

n

nlogn+ LIk-1(i,m-1)+Ik(n,m-1) ~ nlogn+nlk_1(n,m-1)+fk(n,m-1), i=1

if Algorithm 2.3.28 given below is used. Therefore, h(n, m) - (';')n2 Iog n and, in general, !k(n, m) - V~\)nk-llog n (k = 3, ... , m).


2.3.28. Algorithm (E, n, m, k, am-k) Input: mEN, kEN, k 2: 2, m 2: k, n EN, E is nxm-matrix with

degwE:::; m - 2 Output: am-k(E)

Begin

am-k := ° if k = 2 then Algorithm 2.3.27(E, n, m, am -2) else E : omit zero columns

end

m := the number of columns of E Ns :=0 Eo:= 0 do for every nonzero eil in ascending order

NR := eil

Eo: add sequence of rows {(ej2, ... ,ejm)lejl = Ns} No := number of vectors in Eo Algorithm 2.3.28 (Eo, No, m - 1, k - 1, P) am-k := am-k + (NR - Ns)P Ns:= NR

E : omit the first column Algorithm 2.3.28 (E, n, m - 1, k - 1, P) am-k := am-k + P

2.4. Ordering of Kolchin Dimension Polynomials

In this section we introduce some natural ordering of the set of all univariate numerical polynomials. It will be shown that this ordering induces a well-ordering of the set W of all Kolchin polynomials of matrices with non-negative integer entries. Later on, we shall use this result to determine the m:nimal differential and difference polynomials. Furthermore, we shall extend some results concerning Kolchin dimension polynomials to the multivariate dimension polynomials.

2.4.1. DEFINITION. Let Wl(t) and W2(t) be numerical polynomials in one variable t. We say that Wl(t) does not exceed W2(t) (and write wdt) :::; W2(t)) if Wl (s) :::; W2 (s) for all sufficiently large sEN, i.e. there exists r E N such that the last inequality holds for all sEN, s 2: r.

(In spite of the fact that the symbol :::; denotes the product order on Nm and the natural order on IZ, we prefer not to introduce a new symbol for comparisons of numerical polynomials. It will be always clear what objects are compared.)

It is easy to see that:::; is an ordering of the set of all univariate numerical polynomials. Furthermore, if wdt) = 2:7:::0 ai eti) -::j:. W2(t) = 2:7:::0 bi Cti) (ai, b; E IZ foralli=O,I ... ,m),thenwdt ) :::;w2(t)iff(am,am-l, ... ,ao) ~ (bm,bm-1, ... ,bo) (as before, ~ denotes the lexicographic order on IZm+l).

Let us consider the set of all Kolchin polynomials W = {f(t) E Q(t) I f(t) = WE(t) for some matrix E over N} as an ordered set, whose order is induced by the relation:::; on the set of all univariate numerical polynomials. We shall show that

2.4. ORDERING OF KOLCHIN DIMENSION POLYNOMIALS 93

the set W is well ordered, i.e. every descending sequence W1 ~ W2 ~ ... ~ Wn ~ .. . (Wi E W for all i = 1,2, ... ) contains a polynomial Wn such that Wn = wn+1 = ... . We shall also describe the set W and show how, for arbitrary numerical polynomial w, one can find out whether W is the Kolchin polynomial of a matrix over N (see Proposition 2.4.10).

First, we need some preliminary results.

2.4.2. LEMMA. Let E be a n x m-matrix over N. Then there exist matrices E1, ... ,Ek (k EN, k ~ 1) such that

(i) each matrix E; (1 ~ i ~ k) has m columns;

(ii) WE(t + 1) = E;=l WEj (t); (iii) degwE; ~ degwE for all j = 1, ... , k.

PROOF. Let ei E Nm be the vector, whose ith coordinate is equal to one, and the other coordinates are equal to zero. By (2.2.3) we have

for some n x m-matrix E1 over N. Applying (2.2.3) m times, we obtain that

m

WE(t + 1) = wEue,u ... uem(t + 1) + LWE;(t) j=l

for some n x m-matrices E1"'" Em over N. Furthermore, WEue,u ... uem(t) = 1 if E does not contain a zero row, and

WEue,U ... Uem (t) = 0 if all entries of some row of E are equal to zero. In both cases we have WEue,u ... uem(t + 1) = WEue,u ... uem(t), hence WE(t + 1) = E.7'=oWEj(t) (where Eo = EUe1 U·· ·1Jem ) and degwEj ~ degwE (0 ~ j ~ m) since WEj(t) ~ 0 for all j = 0, 1, ... , m. 0

2.4.3. LEMMA. Let E be a non-empty n x m-matrix over N and let WE f. O. Then there exists i E Nm such that

(i) degwE = degwEue;, where ei = (0, ... ,0,1,0, ... ,0) is the vector whose ith coordinate is equal to one and the other coordinates are equal to zero;

(ii) there is a nonzero element in ith column of E.

PROOF. First, note that if degwE = m - k for some kEN, then k ~ 1 (since degwE ~ m for any n x m-matrix E, and degwE = m iff the matrix E is empty). Furthermore, by Corollary 2.2.20, the leading coefficient ad(E) of the Kolchin polynomial of E may be written in the form ad(E) = EOEA(k.m) wE ... Since ad(E) > 0, we have WE" f. 0 for some (1' E A(k, m). By Theorem 2.2.7(3)' we can choose i E (1'

such that ith column of Eo contains a nonzero element. Let ei be the vector, whose ith coordinate is equal to one and the other coordinates are equal to zero. Then WE(t) = WEue;(t)+wH(t-l), where H = (hrsh<r<n is the n x m-matrix with the

l<.<m entries - -

ifsii,

if s = i (1 ~ r ~ n, 1 ~ s ~ m).


Let us show that deg W H :s deg W EUei' (This inequality implies the statement of the lemma, since degwEuei :s degwE.) Assume the contrary: degwH > degwEuei' In this case the leading coefficient ad(E) of the polynomial WE(t) coincides with the leading coefficient of WH(t):

ad(E) = ad (H) = L WHu "EA(k,m)

(see Corollary 2.2.20(5); as in the corollary, H" where () = {il,"" id E A(k, m) denotes the matrix consisting of il, ... , ik th columns of H).

Since ers = hr. for all 1 :s r :s n, 1 :s s :s m, s i= i, the last equality shows that

"EA(k-l,m) irt"

"EA(k-l,m) irt"

(2.4.1)

where Ei , Hi are ith columns of the matrices E and H, respectively, and the matrices E"Ei, EaHi are obtained by attaching to Ea the columns Ei and Hi, respectively.

Now, it follows from Definition 2.2.8 and from the construction of Hi that WEulEi 2: WEulHi for all ()' E A(k - 1, m). By the choice of i, we have WEulEi = WEu > WEulHi for some ()' E A(k -I,m). Thus, relationship (2.4.1) does not hold, hence deg W H :s deg W EUei' This completes the proof. 0

2.4.4. LEMMA. Let E = (eij) l<i<n be a nxm-matrixover N. Then there exists l<j<m

a n x m-matrix El over N such that-

( t + m -1) WE,(t) = m-l +wE(t-l).

PROOF. Consider the matrix El = (eij) l<i<n with the entries lSj~m

-e" _ { eil + 1, 'J -

eij,

if i = 1,

if i i= 1

By (2.2.3) we have

WE,(t) = WE,U(l,O, .. ,O)(t) +WE(t -1) = C -: r:. ~ 1) +WE(t -1),

so that El is the desired matrix. 0

2.4.5. THEOREM. Let E and H be subsets of Nm (m E N, m 2: 1) and let deg W H :s deg WE = m - 1. Then there exists a subset El C Nm such that

WE, (t) = WE(t) + WH(t - 1).


2.4.6. COROLLARY. Let F be a subset ofNm +l and let degwF < m (m E N,

m ~ 1). Then there exists a subset F c Nm such that W F = W F'

PROOF OF THE THEOREM AND THE COROLLARY. We will prove the theorem and its corollary simultaneously by induction on m. Since statements 2.4.5, 2.4.6 concern only dimension polynomials, we may suppose that all subsets of Nm (or of Nt, where lEN) considered in the proof are finite. If m = 1, then W H, wEE N and WEI = wE + WH for El = {WH + WE} ~ N. Obviously, the statement of the corollary is also valid for m = 1. Now we suppose that m > 1 and Statements 2.4.5, 2.4.6 hold for all subsets of Nm' (or of Nm'+l, if we consider the corollary) with m l < m. First, let us consider the case degwH < m - 1 in the theorem. Let E = {el," .,en} where ei = (eil, .. . ,eim) (1:'5 i:'5 n), and let E' = (eij) l<i<n be

l<j<:m the matrix, whose rows are el, ... , en. Since deg WE = m - 1, there exists- i-E Nm

such that ith column of E' consists of nonzero elements (see Corollary 2.2.20(3)). Applying (2.2.3) to E' and e = (0, ... ,0,1,0, ... ,0) (ith coordinate of e is equal to one and its other coordinates are equal to zero) we obtain that

( t + m -1) WE(t) = +wH(t - 1), m-l

(2.4.2)

for some subset H c Nm . Now we will proceed by induction on the leading coefficient r = am_l(wE). If r = 1, then degwH < m - 1 (see (2.4.2)). By our assumption, degwH < m - 1 and the statement of the corollary holds for m l = m - 1, so we may suppose that H C Nt and H C Nt for some I < m (we can also suppose that the statements of the theorem are valid when m is replaced by /). Let deg W H ~ deg W H (the case deg H :'5 deg W H can be considered in the same way). Then we may assume that I = degwH + 1. Now, Lemma 2.4.2 shows that

k

WH = LWHj(t - 1) j=l

for some subsets Hj C Nt such that degwHj :'5 degwH :'5 degwH (j = 1, ... , k). Applying Theorem 2.4.5 for m = I we obtain that

wH(t) +WH(t) = (wH(t) +WHI(t -1)) +WH2 (t -1) + .. ·+WH.(t -1)

= (WHI (t) + WH2 (t - 1)) + ... + WHo (t - 1) = ... = W H. (t).

for some subsets Hl , ... , Hk C Nt. Note that if we write (m - k) zeros to the left of every vector of Hk and adjoin I vectors

(1,0, ... ,0), (0, 1,0, ... ,0), ... , (0, ... ,0,1,0, ... ,0) E Nm ,

then the Kolchin polynomial of the set obtained is the same as the Kolchin polynomial of Hk (see Theorem 2.2.10(8)). Therefore, we may suppose that Hk C N m

and


for some subset El C Nm (see Lemma 2.4.4). Assume now that r = am-I(wE) > 1. Since 0 "# am-l(Wjj) < am-l(wE), we

may suppose (by the inductive hypothesis on r) that the statement of Theorem 2.2.5 is valid for H (instead of E). By Lemma 2.4.2, there exist subsets Hl,"" Hie of Nm such that WH(t) = 2:~=l WHj (t - 1) and

Therefore,

Wjj(t) + WH(t) = Wjj(t) + WHI (t - 1) + ... + WH" (t - 1) = Wjj, (t) + WH2 (t - 1) + ... + WH" (t - 1)

for some Hl C Nm . Since degwH, ~ degwH < m - 1, we have am_l(Wjj.)

am-t{wjj) + am-l(wH1 ) = r - 1, so the statement of the theorem is valid for Hl (instead of E). Finally, we obtain that Wjj(t) + WH(t) = Wjj" (t) for some subset

HIc C Nm . Then we can complete the proof of the theorem as in the case when r = 1 (using Lemma 2.4.4).

Let us prove the theorem in the case when degwH = m - 1. As above, we will proceed by induction on r = am-l(wE). If r = 1, then Lemma 2.4.4 shows that

for some subsets H,Hl,H2 C Nm. (Since degwjj < m - 1, we have degwH, m - 1.) To implement the step of induction we need the following statement.

2.4.7. LEMMA. Let E be a subset ofNm such that degwE = m-1. Then there exist iE, El, ... , Elc ~ Nm (k E N, k 2:: 1) such that degwEj < m - 1 (j = 1, ... , k)

and WE(t + 1) = wEj(t) + 2:~=l WEj(t).

PROOF OF LEMMA 2.4.7. We will proceed by induction on the value of the leading coefficient am-l(wE) of WE(t). If am-l(wE) = 1, then Lemma 2.4.2 shows that there exist subsets El, ... ,EIc ~ Nm such that WE(t + 1) = 2:~=lWEj(t). Since am-l (WE) = 1 = 2:~=l am-l (WE;), the degree of only one of the polynomials WE" ... , WE" is equal to m - 1 (the degrees of the other polynomials are less than m - 1), so that the statement of the lemma is proved for the case am-l (w E) = 1.

Let am-l(wE) > 1. As above (see (2.4.2)), there exists a set iE ~ Nm such that

( t + m -1) WE(t) = +wEj(t -1). m-l

whence am_l(wEj) = am-l(wE) :: 1 < am-t{wE). Furthermore, by the inductive

hypothesis, there exist subsets iE, Eb"" Elc ~ Nm (k E N, k 2:: 1) such that wEj(t + 1) = wi}(t) + 2:~=l WEj(t), where degwEj < m - 1.


Therefore,

( t+m) WE(t + 1) = + WE(t) m-l

( 1) k m ( .) t+m- t+m-z = +w=.(t-l)+"wEj(t-l)+". m-l E L...J L...J m-z

j=l i=2

(see (2.1.5)). By Lemma 2.4.4, there exists a set H ~ Nm such that et,,~~l) + wE(t - 1) =

WH(t). Since degwEj < m-l, we may apply the proven part of Theorem 2.4.5 to the sets Hand El (instead of E and H, respectively, in the conditions of the theorem) and obtain the existence of a set E~ C Nm such that WE: (t) = WH(t) + WE, (t - 1). Now, we may apply the proven part of the theorem to the sets E~ and E2 and so on. Finally we obtain the relationship

k

WH(t) + LWEj(t - 1) = WE·(t) j=l

Ei = {el = (1,0, ... ,0)' e2 = (0,1,0, ... ,0)' ... , ei = (0, ... ,0,1,0, ... , On c Nm

(i = 2, ... , m), we get

m

WE(t + 1) = WE· (t) + LWEi (t). i=2

By Corollary 2.2.20(3)' we have degwEi < m-l, so that Lemma 2.4.7 is proved. 0

Let us return to the proof of Theorem 2.4.5. Now we have to consider the case r = am-dwE) > 1, degwH = m - 1. The foregoing lemma shows that there exist sets E, El, ... , Ek ~ Nm (k EN, k ~ 1) such that

k

wii(t) +WH(t) = (wii(t) +wE(t -1)) + LWEj(t -1) j=l

and degwEj < m - 1 for all j = 1, ... , k. Since am-l(wii) < r, we may assume that the statement of the theorem holds for the matrices ii and E (instead of E and H, respectively, in the conditions of the theorem). Therefore, there exists a subset Hl ~ Nm such that wii(t) + wE(t - 1) = WH, (t).

Since Theorem 2.4.5 has been already proved for the case degwH < m - 1, we have

k

WH, (t) + L WEj (t - 1) = WHo (t) j=l


for some set H2 C f:l m. It remains to apply Lemma 2.4.4 to H2 to complete the proof of the theorem.

Now we are going to complete the proof of Corollary 2.4.6 by induction on IFI = 2::7=1 2::7=i1 /;j where F = {It, ... , In} C f:lm+1 and Ii = (/;1, ... , /;,m+d for all i = 1, ... , n (see the conditions of the corollary). The following considerations are based on the fact that the statement of the corollary holds for all subsets of f:lm' with m' < m + 1, and Theorem 2.4.5 has been proven for all m' ::::; m. Clearly, the statement of the corollary holds for IFI = 0. If IFI > ° then, by Lemma 2.4.3, there exists a vector e = (0, ... ,0, 1,0, ... ,0) E f:lm+1 such that degwF = degwFue. By (2.2.3) we may write

WF(t) = WFue(t) + WH(t -1)

for some set H ~ f:lm+1 with IHI < IFI and degwH ::::; degwFue. By the inductive hypothesis (on IFI) there exists a set H1 ~ f:lm such that WH = WH" and by Theorem 2.2.10(8) we have WFUe = WF, for some set F1 ~ f:lm. Since am(wF) = 0, degwF, < m, hence F1 # 0.

Thus, we may suppose that F1 and H 1 are subsets of f:ll, where l = deg W F, + 1 ::::; m (indeed, if degwF, < m - 1, then, by Corollary 2.4.6 for m' = m - 1, one may consider F1 and H1 as subsets of f:jm-1 and continue in this manner until the dimension of the space in which F1 , H 1 may be considered as subsets attains degwF, + 1). Since the statement of Theorem 2.4.5 holds for subsets of f:ll (if we take F1, H1 instead E and H, respectively), we have

WF(t) = WF, (t) + WH, (t - 1) = wp(t)

for some set F ~ f:jl ~ f:lm. This completes the proof of Theorem 2.4.5 and Corollary 2.4.6. 0

Now we intend to study the properties of the set W of all Kolchin polynomials.

2.4.8. COROLLARY. Let w(t) be a nonzero numerical polynomial of degree d:

d (t + i) w(t) = f;a i i .

Then W E W iff ad > ° and the polynomial

( ) _ ( ) (t + d + 1 + ad) (t + d + 1) v t - W t + ad - d + d ' + 1 + 1

(whose degree does not exceed d - 1) also belongs to W.

PROOF. Corollary 2.4.6 shows that W E W iff there exists a finite set E ~ f:jd+1 such that W = WE. It is obvious that if W E W, then W > 0, so that ad > 0. Let e = (e1' ... ,ed+d E f:ld+1 be the vector, whose ith coordinate is equal to the minimal ith coordinate of elements of E (1 ::::; i ::::; d + 1). Then, by Corollary 2.2.20(3) and (2.2.3), we may write

WE(t) = WEue(t) + WH(t - ad),


hence WEUe = C~!il) - C+dt:;a d ) (see Theorem 2.2.10(9)). Therefore,

and

(t + d + 1 + ad) (t + d + 1) V(t)=wE(t+ad)- d+l + d+l =wH(t)EW.

Conversely, let v(t) = WH(t) for some n x d-matrix Hover N (clearly, for our purposes such a matrix may be identified with the corresponding finite subset of

( ada~ 1 0 ... 0) Nd). Let jj denote the (n + 1) x (d + I)-matrix : H' Applying

ad

(2.2.3) to jj and (ad, 0, ... ,0) E N d+1 we obtain

() (t + d + 1) (t + d + 1 -ad) ( ) W - t = - + WH t - ad H d+l d+l

whence

W(t) = Wji(t) E w.

o 2.4.9. DEFINITION. Let W = w(t) be a univariate numerical polynomial in t

and let d = degw. The sequence of minimizing coefficients of W is the vector b(w) = (bd, ... , bo) E &:;d+l defined by induction on d as follows. If d = 0 (i.e.

W is a constant), then b(w) = (w). Let d > 0 and w(t) = E:=OaiCti). Let

v(t) = w(t + ad) - e+1t::ad ) + C~!il). Since deg v < d, one may suppose that the sequence of minimizing coefficients b(v) = (bk, ... ,bo) (0 :::; k < d) of the polynomial v(t) has been also defined. To define the same for w we set b(w) = (ad, 0, ... ,0, bk, ... , bo) E &:;d+l.

In the terms of minimizing coefficients, Corollary 2.4.8 may be reformulated as follows.

2.4.10. PROPOSITION. A numerical polynomial w = w(t) belongs to W iff its sequence of minimizing coefficients consists of only non-negative integers.

2.4 .11. REMARK. It is easy to see that a numerical polynomial w = w (t) is defined in a unique fashion by its sequence of minimizing coefficients b(w). We also observe that with the help of Proposition 2.4.10 one may replace the checking of the inclusion w E W by the checking of the inclusion v E W, where v = v(t) is a numerical polynomial whose degree is less than degw. Thus, Proposition 2.4.10 gives us a description of the set W.


2.4.12. LEMMA. Let WI = Wl(t), W2 = W2(t) be distinct numerical polynomials, let degwl ~ d, degw2 ~ d for some dEN, and let c(Wj) E N d be the vector obtained by attaching to the vector b(wj) some zeros from the left (j = 1,2). Then Wl(t) ~ W2(t) iff c(wd -< C(W2) (as above ~ denotes the order on the set of all numerical polynomials (see Definition 2.4.1) and -< denotes the lexicographic order on ~d).

PROOF. The statement can be easily proved by induction on d. 0

2.4.13. PROPOSITION. Let Wl(t),W2(t) E W. Then the polynomial w(t) Wl(t) +W2(t) also belongs to W.

PROOF. Let di = degwi (i = 1,2) and let d1 ~ d2 • By Corollary 2.4.6, we may suppose that WI = WE, and W2 = WE2 for some E1 ,E2 ~ Nd,+l. By Lemma 2.4.2, there exist sets HI, ... , Hk C Nd,+l (k E N, k ~ 1) such that W2(t) = E;=l W Hj (t - 1) and deg W Hj ~ d1 (j = 1, ... , k). Furthermore, by Theorem 2.4.5, there exists a set E C N d,+1 such that

k

Wl(t) +W2(t) =WE,(t) + LWHj(t -1) =WE(t) E w. 0 j=l

2.4.14. PROPOSITION. The set W is well-ordered with respect to the ordering introduced in Definition 2.4.1.

PROOF. Since wE W iff b(w) E N d+1, Proposition 2.4.14 is a consequence of the fact that the set Nm is well-ordered with respect to the lexicographic order. 0

2.4.15. PROPOSITION. Let W = w(t) = E~=o ai(w)Cti) E W and let d = degw > o. Then ad-l(w) ~ _cad~W»).

PROOF. Let v(t) = w(t + ad(w)) - C+d+ittd(w») + e~!tl). By Corollary 2.4.8, we have v(t) ~ 0, hence

where O(td- 2 ) denotes a polynomial of degree at most d - 2. Thus, ad-l(w) + (ad~w») ~ 0, so that the proposition is proved. 0

2.4.16. REMARK. As the following example shows, Corollary 2.4.6 cannot be generalized for numerical polynomials in p variables with p > 1.

2.4. ORDERING OF KOLCHIN DIMENSION POLYNOMIALS

2.4.17. EXAMPLE. Consider the partition N4 = IT1 U IT2 where IT1

IT2 = {3, 4}. Let

(

0 1

E = 0 1 1 0 1 0

~ ~) o 1 1 0

101

{1,2},

and let WE(t1, t2) be the dimension polynomial of the matrix E (or of the set ofrows of E which is considered as a subset of N4). Then WE(t1, t2) = WEU(1000)(t1 , t2) +

wH(t1-1,t2)' where H = (~ ~ ~ ~) and the polynomialwH(t1,t2) is asso

ciated with the partition N4 = {I, 2} U {3, 4}. Furthermore, WEU(1000)(t1, t 2 ) = W( 1 0 1) (t l , t2)' where the latter polynomial is associated with the partition N3 =

1 1 0 {I} U {2, 3}.

Thus,

WE(t1,t 2) = W(100)(t 1,t2) +W(~ ~~) (tl -1,t2) + C1; 1) = C2; 2) + t1 + C1 ; 1) = C1 ; 2) + C2 ; 2) - 1.

We see that degwE = 2, but the equality WH(t 1,t2) = WE(t1,t2) does not hold for any set H C N3 and for any partition N3 = (11 U (12. Indeed, for any such partition we have either Card (11 = 1 or Card (12 = 1, so that Theorem 2.2.5(3) implies that either deg t1 WH :S 1 or deg t2 WH :S 1. 0

2.4.18. REMARK. Macauley (see [Mac27)) has found necessary and sufficient conditions for a given function H : N -+ N to be a graded Hilbert function of some set E ~ Nm (m EN, m 2: 1), i.e. a function with the property

m

H(s) = hVE(S) = Card{(i1 , ... , im ) EVE I L ik = s} k=1

for all sufficiently large sEN. The original Macauley's proof of this statement was generalized and simplified by many authors (see, for example, [Stan78]). It has been also shown that Macauley's theorem implies Corollary 2.4.6 (see [Kon88]). In [Mac271 Macauley also proved that the set W coincides with the set of all Hilbert polynomials of matrices over N, i.e. W = {w(t) E Q[tll w{t) = d1WE(t) for some n x m-matrix E over N with m, n EN}.

Let us consider the set WP of all dimension polynomials in p variables (p EN, p 2: 1) associated with different partitions ofNm into p subsets. We shall prove that the ascending chain condition holds for the set WP whose ordering is similar to the ordering :S of all numerical polynomials introduced by Definition 2.4.1 (however, such an ordering of WP is not linear).

2.4.19. LEMMA. Let C1,.·.,Ck EN (k EN, k 2: 1) and Ci 2: 1 (i = 1, ... ,k). Then the polynomial


belongs to W.

PROOF. We will proceed by induction on k. If k = 1, then w(t) = c"!;.m) + WI(t -1), where WI(t) = CIe"!;.m). By Proposition 2.4.13, we have WI E Whence w(t) E W (see Theorem 2.4.5). Now, let k > 1. By the inductive hypothesis,

Since (CI -1)e"!;.m) E W, Theorem 2.4.5 shows that also

(t+m) W2(t) = wI(t) + (CI - 1) m

( t + m) (t + m - 1) (t + m - (k - 1)) W = CI + C2 + ... + Ck E . m m m

Applying Theorem 2.4.5 once again, we obtain that e"!;.m) + W2(t - 1) E W. This completes the proof. 0

2.4.20. LEMMA. Let PI, ... ,Pk EN (k EN, k ~ 1) and Pj ~ 1 (j = 1, ... , k). Then the polynomial w(t) = e!~l) ... e!~k) belongs to W.

PROOF. By [Rio68, p. 25, formula (12)] we have

(m) (n) = t (n + j) (n -~ + q) (n - m -: q), (p, q, m, n E N) P q j=O P + q J P - J

hence

( s + p) (s + h) = t (p + ~ - h) (q + h ~ p) (s + h + j) P q j=O J P - J P + q

(2.4.3)

and

(s + p) (s + q) = t (~) (2q - ~) (8 + q + j) P q j=O J P - J P + q

= (8 + P + q) + t Cj (s + P + q - j) p+ q j=l p+ q (2.4.4)

where Cj = (p~j)(2q7) (j = 1, ... ,p) for all S,p, q, hEN. Now we are going to use induction on k to prove that, for any PI, ... ,Pk E N such that PI ~ ... ~ Pk, there exist numbers al, ... , ap1+ ... +Pk EN such that


and aj 2: 1 (j = 1, ... , ,£7~11 Pi). Indeed, for k = 1 our statement is trivial. Let k > l. By the inductive hypothesis

and by (2.4.4), we obtain

w(s) = (s + P1) [(s + F7=2Pi) + P2+·fk- 1 aj (S + '£~=2 Pi - j)] P1 '£i=2 P. j=l '£i=2 P.

_ (S + ,£7=1 Pi) ~. (S + ,£7=1 Pi - j) - k +~S k

'£i=1 Pi j=l '£i=l Pi

+ P2+·fk- 1 aj (s + P1) (S + '£~=2 Pi - j) j=l P1 '£;=2 P.

for all sEN, where aj,cj E IZ, aj 2: 1, Cj = (p;~)eL:~=jP'-Pl) (j = 1, ... ,Pl).

Now (2.2.1) shows that if x 2: j 2: 0, then (j) 2: 1, hence Cj 2: 1 for all i = 1, ... ,Pl· Furthermore, by (2.4.3) we have

( s + P1) (S + '£~=2P~ - j) = P1 '£i=2P,

= t (P1 ~ j) (2,£7=2 Pi_-I P1 - j) (S + 1 - ~7=2Pi - j) 1=0 P1 '£,=2 P,

= jf ( P1 ~ ~ _) (2 ,£7=2!i ~ PI - j) (S + '£!=1 Pi - I) l=j \PI + J 1 1 - J '£i=l P.

= jf ~C + '£!=1 P, - I) I=j '£.=1 P.

for all S ~ N (1= P1 + j -I). If j :S 1 :S j + P1 and 1 :S j :S P2 + ... + Pk -1, then

1 - j :S PI = 2Pl - P1 k

:S 2Pk - PI + (P2 + ... + Pk-1) = 2(LPi) - PI - (P2 + ... + Pk-I) i=2

k

:S 2(LP;) - P1 - j ;=2

and PI + j > PI + j - T, hence (2 L:~=~I p'.-Pl- j) > 1 and ( P+' ~j -1\ > l. Thus, for all -J - PI J- 1 -

sEN we have

( ) _ (S + '£~=1 Pi) ~. (S + '£~=1 Pi - j) W S - k + ~CJ k

'£i=l Pi j=l '£i=l Pi

P2+···+Pk-1 j+PI (+ "k . _ I) ~ ~ b S L...,;=lP, + ~ aj ~ I k j=l I=j '£;=1 Pi


where Cj ~ 1, aj ~ 1, hi ~ 1 (j = 1, ... ,P2 + ... + Pk-l; I = 1, ... ,PI + ... + Pk-l). Therefore,

where'iij EN, 'iij ~ 1 (j = 1, ... ,L:~':}p;). Now, Lemma 2.4.19 shows that w(t) E W. 0

2.4.21. LEMMA. Let kl, ... ,kp EN, k; ~ 1 (i = 1, ... ,p) and (ml, ... ,mp ) E NP. Then ( ) _ (t + kl + ml) (t + kp + mp) W

w t - kl ... kp E.

PROOF. We will proceed by induction on (ml, ... , mp) with respect to the lexi-cographic ordering of NP. If (ml, ... , mp) = (0, ... , 0), then our statement is the direct consequence of Lemma 2.4.20. Let (ml, ... , mp) > 0, i.e. m; > 0 for some i (1::; i::; p). By (2.1.5) we have

Applying the inductive hypothesis, we obtain that

for alII = 0,1, ... , ki , therefore (see Proposition 2.4.14) L:~~o WI(t) = w(t) E W. 0

2.4.22. THEOREM. Let E be a subset ofNm (m E N, m ~ 1) and let Nm = U~=l Uk be a partition of Nm. If WE(tI, ... , tp ) is the dimension polynomial of E associated with the given partition of Nm , then the polynomial w(t) = WE(t + ml, ... , t + mp) belongs to W for any vector (ml' ... ' mp) E NP.

PROOF. We will proceed by induction on m. If m = 1, then there exists a single partition ofN I and (by Lemma 2.4.2 and Proposition 2.4.13) we have wE(t+mt} E W for all ml EN. Let m > 1 and the statement of the theorem holds for the subsets of Nm' with m' < m. In the case of subsets of Nm we will proceed by induction on lEI = L:~=l L:j:l eij (as above, we consider E as a finite set {el, ... , en}, where ei = (e;1, ... , e;m) E Nm for all i = 1, ... , m). If lEI = 0, then either E = {(O, ... , On and WE(t) = 0 E WorE = 0 and w(t) E W by Theorem 2.2.1O(7) and Lemma 2.4.21.


Let lEI> O. By Lemma 2.4.3 there exist a number kEN, k ~ 1, and a vector e = (0, ... ,0,1,0, ... ,0) (kth coordinate of e is equal to one and the other coordinates of e are equal to zero) such that kth column of (eij) l<i<n contains a

l<j<m nonzero element and degwE = degwEue. Applying (2.2.3) to (eij)~<i<n and ewe

l:5j~m obtain

where degwH ~ degwEue and IHI < lEI. It follows that

W(t) = WEue(t + ml,"" t + mp) + Wl(t - 1),

where

Wl(t) = WH(t + (ml + 1), ... , t + (mi-l + 1), t + mi, t + (mi + 1), ... , t + (mp + 1)).

By Theorem 2.2.10(8), there exists a set El ~ Nm - l such that WEue = WE,. Since we also have IHI < lEI, the inductive hypothesis allows to state that the theorem holds for the sets E U e and H, that is WEue(t + ml, ... , t + mp) E W, Wl(t) E W. Applying Theorem 2.4.5, we obtain that w(t) E W. 0

In what follows we shall often consider orderings ~o of the set WP of all numerical polynomials in p variables tl, ... , tp that satisfy the following condition: if Wl(tl, ... , tp) ~o W2(tl, ... ,tp), then

(2.4.5)

for all (al"'" ap ) E NP (by ~ we denote the ordering of all numerical polynomials in one variable t in the sense of Definition 2.4.1).

2.4.23. EXAMPLE. Let Wl(tl, ... ,tp) ~p W2(t l , ... ,tp) mean that there exist numbers Tl, ... , Tp EN such that WI (SI, ... , Sp) ~ W2(S1, ... , sp) for all S1, ... , sp E N such that Sj ~ Ti (i = 1, ... , p). It is easy to see that ~p is an ordering of the set WP that satisfy (2.4.5). In general, this ordering is not linear. Indeed, let N2 = {I} U {2} be a partition of N2 and let (a, b) E N2. Then

for all sufficiently large (SI, S2) E N2. If al > a2 and bl < b2, then the dimension polynomials W(a"b,) (tl, t2) = bl (tl + 1) + al (t2 + 1) and W(a2,b2) (tl, t2) = b2(tl + 1) + a2(t2 + 1) are incomparable with respect to ~p (if tl = SI is a constant, then W(a"b,)(SI,S2) > W(a2,b2)(SI,S2) for all sufficiently large S2 EN, and ift2 = S2 is a constant, then W(a"b,)(SI, 82) < W(a2,b2)(SI, S2) for all sufficiently large 81 EN).

We will show that if an ordering of WP satisfies (2.4.5), then it also satisfies the descending chain condition. First, we shall prove two lemmas.


2.4.24. LEMMA. Let W(tl, ... ,tp) be a numerical polynomial in p variables tl, ... ,tp and let degtjw = kj (j = l, ... ,p). Suppose that w has the follow-ing property: w(-al, ... ,-ap) = 0 for any vector (al, ... ,ap) E NP such that (al, ... ,ap ) ~ (k l +l, ... ,kp +1). Then

PROOF. By Corollary 2.1.5, we can exprerss w in the form

Let us prove that each coefficient ai" ... ,ip in this sum is equal to zero. We will proceed by induction on d = i l + ... + ip . First of all, we have ao ... o = w(-l, ... ,-l) = O. Now, let 0 ~ ij ~ kj for j = 1, ... ,p and let ir < jr ~ kr (1 ~ r ~ p). Then

hence

+ a' . (jl - il - 1) ... (jp - ip - 1) ]l,···,Jp· .

Jl zp

since by the inductive hypothesis aj" ... ,jp = 0 for all coefficients aj" ... ,jp in the latter sum. Therefore, ai" ... ,i p = 0, so the lemma is proved. 0

2.4.25. LEMMA. Let W(tl, ... ,tp) be a numerical polynomial in p variables tl"'" tp and let d = degw. Suppose that the polynomial w has the following property: w(t+al, ... , t +ap) := 0 for all (al, . .. , ap) E NP such that (al, ... , ap) ~ (d+1, ... ,d+1). Thenw(tl, ... ,tp):=O.

PROOF. In view of Lemma 2.4.24, we only have to prove that w(-al, .. . , -ap) = o for all (al, ... ,ap) E NP such that (al, ... ,ap) ~ (d+ 1, ... ,d+ 1). Let j be an index such that aj = maxl<i<p{a;} and let ai = aj - ai (1 ~ i ~ pl. Then o ~ ai ~ d+ 1 for all i = 1,.~.~p so that w(t+al, ... ,t+ap):= O. Therefore, w(-al, ... , -ap) = w(t + al,.'" t + ap)lt=_oj = 0 hence W(tl, ... , tp) := O. 0

2.5. DIMENSION POLYNOMIALS OF SUBSETS OF zm 107

2.4.26. THEOREM. Let ~o be an ordering of the set of all numerical polynomials in p variables which satisfies condition (2.4.5). Then the set WP satisfies the descending chain condition with respect to this ordering, i.e. for any chain

(2.4.6)

(Ei ~ Nm ; where mi EN, mi ~ 1 for i = 1,2, ... ) there exists j EN, j ~ 1, such thatwE;(t1, ... ,tp ) =WE;+I(t1, ... ,tp) = ....

PROOF. First, note that ifwE, (tl, ... ,tp ) ~o WE~(t1, ... ,tp), then degwE1 ~ degwE2' Indeed, by Lemma 2.2.6, all leading coefficients of the polynomials WEll WE. are non-negative, therefore degwE, (t1, ... ,tp ) = degwE, (t, ... ,t). Since condition (2.4.5) holds for the ordering ~o, we have WE, (t, ... , t) ~ WE.(t, ... , t), hence degwE1 ~ degwE. (see Definition 2.4.1). Thus, we may suppose that degwE; = d for any polynomial WE; (j = 1,2, ... ) in chain (2.4.6). Let (a1, ... ,ap) be an element of NP such that ai ~ d + 1 for j = 1, ... ,po By Theorem 2.4.22, each polynomial WE; (t + a1, ... , t + ap) (j = 1,2, ... ) belongs to W, hence chain (2.4.6) implies the chain

w- (t) > w- (t) > ... > w-.(t) > '" E, - E. - - E. -

that becomes stable on some step (see Proposition 2.4.14). Therefore we may suppose that wE;(t+a1, ... ,t+ap) = wE;(t+a1, ... ,t+ap) for all j = 1,2, ... ; i = 1,2, ... and for all (a1,"" ap) E NP with (a1,"" ap) ~ (d+ 1, ... , d+ 1). Now, let us consider the polynomialwi(t1, ... ,tp) = WE;(tl, ... ,tp) -WE;_1(t1, ... ,tp). For each j = 1,2, ... , this polynomial satisfies the conditions of Lemma 2.4.25, therefore Wi = 0, so chain (2.4.6) becomes stable on some step. 0

2.5. Dimension Polynomials of Subsets of ~m

In this section we extend to subsets of ~m (m E ~, m ~ 1) a number of results about the dimension polynomials of subsets of Nm obtained earlier. In what follows, ~_, ~+ and ~_ denote the sets of negative, positive and nonpositive integers, respectively, and the set ~m will be often considered as the union

u ~~m) J

(2.5.1)

where ~im), ... , ~~'.'.!) are all different Cartesian products of m factors, each one of

which is equal either to ~_ or to N. We assume that ~im) = N, and call each of the sets ~)m) (j = 1, ... , 2m) an ortant of ~m.

We shall consider ~m as an ordered set with the following order relation ~: (el, ... ,em ) ~ (ft"",/m) iff (e1, ... ,em ) and (ft"",/m) belong to the same ortant of ~m and led ~ 1/;1 for all i = 1, ... , m.

Let U be a subset of ~m and Nm = U1 U· . ·Uup be a representation (2.2.1) of the set Nm = {I, ... , m} as an union of non-empty disjoint sets U1, ... , up, consisting of k1' ... , kp elements, respectively, (1 ~ kj ~ m, L~=l kj = m). For any vector (Sl,.'" sp) E NP, we denote by U(Sl, ... , sp) the set of all U = (U1,"" um) E U, for which LiEu,lujl ~ Sl, .. . ,L:iEup IUil ~ sp. Furthermore, by Vu we denote the


set of all v = (V1, ... , vm ) E ~m, which do not exceed any element of U with respect to the order:::! (so the inclusion v E Vu is equivalent to the condition that u :::! v is not valid for any u E U).

Consider the mapping, : ~m -7 ri2m which maps any integer vector (Z1, ... , zm) to a vector (a1, ... , a2m) such that

{ max(z;, 0),

aj= max(-zi_m,O), ifi < m

if m < i:S 2m

If U ~ ~m then by ,(U) we denote the set h(u)lu E U} ~ ri2m. Furthermore, for any given partition (2.2.1) of rim we define the partition

assuming 0'; = O'j U (0'; + m) (i = 1, ... ,p) where 0' + m = UjEu(j + m) for any 0' ~ rim.

2.5.1. LEMMA. IfU ~ ~m, then CardU(s1,".,sp) = Card,(U)(s1, .. "sp) and ,(Vu) = V,,/(U) n ,(~m). (The set VE was defined in Section 2.2 before Lemma 2.2.2')

PROOF. The mapping, is injective: if ,(Z1, ... ,Zm) = (a1, ... ,a2m), then IZil = max(ai,a;+m); the condition ai > 0 implies Zj > 0, and ai = 0 implies Zj :S o (i = 1, ... , m). Furthermore, if ,(Z1, ... , zm) = (a1, ... , a2m), then LjEu. IZjl = LjEa;aj for all i = 1, ... ,p, hence -y(U(S1'''''Sp)) = -y(U)(S1'''''Sp). It is also easy to see, that the condition (Z1, ... ,Zm) :::! (h1, ... ,hm) is equivalent to ,(Z1, ... , zm) :S -y(h1' ... , hm), so that -y(Vu) = V,,/(U) n ,(~m). 0

2.5.2. DEFINITION. A subset V ~ rim (a subset V ~ ~m), mE ri,m ~ 1, is called an initial subset of rim (of ~m, respectively) if the inclusion v E V implies Vi E V for any element Vi E rim such that Vi :S v (for any element Vi E zm such that Vi :::! v, respectively).

2.5.3. LEMMA. A subset V is initial in rim (in ~m) iff there exists a finite set E C rim (respectively, a set E C ~m) such that V = VE (respectively, V = 'liE).

PROOF. The fact that VE is an initial set follows immediately from the definitions. Let V be an initial subset of rim (the case of ~m can be treated similarly) and let E be the set of all minimal elements of rim \ V. Then V = VE. Indeed, it is easy to see that v E rim \ V iff there exists Vi E E such that Vi :S v i.e. v ~ VE. 0

2.5.4. LEMMA. If V is an initial subset of~m, then ,(V) is an initial subset of ri2m.

PROOF. By Lemma 2.5.3, every initial subset of ~m has the form 'liE, and by Lemma 2.5.1 -Y(VE) = V,,/(E) n,(~m). Thus, it remainds to prove that -y(~m) is an initial subset of ri2m. Since (a1, ... , a2m) E ,(~m) iff ai . aHm = 0 (i = 1, ... , m), the condition (b 1, ... , b2m ) :S (a1, ... , a2m) E -y(~m) implies (b1, ... , b2m ) E -y(~m). This completes the proof. 0


2.5.5. THEOREM. Let C be a subset of ~m (m E W, m 2: 1) and let Wm = 0"1 U ... U O"p be a partition of Wm into p non-empty disjoint subsets, consisting of k1, ... ,kp elements, respectively. Then there exists a numerical polynomial We (t l, ... , tp) in p independent variables t 1, ... , tp, with the following properties:

(1) W e(Sl, ... , sp) = Card Ve(Sl, ... , sp) for all sufficiently large (S1' ... , sp) E WP;

(2) the total degree deg W c of we(t 1 , ... , tp) does not exceed m, and degt j W c ~ kj (j = 1, .. . ,p);

(3) let /(C) = E C W2m, and let ei E Wm be a vector whose ith and (i + m)th coordinates are equal to 1, and all other coordinates are equal to O. Then We(Sl, ... ,Sp) = Wi;(Sl, ... ,Sp), where E = U~de;}UE and wi; is the dimension polynomial of the set E associated with the partition (*);

(4) there exists a finite sequence {Ejh<j<2m of subsets ofWm such that

~dt" ... ,Ip) ~ ~ WEi (t" ... ,tp) + ['~' (-I)" -'2' (",') (" 7 i) 1

x tl [t(-1)kA-i2i et) eA 7 i)] + ~ 2" ++k. ll, (', t, k, ) . ['.~ 'r -I )'.w' 2' (\+') ('e+: + i) 1

x JJ+Jt(-1)kA-i2iet)eA7i)] + 2m-'. TI ('e te ke) [~'(-I)"-'2' e:) ('p 7 i) l' (2.5.2)

whereWEj(tl, .. " tp) are the dimension polynomials of the subsets E j ~ wm (1 ~ j ~ 2m), associated with the given partition of Wm (see Definition 2.2.6);

(5) if C = 0, then deg Wc = m and

PROOF. By Lemma 2.5.4, /(\1c) is an initial subset of W2m. Therefore (see Lemma 2.5.3) there exists a subset fj; ~ W2m such that /(V£) = Vi;. Now, Lemma


2.5.1 implies the equality Card Ve(Sl' ... , Sp) = Card Vt(Sl, ... , sp) that proves the first statement of the theorem.

We shall find now the set E. By Lemma 2.5.1, ,(ife) = V-y(e) n ,(~m). As the proof of Lemma 2.5.3 shows, we can take the set of minimal elements of N2m \ ,(ife) = (N2m\ V-y(e»)U(N2m\,(~m)) as the set E. It is easy to see that E contains the set of the minimal elements of N2m \ V-y(e) and if bE N2m \ ,(~m), then b ~ e, for some 1 ~ i ~ m. Indeed, if b = (bl , ... , b2m ) ti. ,(~m), then bibi+m "# 0 for some i, hence b ~ e,. Thus, E = E U el U ... U em, so that the third statement of the theorem is proved.

Statement (2) follows from (3) and Corollary 2.2.18, since the coefficients akl, ... ,ki+l, ... ,k p (1 ~ i ~ p) are leading for \lie and they are equal to O. Indeed, for any i = 1, ... , p and for any j E (1" there exists a row of E, such that all coordinates of this row with the indices from (1~ (k "# i) and from (1: \ {j, j + m} are equal to 0 (in fact, ej is such a row).

(4) Without loss of generality, we may assume that the set £ satisfies the following condition: if £ contains an element e = (el,"" ek-l, 0, ek+l, ... , em) (0 ~ k ~ m), then it also contains the element e' = (el, ... ,ek-l,l,ek+1, ... ,em). Indeed, if e E £, but e' ti. £, then the set £' = £ U { e'} satisfies the conditions Vel = Ve and Vel (S1' ... , sp) = Ve( Sl , ... , sp) for all sufficiently large (Sl, . .. , sp) E NP. Thus, \lI e' (t 1 , ... , tp) = \lie (t 1, ... , tp).

Let us denote the set ~m\ Ve = {e E ~m I e ~ e for some e E £} by t, and let us

consider the set ~m as the union of pairwise disjoint subsets: ~m = Ul <j<2m ~Jm), where ~;m) is the Cartesian product of m sets of the form ~+ or Z_. Th;n the set £ can be also represented as an union of pairwise disjoint sets: £ = U1 <j<2m £j,

where £j = £ n ~;m) (j = 1, ... , 2m). Let us put into correspondence -t~ every set £j (1 ~ j ~ 2m) the subset E j ~ Nm, consisting of all elements of Nm, each of which may be obtained from some element of the set £j by replacing all its coordinates by their absolute values. Let Ej = Nm \ VEj (j = 1, ... , 2m). Then Ej = {e E Nm I e ~ e for some e E Ej} (as before, ~ denote the product order on Nm). We shall show that

- "(m) -Card(£(s1, ... ,sp)n~j )=CardEj(sl, ... ,Sp) (2.5.4)

for all sufficiently large (Sl' ... , sp) E NP.

Indeed, suppose that e = (e1,"" em) E t (Sl, ... , sp) n ~Jm), eil>"" e'k E ~+, and all other coordinates of e belong to Z_. By assumption, there exists e = (el,"" em) E £ such that e ~ e, i.e. e and e are in the same ortant ~;m) and lejl ~ leil for every j = 1, ... ,m. In particular, ei" .. ·,eik EN (all other coordinates of e belong to Z_). Of course, some of the elements ei l , ••. , eik may be equal to zero. Replacing such elements by 1, we obtain an element e' E £ such that leU ~ lei ... I (1 ~ v ~ k). Hence, the element lei = (led,···, lemD, that corresponds to e under the one-to-one correspondence between ~;m) and Nm

(m) m-. (~j 3 (e1' ... , em) t+ (leI I, ... , lemD EN), belongs to the set E.1(Sl' ... , sp).

Conversely, suppose that

lei = (le11,···, lem!) E Ej (S1," .,sp),


where e = (e1, ... , em) is the corresponding element of 7I...Jm). Then there exists an

element 1= (h,···'/m) E t'n~Jm) such that III = (lhl,···,l/mi):S lei· Since

I ~ e, we have e E £(Sl, ... , sp) n ~Jm). This implies (2.5.4) for all sufficiently large (Sl' ... 'Sp) ENP.

Let us note now that

- A (m) -Card(t'(Sl, ... , Sp) n 7I...j ) = Card Ej (Sl, ... , Sp)

= CardNm (Sl, ... , Sp) - Card VEj(Sl, ... , Sp)

= IT (Si t. k;) - WEj(Sl, ... , Sp), ;=1 •

for all sufficiently large (Sl, ... , Sp) E NP and for any j = 1, ... , 2m (we use here statement (3) of Theorem 2.2.5 according to which Card Nm (Sl, ... , Sp) W.0" (Sl, ... , Sp) = TIj=l ('j~kj)). It follows that

Since

Card tie( Sl, ... , Sp) = Card [7I...m (Sl, ... , Sp) \ £ (Sl , ... , Sp)] P

= IIp(kj ,sj)-Card£(sl, ... ,Sp) j=l

for all sufficiently large (Sl, ... , Sp) E NP (as before, p(kj , Sj) denotes the number of solutions (Xl, ... , XkJ E 7I...kj of the inequality IX11 + ... + IXkj I :s Sj (j = 1, ... , p); this number can be obtained by (2.l.3) for m = kj , r = Sj), we have


for all sufficiently large (S1, ... , Sp) E NP (here we use the obvious identity

p p

II(aj +bj ) =a1·· .ap +b1 II(aA + bA) j=1 A=2

p-2 p

+L a1 ... aV· bV+1· II (a A+bA)+a1 ... ap-1 bp v=1 v=A+2

for av = 2k~('"tk~), bv = L7~~1(-1)k~-i2i(kt)(S~ti) (1:S l/:S p)). If £ = 0, then £j = 0 and Ej = 0 for any j = 1, ... ,2m. Therefore, in (2.5.2)

we have wEj(i1, ... , ip) = ITj=1 cjtjkj ) (see statement (3) of Theorem 2.2.5), thus

W,,(i 1, ... ,ip) =2m fl Cj ~ kj ) + [}~1(_1)kl-i2i (kn Cl ~ i) 1

X 11 [~(-1)kA-i2iet) eA ~ i) 1

+ ~ 2kl+"·+k~ C1 :1 k1) ". Cv:v kv)

x ["r (-1)""-'2' (k"t) ('"+: + i) 1

x AfI+J~(-1)kA-i2i(kt) (i>.7i) 1

+ [2m -" Ii, ('" :" k") 't,'t -I )"-'2' (k; ) (" : i) 1


The other representations of the polynomial W 0 (t 1, ... , tp), that are mentioned in (2.5.3), follow from identities (2.1.7) and (2.1.8). In this case, obviously, deg W0 = m.

If the set £. contains (0, ... ,0), then 1f£ = 0, therefore Wdt1, ... ,tp) == 0. Conversely, if W £ == 0, then 1f£ = 0, so (0, ... ,0) E t = zm \ 1f£ that implies (0, ... ,0) E£.. The theorem is proved. 0

2.5.6. DEFINITION. The polynomial W £ (t1, ... , tp), whose existence is stated by Theorem 2.5.5, is called Z-dimensional polynomial of the set £. ~ zm, associated with the partition of Nm into non-empty pairwise disjoint subsets U1, ... , up.

For p = 1, Theorem 2.5.5 implies the following result.

2.5.7. THEOREM. For any subset £. ~ zm (m E N, m ~ 1), there exists a numerical polynomial w£(t) in one variable t, with the following properties:

(1) W d s) = Card 1fd s) for all sufficiently large sEN; (2) deg w£ ~ m; (3) there exists a finite sequence {Ej h~j9m of subsets ofNm such that

(2.5.5)

where wEj(t)(1 ~ j ~ 2m) are the Kolchin dimension polynomials of the subsets E j ~ Nm (see Definition 2.2.8);

(4) if [. = 0, then deg w£ = m and

W0(t) = ~(_I)m-i2i (7) C ~ i)

= ~2i(7) G) = ~ (7) C ~i} (2.5.6)

(5) wdt) == ° iff (0, ... ,0) E£..

2.5.8. DEFINITION. The polynomial wdt), whose existence is stated by Theorem 2.5.7, is called a standard Z-dimensional polynomial of the set [. ~ ;;zm.

2.5.9. REMARK. Let E ~ Nm (m E N, m ~ 1) and let a partition of the form (2.2.1) of Nm in non-empty pairwise disjoint subsets U1, ... ,up (Carduj = kj ~ 1 for j = 1, ... ,p) be fixed. Then Theorem 2.2.5 shows that there exists a dimension polynomial WE(t1, ... , tp) associated with the set E. At the same time, the set E may be treated as a subset of zm and we can find its Z-dimensional polynomial WE(t1, ... , tp). It is easy to see that, in general, the polynomials WE and WE do


not coincide. Indeed, if the set E C Nm does not contain elements with zero coordinates, then VE = (/Zm \ Nm) U VE whence

+ Card VE(81, ... , 8p )

for all sufficiently large (81, ... , 8 p ) E NP. Therefore,

k-i i k j tj + l tj + k j P [kj 'J P We(t 1, ... ,tp )=WE(t1, ... ,tp )+}1 ~(-l) J 2 (i)( i ) -}1( kj )

(2.5.7) In particular, the standard /Z-dimensional polynomial of E is connected with the Kolchin dimension polynomial of this set in the following way:

WE(t) = ~(-1)m-i2i(7) C~i) - C:m) +WE(t)

=We(t)+(2m -l)C:m) + ~(-1)m-i2i(7)C~i) (2.5.8)

2.5.10. EXAMPLE. Let m = 2, let N2 = 0"1 U 0"2 be a partition with 0"1 = {I}, 0"2 = {2}, and let E = {e1 = (1,2), e2 = (3, I)}. The corresponding dimension polynomial WE(t 1, t2) = t1 + t2 + 3 has been found in Example 2.2.12. Using relation (2.5.7), we can find /Z-dimensional polynomial of E that corresponds to the given partition N2 = 0"1 U 0"2:

[1 l' .(I)(t1+i)] WE(t 1,t2) =WE(t1,t2) + ~(-1) -'2' i i

x [~(-1)1-i2iG)C2ti)]- C171)(t271)

=t1 + t2 + 3 + [-1 + 2(t1 + 1)] . [-1 + 2(t2 + 1)]- (t1 + l)(t2 + 1)

=3t1t2 + 2t1 + 2t2 + 3.

This result, as it is easy to check, coincides with the polynomial obtained by the direct computation of the cardinality

Card VE (81,82) = Card{ v = (V1' v2)lv E (/Z2 \ N2)

U {(O, 0), (1,0), (2,0), ... ; (0, 1), (0, 2), ... ; (1, 1), (2, I)}, IV11 ~ 81, IV21 ~ 82}

for all sufficiently large (81,82) E N2. Indeed, Card VE (81,82) = Card(/Z2(s1' 82) \ E(S1' S2)), where E = /Z~ \ {(I, 1), (2, I)} is the set of all elements v E /Z2 that


exceed at least one element of E relative to the order <l. Since

Card~2(81,82)= [~(-I)I-i2iG)(817i)l' [~(-I)I-i2;G)(827i)l = (281 + 1)(282 + 1),

and

CardE(81, 82) = 8182 - 2,

Card VE(81, 82) = (281 + 1)(282 + 1) - (8182 - 2) = 38182 + 281 + 282 + 3

for all sufficiently large (81,82) E N2,

WE(tl, t2) = 3tlt2 + 2tl + 2t2 + 3.

Let E be a subset of ~m (m EN, m ~ 1). As above, for any lEN, 0 ~ l ~ m, we shall denote by A(l, m) the set of alll element subsets of Nm = {I, ... , m}. If e E A(l, m), we shall denote by E{ the set of all elements of E, whose coordinates with indices from e are equal to zero. Furthermore, we shall denote by t{ the subset of ~m-l, obtained by omitting in every element of E{ coordinates with indices from

e (these coordinates are equal to zero). Let t{j = t{ n~)m-l) (j = 1, ... , 2m - I ) and let E{j be the subset of Nm-l consisting of elements e E N m - 1, each of which can be obtained from an element c E t{j by replacing its coordinates by their absolute values.

The following result is similar to relation (2.5.5); it can be used for computing Z-dimensional polynomial of E.

2.5.11. PROPOSITION. With the preceding notation and with a fixed partition Nm = 0"1 U··· U O"p ofNm into non-empty pairwise disjoint subsets 0"1,.'" O"p, we have

m 2m - I

Wdtl, ... ,tp ) = LC-I)l L L WE(j(tl, ... ,tp ) (2.5.9) 1=0 j=1 (EA(I,m)

where WE(j (tt, ... , tp ) is the dimension polynomial of E{j ~ N m - 1, (see Definition 2.2.6) that corresponds to the partition of Nm-l into disjoint subsets O"~ = 0"1 \

e, ... , O"~ = 0" P \ e (if one of the sets O"~ is empty then the polynomial WEb does not depend on the corresponding variable t;).

PROOF. Obviously, in order to prove (2.5.9)' it is sufficient to show that

m 2m - I

CardVd81, ... ,8p)=L(-I)IL L WE(j(81, ... ,8p ) (2.5.10) 1=0 j=1 (EA(I,m)

for all sufficiently large (81, ... , 8 p ) E NP. To do it, we compute how many times every element a = (al,"" am) E ~m is counted in the right-hand side of relationship (2.5.10). In this process we have to remember that for all sufficiently large (8t, ... ,8p ) ENP,


and the latter quantity is equal to the number of elements b = (b1 , ... , bm ) E ~m, with the following properties:

(1) LiEu; Ibil ~ Sj (j = 1, ... ,p); (2) coordinates of b with indices in ~ are equal to zero; if we omit these coordi

nates we obtain an element of ~}m-I) that does not exceed (relative to the

order ~) any element of the set t~j. Let us consider the following two cases: 1) The inequality e ~ a does not hold for any element e E e. In this case a E lie

and a E lie (S1 , ... , sp) for all sufficiently large (S1' ... , sp) E NP. Suppose that a

has r zero coordinates (0 ~ r ~ m), and let ~ be a subset of Nm consisting of 1 elements (0 ~ 1 ~ r) such that the coordinates of a with indices from ~ are equal to zero. If we omit these zero coordinates then we obtain an element a~ E ~m-I, belonging, obviously, just to 2r - 1 ortants of the form ~jm-I) (because exactly r-I

coordinates of a~ are equal to zero). If these ortants correspond to the sets t~j and E~j (the latter set is obtained from the set t~j using the method described above), then a~ is counted in each corresponding summand WEe; (S1' ... , sp) (for sufficiently large (S1,"" sp) E NP) just one time. Since there are precisely (~) I element subsets ~ such that coordinates of a with indices in ~ are equal to 0, each element a E lIe(S1, ... ,sp) is counted in the right-hand side of (2.5.10) just L~=o ( _1)1 (D 2r - 1 =(2 - 1 r = 1 times.

2) Suppose that there exists an element e E e such that e ~ a. Then e is contained in the same ortants ~}m) which contain a. Suppose that a has r zero coordinates (0 ~ r ~ m) at the places i1 , ... , ir . Then e has also zeros at the same places. If ~ ~ {il, ... , ir }, Card ~ = 1 and e~, a~ are elements of ~m-I obtained by omitting zero coordinates with indices from ~ in e and a, respectively, then e~

is contained in the same 2r - 1 ortants ~)m-I) as a~. It implies that the element of Nm - I , whose coordinates are equal to the absolute values of the corresponding coordinates of a~, does not belong to VEE; for any set Eej, i.e. a is not counted in the right-hand side of (2.5.10).

Thus, the right-hand side of (2.5.10) is equal to Card lIds1' ... , sp) for all sufficiently large (S1' ... , Sp) E NP, that implies (2.5.9). The proposition is proved. 0

2.5.12. EXAMPLE. Applying (2.5.9) to the computation of ~-dimensional polynomial WE(t1, t2) of the set E = {e1 = (1,2), e2 = (3, I)} ~ ~2 in Example 2.5.10 (with fixed partition N 2 = (71 U (72 of N 2 such that (71 = {I}, (72 = {2}), we obtain

2 2(2-1)

WE(S1,S2) = ~)-I)1 L L WEE; (Sl, S2) (2.5.11) 1=0 j=1 ~EA(I,2)

for all sufficiently large (S1' S2) E N2 .

In sum (2.5.11) the value I = 0 corresponds to only one set ~ = 0 E A(0,2). In this case (7~ = (71, (7~ = (72, te = e, t~1 = e, t~j = 0 for j = 2,3,4, hence WEE1 (S1' S2) = S1 + 82 + 3 (see Example 2.2.12),

(j = 2,3,4).


The value l = 1 in the sum (2.5.10) corresponds to two subsets of A(l, 2) : e(l) = {I} and e(2) = {2}. For these subsets we have ui = 001 \e(l) = 0, u~ = 002 \e(l) = 002' and ui = 001, 00; = 0, respectively. Therefore, tf. = 0, tf.j = 0 (j = 1,2), hence WEE(1)j(Sl,S2) = ("2t1), WEE(2)j(Sl,S2) = ("lt1) (j = 1,2). Furthermore, the value l = 2 in the sum (2.5.10) corresponds to the subset e = {I, 2} E A(2, 2), for which ui = u~ = 0, so we have only one summand WEEi (Sl, S2) = l.

Thus,

WE (S1, S2) = [3 C1 7 1) (S2 7 1) + (Sl + S2 + 3)] - 2 [ (S2 7 1) (Sl 7 1)] + 1

= 3s1 S2 + 2S1 + 2S2 + 3

for all sufficiently large (Sl, S2) E W2, hence W E(tl, t2) = 3t1t2 + 2t1 + 2t2 + 3. The same result was obtained earlier in Example 2.5.10 by using formula (2.5.4).

Let E S; zm and let the set EnZ;m) be non-empty for every ortant Z;m) (1 ~ j ~ 2m). We denote by E' the subset of zm, obtained by adjoining to E the elements of the form e' = (C1' ... ' 1, ... , em) corresponding to C = (Cl, ... , 0, ... , cm) E E (elements e' and e have 0 and 1 at the same place i (1 ~ i ~ m)). Let us fix a partition of Wm into non-empty pairwise disjoint subsets 001, ... , Up. Then, as we mentioned in the proof of Theorem 2.5.5, WC(t1, ... ,tp) == WCI(t1, ... ,tp) and,

obviously, E' n Z;m) -::j:. 0 for any j = 1, ... , 2m. Let Ej be the subset of wm, consisting of all elements of the form (le11. ... , lem I), where

( ) E' 7l(m) e1, ... , em E n fL.j

Then Ej -::j:. 0, therefore degwEj < m for all j = 1, ... , 2m (see Theorem 2.2.5(3)). In this case relation (2.5.2) implies degwc = degwcl < m. In particular, if E n

Z;m) -::j:. 0 for every j = 1, ... , 2m, then the standard Z-dimensional polynomial of

E may be represented in the form wc(t) = L:~~1 aieti), where ao, ... , am-1 E Z. The following proposition gives the form of the leading coefficient a m-1 of such

a polynomial when m = 2.

2.5.13. PROPOSITION. Let E be a subset of Z2 whose elements are pairwise incomparable (relative to the given above order ::9), and let T be the number of elements of E with at least one zero coordinate. Suppose that each of the sets Ej = E n zY) (j = 1, ... ,4) is non-empty and Ej = {gj1, ... , gjn(j)}, where n(j) E N, n(j) 2:: 1, gj£l = (ej£ll, ej£l2) (j = 1, ... ,4, 1 ~ 1/ ~ n(j)). Furthermore, let ej1 = minlS£lSn(j){lej£lll}, ej2 = min1S£lSn(j){lej£l21}, and let ej denote the element (Cj1, Cj2) E Z2. Then

where

Wc(t) = at + b,

4

a = ~)ej1 + ej2) - 4 + T.

j=l


PROOF. Theorem 2.5.5 shows that there exist E1,"" E4 ~ N2 such that

4 1 (2) (t .) 4 We(t) = ~WEi(t) + t;(-1)2-i2i i ~ 1 = ~WEi(t) - 4t - 3.

We also know that Ej = {(led, hI) I (el, e2) E £ n zyl} for every j = 1, ... ,4 (as before, the different Cartesian products of two sets of the form;:Z+ or Z_ are denoted

by zyl (1 :S j :S 4)). If £ = {e E ;:z2 I e ~ e for atleast one element e E £} = ;:z2\ V£, then (see Theorem 2.2.5) there exist numerical polynomials pj(t) (j = 1, ... ,4) in

one variable t such that pj(s) = {z = (Zl,Z2) E £ n ;:z)2l I IZ11 + IZ21 :S s} for all sufficiently large sEN. Proposition 2.1.3 implies that polynomial Pj (t) may be represented in the form pj(t) = e~2) -[(ej1 + ej2)t + bj]' where bj E;:Z (1 :S j :S 4).

Obviously Ej ~ £ n ;:z?l (1 :S j :S 4), hence W£nz(2)(t) :S WEj(t) and J

2:;=1 W£nz(2)(t) :S 2:;=1 WEj(t) (if £ contains no elements with zero coordinates J

then ":S" in this inequality becomes in fact "="). Suppose that an element of the form (e,O) belongs to £ (we may assume that e =j:. 0, because £"# 0 and the elements of £ are pairwise incomparable, so that (0,0) fi. £). This element belongs to two different sets £j and £k, hence for all sufficiently large sEN elements of

£(s) n;:z)2ln;:z~2l are counted in the process of computing Card(£(s) n;:z?l) = pj(s)

as well as in computing Card(£(s) n ;:z~2l) = Pk(S). The number of such elements is equal to (s + 1) - lei- Thus, for all sufficiently large sEN, we have we(s) = Card(;:z2(s)) - (2:;=1 pj(s) - r(s + 1) + b) (b E ;:Z), that implies

We(t) = t(-1)2-i2i G) C ~ i) - {~ [(' ~ 2) _ ('id 'i,)1 +bi]- ,(1+ 1) +b}

= [t(ej1 +ej2 )-4+r] t+c, J=l

where c = 1 - b - 2:;=1 bj E ;:Z. The proposition is proved. 0

2.5.14. REMARK. Let £ c ;:zm and let F denote the set of all minimal elements of £ (relative to the order ~). Obviously, F is finite, its elements are pairwise incomparable and V£ = VF. Therefore, if W e(t1, ... , tp ) and W F (t1, ... , tp ) are ;:zdimensional polynomials of £ and F, respectively (that correspond to a partition of the form (2.2.1) ofNm), then we(t1,,,.,tp ) == WF(tl, ... ,tp )' This, in particular, implies, that Proposition 2.5.13 can be used for computation of the leading coefficient of the standard ;:Z-dimensional polynomial of the set £ C ;:z2 even in the case when £ contains comparable elements (under the condition '[ n ;:z?l =j:. 0 for j=1, ... ,4).


2.5.15. EXAMPLE. Compute the standard Z-dimensional polynomial of the set H = {(I, 3); (2,1); (0,5); (-2, 0); (I, -2); (2,2); (2, -3); (O, -3); {-4, In.

First of all, we replace H by the set of all its minimal elements i.e. by the set £ = {(1,3);(2,1);(0,5); (-2,0);(I,-2);{0,-3n (it has been mentioned that such a replacement does not change ~-dimensional polynomial). Let ~)1) = N2,

(2) - (2) - - (2) - . . ~2 = ~- x N, ~3 = ~- x ~_, ~4 = N x ~_. Then (w1th the notatlOn of Proposition 2.5.13) we have: £1 = {(I, 3); (2, 1); {O, 5n, £2 = {(O, 5); {-2, On, £3 = {(-2,0);(0,-3n, £4 = {(0,-3);(I,-2n· Hence, ell = min{I,2,0} = 0; e12 = min{3, 1, 5} = 1; e21 = min{0,2} = 0; e22 = min{5, O} = 0; e31 = min{2,0} = 0; e32 = min{O, 3} = 0; e41 = min{O, I} = 0; e42 = min{3, 2} = 2. Since £ n ~y) #o for j = 1, ... ,4, Proposition 2.5.13 implies that the standard ~-dimensional polynomial of £ has the form we(t) = at + b, where a = Z=;:=1(ej1 + ej2) - 4 + r, b E ~ (r is the number of elements of £ with a zero coordinate; in our case r = 3). Thus a = 3 - 4 + 3 = 2, i.e. WH(t) == 'lic(t) = 2t + b for some b E ~.

In order to obtain the explicit expression of the polynomial wc(t) (hence, of 'liH(t)), we use (2.5.9):

2 22 - 1

Wc(t) = L)-I)1 L L WE(j(t) (2.5.12) 1:=0 j:=l {EA(I,2)

The value l = 0 corresponds to the set e = 0 C A(0,2). It is easy to see that t{ = £ (with the notation of Proposition 2.5.11) E{l = {(I, 3); (2, 1); {O, 5n, E{2 = {(O, 5); (2, On, E{3 = {(2, 0); (0, 3n, E{4 = {(O, 3); (1, 2n. We can find the Kolchin dimension polynomials WE(j(t) (1 ~ j ~ 4) using relation (2.2.6). For the Kolchin polynomial of the set E{l we have (with the notation of Proposition 2.2.11)

(2.5.13)

Here elll = (0,0), and if 8 #- 0, then eo = (en, e02) is an element of N2 such that eOi = maXjEo{eji} (i = 1,2) where {eji}jEO is the set of ith coordinates of elements of Eel belonging to Eo. The value q = 0 in the sum (2.5.13) corresponds to the set Eo = 0, for which eo = (0,0). The value q = 1 corresponds to the sets EOll = {(1,3n, E012 = {(2, In, E013 = {(0,5n hence eO ll = {(1,3n, e012 = {(2, In, e013 = {(0,5n· Furthermore, A(2,3) = {82l , 822 , 823} where EO' l = {(I, 3), (2, In, Eo •• = {(I, 3), (0, 5)}, EO'3 = {(2, 1), (0, 5n. For these sets we obtain eO. l = (2,3), eo •• = (1,5), e0 23 = (2,5), respectively, and the value q = 3 in (2.5.12) corresponds to the set E03 = E{l, for which e03 = (2, 5). Thus,

WE(l (t) = e ~ 2) - [(t + ~ - 4) + e + ~ - 3) + e + ~ - 5)]

+ [e+~-5) + e+~-6) + C+~-7)] - C+~-7) =t+7. Similarly we find the Kolchin polynomials of the sets E{2, E{3 and E{4:


W{2(t) = t(-I)q L (t + 2 - (~91 + e92))

q=O 9EA(q,2)

= (t ~ 2) _ [e + ~ - 5) + e + ~ - 2)] + e + ~ - 7) = 10,

2

W{3(t) = L( -1)q L (t + 2 - (~91 + e92))

q=O 9EA(q,2)

Therefore, the value 1 = 0 in sum (2.5.12) corresponds to the summand (t + 7) + 10 + 6 + (2t + 2) = 3t + 25.

The value 1 = 1 in the inner sum of (2.5.12) corresponds to the summands associated with the sets e(l) = {I} E A(I, 2) and e(2) = {2} E A(I, 2). In this case we have £{(1) = {(O, 5); (0, -3n, t{(1) = {(5, -3n, E{(I)l = {5}, E{(1)2 = {3}, hence WEE(lh (t) = 5 (because VEE(1h = {O, 1, ... , 4}) and WEE(1), (t) = 3 (VEE(I)' = {O, 1, 2}). We also obtain £{(') = {( -2, On, t{(.) = {-2}, E{(2)1 = 0, E{(2)2 = {2}, hence WEE(2l! (t) = Cil) = t + 1, WEE(2),(t) = 2, (because VEE(2), = {O, I}). Therefore, the value I = 1 in the sum (2.5.12) corresponds to the summand - [5 + 3 + (t + 1) + 2] = -(t + 11).

Furthermore, if 1= 2, then A(l, 2) = {(I, 2n, £{ = 0 and the value 1 = 2 in sum (2.5.12) corresponds to the summand that is equal to 1 (as usual, we consider 1Z0

as {O}). Now, it follows from (2.5.12) that qie(t) = (3t + 25) - (t + 11) + 1 = 2t + 15. Now, let us find the same polynomial qie(t) with the help of formula (3) of

1 0 1 0 100 1 1300 2 1 00 0500

Theorem 2.5.5: qie(t) = qiE(t), where E = 0020 . It is clear that the vectors 1002 2200 2003 0003 0140

(2200), (2003), (0140) are superfluous and may be omitted. Applying formula

(2.2.3) to E and (0100), we get WE(t) = WEI (t) +WH(t - 1), where El = (~~ ~) , 003

(101) 1 20

H = 200 040 002


In order to compute WE" and WH, we once again apply formula (2) in which v is equal to (100). We obtain

Let cI> denote the set of all standard ~-dimensional polynomials of subsets of ~m (m = 1,2, ... ), i.e. the set of all numerical polynomials in one variable t which may be represented in the form qi e (t) for some t: ~ ~m. Since cI> ~ W (see Theorem 2.5.5(3)), Proposition 2.4.14 implies the following statement.

2.5.16. COROLLARY. The set cI> is well-ordered relative to the order introduced in Definition 2.4.1.

2.5.17. PROPOSITION. Any Kolchin dimension polynomial WE, E ~ Nm , is equal to a standard ~-dimensional polynomial of some set t: ~ ~m, i.e. W = cI>.

PROOF. Let t: ~ ~m, where t: = Eu{hdu" ·U{hm }, hi = (0, ... , -1, 0, ... ,0), i = 1, ... , m (the set E is considered as a subset of ~m). By Theorem 2.5.5(3), we have qie(t) = wE(t), where

m m

'E' ------.. 0 1 0

E= 0 0 1

1 0 1 0

0 1 0 1

Clearly, the last m rows of E are superfluous, and Theorem 2.2.10(5) shows that wE(t) = WE(t). 0

Now, Propositions 2.4.14 and 2.5.17 imply the following result.

2.5.18. COROLLARY. The set cI> is closed relative to addition.

CHAPTER III

BASIC NOTIONS OF DIFFERENTIAL

AND DIFFERENCE ALGEBRA

3.1. Rings with Operators

Let R be a ring and let ~ be a set of operators acting on R. In this case R is said to be a ~-ring and ~ is called its basic set of operators. In the following sections the operators in ~ will be either derivation operators or endomorphisms but now we do not impose any restrictions on ~.

If a ~-ring is a field, it will be called a ~-field. An ideal I of a ~-ring will be called a ~-ideal, if I is closed under the action of any operator from~. If a ~-ideal I is a prime ideal (i.e. a, b r!. I implies ab r!. I), then we shall speak about a prime ~-ideal. If a ~-ideal I is a perfect l ideal (i.e. an E I implies a E I), then we shall speak about a perfect ~-ideal. We have to be careful while speaking about maximal ~-ideals: they are maximal elements in the set of all proper ~-ideals (generally speaking, a maximal ~-ideal is not a maximal ideal in the usual sense).

Let Rand R' be ~-rings with the same basic set of operators~. A ring homomorphism <p : R -t R' will be called ~-homomorphism if <p commutes with every operator from ~, i.e. d<pb = <pdb for any b E Rand d E ~2. The notions of ~-monomorphisms, ~-epimorphisms, ~-endomorphisms, ~-isomorphisms, ~automorphisms and so on are defined in the natural way (as the corresponding ring homomorphisms which are ~-homomorphisms).

Let R be a ~-ring and R' a subring of R that is closed with respect to the action of any operator from~. In this case R' is a ~-ring that is called a ~-subring of Rj the ring R is called a ~-overring of R'. If, in addition, Rand R' are ~-fields, we say that R' is a ~-subfield of Rand R is a ~-overfield of R' or a ~-extension of R'. Sometimes (it will be clear from the context) we shall speak about ~-extensions when Rand R' are ~-rings.

Let Ro be a ~-subring of a ~-ring R and let B be a subset of R. Since the intersection of any set of ~-subrings of R is a ~-subring, there exists a smallest ~-subring of R containing Ro and Bj it is called ~-subring generated by B over Ro and is denoted by Ro{B}. In this case B is said to be a set of generators of the ~-ring Ro{B} over Ro. A ~-overring of a ~-ring Ro is said to be finitely generated over Ro if it has a finite set of generators over Ro.

lSometimes such ideals are called radical. 2More precisely, we consider ti-ring R and ti'-ring R' such that there is a bijective mapping

v : ti ""i- ti' (so that Card ti = Card ti'). With this notation, a ti-homomorphism cp : R ""i- R' is a ring homomorphism such that cp(fib) = v(fi)cp(b) for every b E R. However, it is reasonable (and convenient) to identify elements 5 E ti with the appropriate elements 1'(5) E ti' and write the basic set of R' as ti. In what follows we will keep this convention.

123


124 III. BASIC NOTIONS OF DIFFERENTIAL AND DIFFERENCE ALGEBRA

Let Fa be a .6.-subfield of a .6.-field F and let B be a subset of F. Since the intersection of any set of .6.-subfields of F is a .6.-subfield of F, there exists the smallest .6.-subfield of F containing Fa and B. We denote this .6.-field by Fa(B) and call it the .6.-field obtained by adjoining the elements of B to Fa, or the extension of Fa generated by B; the set B is called a set of generators of the extension Fa(B) over Fa. An extension is said to be finitely generated if it has a finite set of generators; an extension is said to be simply generated if it has a set of generators consisting of a single element.

Let R be a .6.-ring. Suppose that the set .6. is finite and that operators from .6. pairwise commute, i.e. o;ojb = ojo;b for any b E Rand 0;, OJ E.6.. Suppose also that for any 8 E .6. there exist functions 'PI, 'P2 : R -+ R such that

(3.1.1)

We shall write it as

O· (a id) = 'Pda)o + 'P2(a) id, (3.1.1')

where id is the identical operator on R. Let R[.6.] denote the ring of operators on R generated by operators of the form a id, where a E R, and by operators from .6.. If T = T(.6.) denotes the free commutative unitary semigroup generated by elements from .6., then every element of R[.6.] can be espressed as a finite sum LOET aoO; the product of two such sums are uniquely defined by (3.1.1'). The ring R[.6.] will be called the ring of .6.-operators over R. Any left module over R[.6.] will be called .6.-R-module . .6.-R-submodules, .6.-R-homomorphisms, etc., are defined as R[.6.]-submodules, R[.6.]-homomorphisms, etc.

Let R be a .6.-ring, .6. = {d1 , ... , dm }, and T = T(.6.) the free commutative unitary semigroup with generators d l , ... , dm . Furthermore, if 0 = d~l ... d~ E T (iI, ... ,im E N), then let ord 0 = i 1 + ... + im. The following definition describes a construction that plays the key role in differential and difference algebra.

3.1.1. DEFINITION. The ring of polynomials R{y} = R{Yl, ... , Yr} in denumerable set of variables {OYj }OET, l~j~r over R is called a .6.-polynomial ring over a .6.-ring R. The ring R{YI, ... , Yr} is naturally considered as a .6.-ring where the action of the operators d E .6. on the coefficients of .6.-polynomials is defined as in the ring Rand d(OYj) = (dO)Yj for any indeterminate OYj (0 E T, 1 ~ j ~ r). The elements of R{YI, ... , Yr} are called .6.-polynomials. By the definition, the degree of f E R{YI, ... , Yr}, deg f is the degree of f as a polynomial in variables {OYj }OET, 19~r, and the order, ord f, is the maximal order of operators 0 E T contained in f: if f = LOET,I~j~r aOjOYj, then

ord f = max {ord O}. OET, I~j~r, a8j;>!a

A .6.-polynomial ring, from the viewpoint of the ring theory, is a commutative polynomial ring in a denumerable set of variables (every fixed polynomial depends only on a finite set of variables). At the same time, a .6.-polynomial ring is a .6.-ring, where operators from .6. map the set of ring generators into itself.

3.2. BASIC NOTIONS OF DIFFERENTIAL ALGEBRA 125

3.2. Basic Notions of Differential Algebra

As usual, if the contrary is not said explicitly, all rings are supposed to be commutative.

3.21. DEFINITION. An operator d on a ring R is called a derivation opemtor (or differentiation) iff d(a + b) = da + db and d(ab) = (da)b + adb for all a, bE R.

In this case the element d(a) is called a derivative of a. Denoting the successive derivatives by d2 (a), ... , dn(a), ... , we obtain by induction on n the Leibniz rule

It is easy to see that d(an) = nan-1d(a), d(l) = 0, and d(a- 1) = -d(a)(a- 1)2.

3.2.2. EXERCISE. Prove that any differentiation d of a commutative integral domain R can be extended to the differentiation of the corresponding quotient field. Show that this extension is unique and it is determined by the formula d( f!) _ bd(a)-ad(b)

b - b2 .

3.2.3. DEFINITION. A commutative ring R with a finite set ~ = {d1 , ... , dm }

of mutually commuting derivation operators on R is called a differential ring. The set ~ is called the basic set of derivations (or the basic set of differentiations) of R.

Below we shall call a differential ring with a set of derivation operators ~ by a ~-ring. If Card ~ = 1 then the ~-ring R is called ordinary and the derivative of y is denoted y'. If Card ~ > 1 then the ~-ring R is called partial. We shall often use the symbol ~ instead of adjective "differential". The notions of ~-homomorphism, ~-ideal, ~-field, ~-subring, ~-subfield and ~-extension are defined in accordance with the corresponding definitions in Section 3.1. The elements a E R such that d(a) = 0 for any d E ~ form a ring that is called the ring of constants of the ~-ring R.

3.2.4. EXAMPLE. Any commutative ring may be considered as a differential ring with zero derivation operator. Every ideal of such a ring is differential. The rings :£:, Q, :£:n have no other derivation operators.

3.2.5. EXERCISE. Show that the ring of univariate polynomials R[x] over an ordinary differential ring with a derivation operator d can be transformed into an ordinary differential ring by assigning an arbitrary fixed value to d(x). Prove that the value of d(x) uniquely defines the extension of d to R[x] and show that if R is a field of zero characteristic and d(x) = 1, then R[x] contains no nontrivial ~-ideals.

3.2.6. EXAMPLE. The ring of infinitely differentiable functions of one real variable x is an ordinary differential ring with the derivation operator d/ dx.

3.2.7. EXAMPLE. Let F be a field of functions of n real variables Xl, ... , xn that are meromorphic in a region of the real n-dimensional space. Then F can be considered as a differential field with the set of derivation operators ~ = {a/axil (i = 1, ... , n).


3.2.8. DEFINITION. Let R be a ~-ring and {fa}aE/ be a system of its elements. Then the smallest ~-ideal of R containing all elements fa is called a ~-ideal of R generated by the family {fa}aEI and is denoted by [{fa}aE/] or [fa]aEI. The smallest perfect ~-ideal containing all elements fa is called a perfect ~-ideal of R generated by the family {fa}aEI and is denoted by {{fa}aEr} or by {fa}aE/. (As usual, the smallest ideal of R containing the family {fa }aE/, is denoted by ({fa}aEI) or (fa)aEI.)

The radical of a differential ideal is not necessary a differential ideal as the following example shows.

3.2.9. EXAMPLE. Consider the derivation operator d of the ring Z2[X,y] such that dx = y and dy = O. Let J = [x 2]. The equality d(x 2) = 0 implies J = (x2) but at the same time rad J = (x) and y f/:. rad J.

The following example shows that a maximal ~-ideal is not necessary prime.

3.2.10. EXAMPLE. Consider the derivation operator d of the ring Z2[X] such that dx = 1. As in the previous example, let J = [x2] = (x 2). If J C J for some ~-ideal J f:. Z2[X] then J contains an element f = ao + x (ao E Z2). Obviously, df = 1 E J, so that J is a maxial ~-ideal that is not prime.

However, if the ring R contains the field of rationals Q, then the radical of any differential ideal is a differential ideal. In particular, {fa}aEI = rad[Ja]aEI for any family of elements of such ring (see Theorem 3.2.12).

3.2.11. DEFINITION. A differential ring is called Ritt's algebra if it contains the field of rationals Q (which is a subfield of the ring of constants).

3.2.12. THEOREM. The radical of any ~-ideal in Ritt's algebra is a ~-ideal.

This theorem is a consequence of the following lemma.

3.2.13. LEMMA. Let I be a ~-ideal of a Ritt's algebra R and a be an element of R such that an E I. Then (d(a)fn-l E I for any derivation operator d E ~.

PROOF. Proceeding by induction on k we can prove that

for k = 1, ... , n. Indeed, the statement holds for k = 1 because

and the ideal I is invariant under multiplication by lin, hence an-1(d(a» E I. Suppose that an - k(d(a»2k-l E I. Then

d(an- k(d(a»2k-l) = (n - k)an - k- 1(d(a»2k + an- k(2k - 1)(d(a»2k-2d(d(a» E I.

Multiplying this relationship by (d(a»2 and applying the inductive hypothesis, we obtain that (n - k)an- k- 1d(a)2k+l E I whence an- k- 1 (d(a»2k+l E I. Thus, the step of induction is fulfilled. If k = n, the formula we just proved gives us the statement of the lemma. 0

Now, we are going to prove the following theorem.


3.2.14. THEOREM. Every perfect differential ideal I of a differential ring R can be represented as an intersection of prime differential ideals.

First we prove some lemmas.

3.2.15. LEMMA. If the product of elements a and b of a Ll.-ring R belongs to a perfect differential ideal I, then d(a)b E I and ad(b) E I for any dEll..

PROOF. By assumption, we have d(ab) = d(a)b + ad(b) E I for any dEll.. Multiplying d(a)b + ad(b) by ad(b) we obtain (ad(b))2 E I, hence ad(b) E I. 0

3.2.16. LEMMA. For any perfect differential ideal I and for any subset S of a differential ring R, the set T of all x E R such that xS ~ I is a perfect differential ideal of R.

PROOF. It is an easy exercise, to prove that T is an ideal. Furthermore, Lemma 3.2.15 implies that the ideal T is differential, so it remains to show that T is perfect. Let xn E T. Then xn s E I for any s E S, hence xn sn E I. Since I is a perfect ideal, xs E I for all s E S, hence x E T. 0

3.2.17. LEMMA. a{S} ~ {as} for any element a and any subset S of a differential ring.

PROOF. Let T be the set of all elements x of R such that ax E {as}. By Lemma 3.2.16, T is a perfect differential ideal. It is evident that this ideal contains S, hence it contains the ideal {S}. 0

3.2.18. LEMMA. {S}{T} ~ {ST} for any subsets Sand T of a differential ring.

PROOF. By Lemma 3.2.17, the set of all x, for which x{T} ~ {ST}, contains S and (see Lemma 3.2.16) is a perfect differential ideal. Hence, this set contains is}. 0

3.2.19. LEMMA. Let T be a multiplicatively closed subset of a differential ring R, and let q be a maximal perfect differential ideal disjoint with T. Then the ideal q is prime.

PROOF. Let, in contradiction with the statement, there exist elements a, b E R such that ab E q, a ~ q, b ~ q. Then {q, a} and {q, b} are perfect differential ideals strictly containing q, so that there exist elements tl E {q, a} n T and t2 E {q, b} n T. By Lemma 3.2.18, we obtain tlt2 E {q, a}{ q, b} ~ q that contradicts the assumption. 0

PROOF OF THEOREM 3.2.14. It is sufficient to construct for any x ~ I a prime differential ideal which contains the ideal I, but does not contain x. Let T be the set of all powers of x. By Zorn's Lemma there exists a maximal perfect differential ideal q which contains I and does not contain any element of T. By Lemma 3.2.19 the ideal q is prime. 0

3.2.20. COROLLARY. Any maximal Ll.-ideal of a Ritt's algebra is prime. The nilradical of a Ritt's algebra is the intersection of all prime Ll.-ideals.

PROOF. Let M be a maximal Ll.-ideal of Ritt's algebra R. Then the inclusions M ~ {M} ~ R hold. Since M is a maximal Ll.-ideal, we have either M = {M},


or {M} = R. By Theorem 3.2.12, the ideal {M} coincides with the radical of M. Hence, either M is a perfect ideal, or the radical of M contains 1, that means M = R. Therefore, M is a perfect differential ideal of R, which (by Theorem 3.2.14) may be represented as an intersection of prime Ll-ideals. The maximality of M implies, that this intersection is trivial, i.e. M is prime. The second assertion of the corollary can be obtained from Theorems 3.2.12 and 3.2.14. 0

If differential rings we consider are differential Ritt's algebras then many constructions of algebraic geometry can be realized in differential case. In particular, the set Ll-Spec A of prime Ll-ideals of A may be considered as a topological space, if we define the closed sets as the sets of the form V (I), where V (I) is the set of all prime Ll-ideals containing Ll-ideal I of A. Some properties of this space are similar to the properties of Spec A. However, the ring A cannot be always reconstructed from the structural sheaf on Ll-SpecA (see [CF90]).

In general case (Q 1:. A), while studying the problem of constructing differentialalgebraic schemes, W. Keigher [Keig73j proposed to consider the class of quasi prime differential ideals instead of prime ideals. (An ideal J of a ring A is called quasiprime if there exists a multiplicative subset S of A such that J is maximal among the differential ideals disjoint with S.) In many aspects the class of quasiprime differential ideals is similar to the class of prime ideals in commutative polynomial rings.

Let R be a differential ring with a basic set Ll = {d l , ... , dm} and let T = T(Ll) be the free commutative semigroup with generators d l , ... , dm . If 0 = d~l ... d!;; E T, (i I , ... , im ) E Nm , we set ord 0 = i I + ... + im .

3.2.21. DEFINITION. The ring of polynomials R{y} = R{YI, ... , Yn} in a denumerable set of variables {OYj }SET, I~j~n over a differential ring R is called differential polynomial ring (Ll-polynomial ring) over R. Its elements are called differential polynomials. The derivation operators of Ll act on the coefficients of a differerntial polynomial as in the differential ring R, and their action on the generators OYj is defined as follows: d(OYj) = (dO)Yj. By the definition, the degree, deg I, of of a Ll-polynomial I E R{YI, ... , Yn} is the degree of I as a polynomial in variables {OYj }SET, I~j~n, and its order, ord I, is the maximal order of derivatives actually present in I.

From the viewpoint of the ring theory, a differential polynomial ring is a commutative polynomial ring in a denumerable set of variables (every fixed polynomial depends only on a finite set of variables). At the same time, a differential polynomial ring is a differential ring, where basic derivation operators from Ll map the set of ring generators into itself.

The role of affine varieties in differential algebraic geometry is played by the sets of zeros of differential ideals in rings .1"{YI, . .. ,Yn} (.1" is a Ll-field). This implies the crucial role of the Ritt-Raudenbush basis theorem according to which every perfect differential ideal of .1"{Yl. ... , Yn} (.1" is a Ll-field of zero characteristic) has a finite system of generators (as a perfect Ll-ideal). In a contrast with the algebraic case, the radicality is essential, as the following example shows.

3.2.22. EXAMPLE. Let .1" be an ordinary differential field and .1"{x} a Ll-polynomial ring in a Ll-indeterminate x. Then the sequence of Ll-ideals


[x 2], [X2, (dx )2], ... , [X2, (dX)2, ... , (di X )2] in .:F {x} is an infinite strictly ascending chain of ~-ideals. We leave the proof of this fact to the reader as an exercise.

3.2.23. THEOREM (RITT - RAUDENBUSH). If a Ritt algebra R satisfies the ascending chain condition for perfect differential ideals then the ring R{ x} of differential polynomials in one differential indeterminate x also satisfies this condition.

The proof of this theorem in the case, when R is a differential field, will be given in Section 5.3 (Theorem 5.3.17). The proof in the general case is similar.

3.2.24. COROLLARY. Let a Ritt algebra R satisfy the ascending chain condition for perfect ~-ideals. Then every perfect ~-ideal I of R{Yl' ... , Yn} is of finite type, i.e. 1= {h, ... , /k} for some elements h,···,!k E R{Yl, ... , Yn}.

It should be noted that the conditions of the corollary do not imply the fact that I = [gl, ... ,gm] for some gl, ... ,gm E R {Yl , ... , Yn}.

3.2.25. EXAMPLE. Let.:F {x, y} be a ~-polynomial ring in two ~-indeterminates x, Y over an ordinary ~-field .:F (~ = {d}). Consider the ideal J of .:F {x, y} generated by all products dix· djy, i,j EN. It is easy to show that J = {x· y}, hence J is a perfect ~-ideal. Suppose that J = [h, ... , !k], then J = [{ dix ·dj Y I 0 :s i,j:S r}], for some r EN. But dr+1x· dr+1 y rI. [{dix. djy I O:s i,j:S r}] because (dr +1x)2 rI. [x 2, (dx)2, ... , (dr X)2] (see Example 3.2.22).

Ritt - Raudenbush theorem implies the following decomposition theorem.

3.2.26. THEOREM. Let R be a differential ring that satisfies the ascending chain condition for perfect differential ideals. Then every perfect differential ideal J of R can be represented as an intersection of a finite number of prime differential ideals. (In particular, such representation exists for any perfect differential ideal of the ring .:F {Yl, ... , Yn} of differential polynomials in differential indeterminates Yl, ... , Yn over a differential field .:F of zero characteristic.) If no ideal can be omitted in the representation, then the prime differential ideals are uniquely determined, they are called prime components of the ideal J.

PROOF. Let the assertion of the theorem be false. Then, by the ascending chain condition, there exists a perfect differential ideal I of R that is maximal among the ideals which cannot be represented as an intersection of a finite number of prime differential ideals. It is evident, that I is not prime. Hence, there exist elements a E Rand b E R such that ab E I but a rI. I, b rI. I. Perfect ~-ideals {I, a} and {I, b} contain I as a proper part, hence they may be represented as an intersection of a finite number of prime ~-ideals. Therefore, in order to get a contradiction that proves the first part of the theorem, it is sufficient to show that I = {I, a} n {I, b}. By Lemma 3.2.18, {I, a}{I, b} ~ {ab, I} ~ I. Furthermore, for any element c of {I, a} n {I, b}, we have c2 E {I, a}{I, b} ~ I, hence c E I. Therefore, {I, a} n {I, b} ~ I. The inverse inclusion is evident.

Let I = PI n ... n Pr = ql n ... q. be two representations of an ideal I of R as an intersection of prime ideals. Suppose, that no ideal may be omitted in these representations. Since PI n P2 n ... n Pr ~ ql, we have PI ..... Pr ~ ql, hence ql contains one of Pi. We may assume, that PI ~ ql' Now, similar arguments show that PI contains one of qi. If i > 1, then qi may be omitted in I = ql n ... q •. Hence,


PI = ql· Similarly, each qi coincides with some Pj and vice versa. In particular, 7' = s. Theorem 3.2.26 is proved. 0

In what follows F denotes a differential (~-) field and F{Yl, ... , Yn} denotes the ring of differential polynomials in differential indeterminates Yl, ... ,Yn over :F.

3.2.27. DEFINITION. A ~-polynomial / E F{Yl, ... ,Yn} is called linear, if deg / = 1. An ideal of F {Yl, ... , Yn} is called linear, if it is generated by a system of linear polynomials.

3.2.28. PROPOSITION. Let a set C F{Yl, ... ,Yn} consist of linear ~-polynomials. Then either [l = F{Yl, ... , Yn} or [l is a prime ideal.

PROOF. We shall prove that if R = F[XilieJ is the polynomial ring over a field F in the variables {XdieJ (where J is a finite or infinite set), then every proper ideal of R generated by a set of linear polynomials {Lil i El is prime. First of all, we reduce the case of an infinite set to the finite one. Indeed, if L = {LdiEI is a linear ideal of R that is not prime, we can consider elements /, 9 E R such that / tJ. L, 9 tJ. L, /g E L, i.e. /g = E~=l hkLk· It is easy to see that in this case the linear ideal L = (Ll' ... , Lr) of the polynomial ring R = F[XiliEJ1 in the finite set of all variables {Xi};eJl, that are present in /, g, or in the decomposition /g = E~=l hkLk, is not prime. The case of a finite set J can be reduced to the trivial case {Li = Xi}iEl by linear transformations of given polynomials {Lil and changing the variables Xi by some linear combinations Xi = EiEJ kiXi, ki E F. 0

It should be noted that F {Yl, ... , Yn} (where F is a differential field) is a factorial rmg.

3.2.29. PROPOSITION. Let F be a ~-field of characteristic zero, let A = A~l . .. . . A~r E F {Yl, ... , Yn}, and let Aj, j = 1, ... ,7', be linear polynomials. Then {A} = [All n··· n [Arl.

PROOF. Obviously, {A} = {AI' .... Ar}, so we can assume that kj = 1 (j = 1, ... , 7'). Furthermore, since {A} ~ [Alln· . ·n[Ar], it remains to prove the opposite inclusion.

Let .x E [Ad n ... n [Arl. It is easy to show that .xr E I, where I is the ideal of F {Yl, ... , Yn} generated by all elements t41 Al ..... d~r Ar (where (iI, ... ,ir) E Nr, and dl , ... ,dr are some (possibly coinciding) derivative operators). We shall use the induction on lexicographically ordered vectors (i l , ... ,ir ) E Nr to prove that d~l Al ..... d~r Ar E {A} for all (il,"" ir) E W. It is evident for (0, ... ,0). If d~l Al ..... d~r Ar E {A}, then

dilA dir-IA dir+lA + d (dilA di2 A dir-1A) dirA E {A} 1 1····· r-l r-l' r r r 1 l' 2 2····· r-l r-l . r r .

Multiplying this relation by d~l Al ..... d~':ll Ar- 1 , we obtain

(d~l Al)2 . (d~2 A2)2 ..... (d~':ll Ar _l )2 . d~r+1 Ar E {A},

hence, d~l Al . d~2 A2 ..... d~r+l Ar E {A}.


If A is a nonlinear irreducible differential polynomial, then the decomposition of {A} may be nontrivial.


3.2.30. EXAMPLE. Let us consider a ~-polynomial A = (y')2 + y in the ordinary ~-polynomial ring over a ~-field. Then {A} = [y] n p, where the ideal p is determined by the following condition: 1 E P iff I· (y')k E [A] for some kEN.

All fields considered below are supposed to have zero characteristic. Let 9 be a ~-extension of a ~-field:F. Then for any ~-polynomial 1 E F{YihEl

and for any family of elements (Oi)iEl of9, the result of substitution of the elements (odiEl instead of {YihEI in the polynomial 1 is defined in the natural way. We shall denote it by I( Oi) (of course, I( Oi) E 9).

3.2.31. DEFINITION. A family (Oi)iEI ~ 9 is called ~-algebraically dependent over a ~-field F ~ 9, if the family (OOi)8ET, iEI is algebraically dependent over F; otherwise the family (Oi)iEl is called Ll-algebraically independent over :F. Obviously, the family (Oi)iEl is Ll-algebraically dependent over F, if there exists a Ll-polynomiall E F{YihEl, f =/; 0, such that I(Oi) = O. An element 'T] of 9 is called ~-algebraic (~-transcendental) over F, if the family, consisting of the only element 'T], is Ll-algebraically dependent (respectively, Ll-algebraically independent) over F.

3.2.32. DEFINITION. All-extension 9 of a ~-field F is called differentially algebraic over F, if every element of 9 is Ll-algebraic over F.

According to the conventions of Section 3.1, if ~ is a set of elements of all-field 9 ;2 F, then the minimal Ll-subfield of 9 containing F and ~ is called a Ll-subfield of 9 generated by the set ~ over F; it is denoted by F(~). A Ll-extension of a Ll-field F is called Ll-finitely generated if it has a finite set of Ll-generators, i.e. it is of the form F(~) for a finite set ~.

The definition of the differential transcendence degree may be introduced by analogy with the notion of the transcendence degree.

3.2.33. DEFINITION. Let B be a subset of all-field 9 that is minimal (with respect to the inclusion) in the family of all those subsets ~ of 9 for which 9 is differentially algebraic over F (~). Then B is called a Ll-transcendence basis of the extension 9 over F. The cardinality of B is called differential transcendence degree of the Ll-extension 9 of F; it is denoted by Ll-trdeg9/:F.

The correctness of Definition 3.2.33 will be proved in Section 5.3 (see Proposition 5.3.13) .

It should be noted that there is a differential version of the theorem on a primitive in the classical field theory (see Theorem 3.2.35 below).

3.2.34. DEFINITION. A set of derivation operators ~ = {dl, ... ,dm } is called independent over all-field F, if there exist elements 111, ... , 11m E F such that det( dj/lj )i=l, .. ,m;j=l, .. ,m =/; O.

In particular, the set ~ = {a/ax 1 , ... , a/ax m} is independent over arbitrary ~extension of the rational function field q xl, ... , Xm ), (as usual, a/aX j (Xi) = Jij ).

3.2.35. THEOREM [KoI73, p. 103]. Let the set ~ = {dl , ... , dm } be independent over a field F. Then every Ll-algebraic extension 9 of F can be generated by a single element, i.e. 9 = F('T]) for some 'T] E 9. Furthermore, if two elements 0 and


(3 of the field 9 are Il-algebraic over F, then there exists an element c E F such that F(o:, (3) = F(o: + c(3).

For every Il-field F, there exists a semiuniversal extension U of F, that is a Il-extension of F in which any Il-finitely generated Il-algebraic extension of F can be embedded (see [KoI73, p. 92]).

The proof of the existence of a semiuniversal extension is similar to the proof of the existence of the similar extension in algebraic case, it is based on the following statement.

3.2.36. PROPOSITION. Let F be a Il-field and let 9 = F(TJI,"" TJn) be a finitely generated Il-extension of F. Then there exists a differential ideal p of F {YI, ... , Yn} such that Il-field 9 is Il-isomorphic to the quotient field of the ring F {YI, ... , Yn} Ip·

PROOF. Let p denote the differential ideal of the ring F{YI, ... , Yn} consisting of the Il-polynomials f, for which f(TJI,"" TJn) = O. It is easy to check, that F {YI, ... , Yn} Ip is a differential ring, hence it may be embedded into its Il-field of quotients. If ~i denotes the image of Yi (mod p) with respect to this embedding, then the mapping TJi >--t 6 produces the desired isomorphism. D

Now it is easy to prove the existence of a Il-compositum of differential fields, that implies the existence of semiuniversal extension.

Let Il-field F and its semiuniversal extension U be fixed, and let

TJ = (TJI, ... , TJn) E Un.

Then the prime Il-ideal p = {f E F{YI, ... , Yn} I f(TJ) = O} is called the defining ideal of the point TJ.

Furthermore, we define the generic zero of a prime Il-ideal p ~ F{YI,"" Yn} as a point ~ = (6, ... ,~n) E un for which p is the defining ideal. It is easy to see, that the generic zero exists for any Il-ideal p. Indeed, let us consider the quotient field of F{YI, ... , Yn}/p as a Il-extension of the field F embedded in U. If ~i is the canonical image in F {YI, ... , Yn} Ip (1 ::; i ::; n) of the Il-indeterminate Yi, then the point (~l' ... '~n) E un is the generic zero of p.

Every subset I: of the Il-polynomial ring F {YI , ... , Yn} determines a system of algebraic differential equations: A = 0, A E I:. A point TJ E un is called a solution (or zero) of this system (or of the set I:), if A(TJ) = 0 for any A E I:. Obviously, the set of zeros of a system I: coincides with the set of zeros of the ideal {I:}. The following theorem is the differential version of the Hilbert Nullstellensatz.

3.2.37. THEOREM. Let I: be a subset of the ring F{YI, ... , Yn}, and let B E F{YI, ... ,Yn}. Then:

(i) if BE {I:}, then B vanishes at every zero ofI:; (ii) con versely, if B (TJI, ... , TJn) = 0 for every zero of the set I:, then B E {I:}.

PROOF. Since {I:} is the radical of [I:], the first statement of the theorem is obvious, so we have to prove only statement (ii). By analogy with the algebraic case, we can reduce statement (ii) of the theorem to the following assertion: if the set of solutions of I: is empty, then the ideal of F {Yl, ... , Yn} generated by I: is not


proper. Indeed, suppose that this assertion is proved and let polynomial B vanish at every zero of E. Introduce a new ~-variable z and consider in :F {Yl, ... , Yn, z} the system E U (1- Z B(YI , ... , Yn)). By condition on B, this system has no zeros in Un+1 . Hence, we may assume that there exist polynomials hi, gi E :F {Yl, ... ,Yn, z} such that 1 = LiEI(hdi + gibi), where I; E [E], bi E [1 - zB] (we use Theorem 3.2.12 and the fact that the radical of a proper ideal cannot coincide with the whole ring). Substituting 1/ B instead of z in the last equation, and multiplying the equation obtained by a sufficiently large power of B, in order to delete the denominators (the operations are in the ~-field of quotients of :F {Yl, ... , Yn, z}), so we obtain B r E [E] for some rEf:!. To complete the proof we have to show that any proper ideal I of :F {Yl, ... , Yn} has a zero in un. Indeed, the ideal I may be embedded in a maximal ~-ideal, which by Corollary 3.2.20 is prime. It has been proved that such an ideal always has a generic zero, which is evidently a zero of I. 0

We can introduce Zariski topology on un, declaring that the closed sets are the sets of zeros of subsets E ~ :F {Yl, ... , Yn}. There is an one-to-one correspondence between the set of perfect differential ideals of :F {Yl, ... , Yn} and the set of closed subsets of the space un. Under this correspondence, the set of prime ~-ideals of :F {Yl, ... , Yn} corresponds to the set of irreducible closed subsets of un. Furthermore, by Ritt-Raudenbush basis theorem (see Theorem 3.2.22), un is a Noetherian space.

It has been already remarked that the differential polynomial ring can be considered as a generalization of the polynomial ring. Another generalization of the polynomial ring is the ring of differential operators.

3.2.38. DEFINITION. Let R be an integral differential domain and let ~ = { d 1, ... , dm } be a basic set of derivation operators on R. The ring D = R[ d 1, ... , dm ]

of skew polynomials in indeterminates d 1 , ... , dm with coefficients in R and the commutation rules djdj = djdi , dja = adi + d;(a) for all a E R, d;, dj E ~ is called a (linear) differential (~-) operator ring. This ring will be also denoted by R[~]. In particular, if derivation operators are trivial on R, then D is isomorphic to the commutative polynomial ring with the same generators.

Every element CT of D = R[~] may be uniquely represented as a finite sum

CT = L aBO = L ajl,.,;~d~l, ... , d~. BET(A) i" ... ,i~

The maximal value of ord 0 among all 0 for which aB # 0, is called the order of CT, it is denoted by ord CT.

The proof of the following theorem can be found in [C071, p. 56].

3.2.39. THEOREM. Let:F be a ~-field and let D = :F[d1 , ... ,dm ] be a ~operator ring over:F. Then D is a right and left Noetherian integral domain, it is also an Ore's ring.

This theorem implies that the ~-operator ring :F[~] has the properties that are similar to the properties of polynomial rings. In particular, this ring may be embedded into its quotient skew field, there exist a greatest common divisor and a


least common multiple of any two elements (right ones and left ones, in general case, they do not coincide). There is a natural filtration of the ring F[d] by the order of monomials: Do ~ D1 ~ '" ~ Di ~ ... , where Di = {f E F[d] lord f ~ i}. Obviously, UiEl'l Di = F[d] and the associated graded ring is a polynomial ring.

3.2.40. EXERCISE. Let R be an ordinary differential field. Show that there is the Euclidean algorithm in the ring R[d], so that for any p, q E R[d], q =I- 0, there exist elements r, bE R[d] such that deg r < deg q and p = bq + r.

Let R be a polynomial ring over a field K, R = K[X1, ... , xm], and d = {d1, ... , dm} = {f)/f)Xl, ... , f)j{):cm}. Then the linear d-operator ring R[d] IS

called Weyl algebm over K and is denoted by Am(K).

3.2.41. THEOREM. Weyl algebra Am(K) is simple.

First we prove the following lemma.

3.2.42. LEMMA. Let Am(K) be the Weyl algebra generated by the elements x1, ... ,xm,d1, ... ,dm over a field K. Suppose that (i1, ... ,im) E Nm and r = i1 + .. ·+im > O. Then ord[x1 .. . xm ,t4' ... d~] = r-l (here we use [,] to denote the commutator of two elements: [a, b] = ab - bay.

PROOF. We can assume that i1 > O. It is easy to check, that

[d~, Xi] = kd~-l + g, where ordg < k - 1

and

[ di, dim] - di, dim di, di2 dim Xl· .. Xm, 1 ... m - Xl ... Xm 1 ... m - 1 . Xl' 2 . X2 ... m • Xm

+

= Xl ... Xm . d~' ... d~ - ([t4', Xd + x1d~') ..... ([d~, Xm] + Xmd~) - 0 -Ii, d'..·m . di,-ldi2 d'..·m - . a1 ... m - Zl . X2 ... Xm ' 1 2' .. m

. . di'd:i2 dim + g12···1m 1 2'" m h · . d:i, d'i m

}'···,}m 1 ... m'

it +···+jm<r-l

PROOF OF THEOREM 3.2.41. Let J be a two-sided ideal of Am(K), and f be an element of J of the least order. We may assume that f = d~' ... d~ + j, where r = ord f = i1 + ... + im > ord j ~ O. Then [Xl ••• X m ,!] E J, and Lemma 3.2.42 shows that ord[ Xl ... Xm,!] = r-l < r. This contradiction completes the proof. 0

Let R be ad-ring, D = R[d]. We define a differential module Mover R as a left D-module. Thus, a differential R-module M is a R-module on which the operators of the set d act by the following rules: d(u+v) = d(u)+d(v), d(au) = d(a)u+ad(u) for all u, v E M, a E R.

We say that a differential module is finitely generated, if it has a finite system of D-generators.

3.2.43. REMARK. Let 9 be a d-field and let :F be a d-subfield of g. Then the module of Kaler differentials o'O/F becomes a d-module over g, if we set dd(a) = d(d(a)) for any a E g, d E d (here II' denotes the canonical differentiation 9 -+ nO/F)'

3.2.44. EXERCISE. Let 9 = F{''l}, 1/ = (1/1, ... , 1/n) E un. Then the set 11'1/1, ... , d1/n generates the module of differentials o'O/F as a module over the differential operators ring Q[d].

3.3. BASIC NOTIONS OF DIFFERENCE ALGEBRA 135

3.3. Basic Notions of Difference Algebra

3.3.1. DEFINITION. A difference ring is a commutative ring R, together with a finite set u = {0:1' ... , o:d of mutually commuting injective endomorph isms of this ring. The set u is called a basic set of R, and its components are called translations.

A difference ring with a basic set u will be also called a u-ring. If Card u = 1, then R is called an ordinary u-ring, and if Card u > 1 then R is called a partial u-ring. The notions of difference (u-) homomorphism of u-rings, difference field (u-field), difference (u-) subring and difference (u-) overring, difference (u-) ideal, difference (u-) subfield and difference (u-) extension are defined in accordance with the general notions introduced in Section 3.1 for rings with operators. Elements a E R, such that o:(a) = a for any 0: E u, form a subring C of R that is called the ring of constants of the difference ring R.

While considering difference ring R with a basic set u = {o: 1, ... , O:n}, we shall denote by Ta the free commutative semigroup generated by elements 0:1, ... , O:n.

The elements of Ta can be considered as endoomorphisms of the ring R of the form 0:7' ... o:~n, where k1' ... , kn E ;;Z (we write the operation in Ta in the multiplicative form). If a is an element of au-ring Rand r ETa, then r(a) will be called a transform of a.

If Ro is a u-subring of au-ring Rand B ~ R, then by Ro{B}17 we shall denote the u-subring of R, generated by the set B over Ro (in the sense of the corresponding definition in Section 3.1). Evidently, Ro{B}" coincides with the ring R[{r(b)lb E B, r E T17}]' Furthermore, if Ro is a u-subfield of au-field Rand B ~ R, then by Ro(B)" we shall denote the u-subfield of R, generated by the set B over Ro (in the sense of Section 3.1). It is easy to see, that Ro(B)17 coincides with the field R({r(b) IbEB,rET,,}).

3.3.2. DEFINITION. A difference ring R with a basic set u = {O:l, ... ,O:n} is called an inversive difference ring, if 0:1, ... , O:n are automorphisms of the ring R.

Let R be an inversive difference ring with a basic set of automorphisms u = {O:l, ... ,O:n}' Let u* = {0:1,"" O:n, 0:11, ... ,0:;;-1} be the set consisting of all translations from u and their inverse automorphisms. Then R is a u*-ring in the sense of Section 3.1. The operators from u* act as automorphisms, and 0:;0:;1 (a) = o:;lo:;(a) = a for any a E R (i = 1, ... , n). Below, an inversive difference ring with a basic set of automorphisms u will be also called a u* -ring.

3.3.3. Examples and Exercises. 1. Any commutative ring may be considered as inversive difference ring with the

basic set u consisting of one or several identical automorphisms. 2. The polynomial ring R[x] in one variable x over an ordinary difference field

R with a translation 0: may be transformed into an ordinary difference ring, if we extend 0: on R[x] setting o:(x) = f(x) for some polynomial f(x) E R[x] \ R.

3. Show that not every automorphism 0: of the ring R[x] can be extended on the formal power series ring R[[xll.

4. Show that the formal (convergent) Laurent series field K((x)) over a field K may be considered as an ordinary difference field with a translation 0: such that o:(x) = ax, where a is an arbitrary nonzero element of K.


5. Let F be a field and let R = F(XI, ... , x.) be the field ofrational fractions over F in s independent variables Xl, ... , X.. Show that if PI, ... , P. are algebraically independent over F elements of R, then there exists a unique injective endomorphism a of the field R identical on F and such that a(xi) = Pi (i = 1, ... , s). It follows that the field R can be treated as an ordinary difference field with the translation a.

Show that if PI, ... , P. are homogeneous linear rational functions in variables Xl,"" x. with a common denominator, then a is an automorphism, so in this case R is an ordinary inversive difference field with the basic set u = {a}.

6. Let Zo be a complex number and let U be a region of the complex plane that satisfies the following condition: if z E U, then z + Zo E U. Let Mu denote the field of functions of one variable meromorphic in U. The field Mu can be treated as an ordinary difference field with a translation a such that aJ(z) = J(z + zo) for any function J(z) EMu. We shall denote this difference field by Mu(zo). Clearly, if z - Zo E U for any z E U, then Mu is an inversive difference field (in this case a-I J(z) = J(z - zo) for any function J(z) EMu).

7. Let Zo be a nonzero complex number and let U be a region of the complex plane such that zZo E U for any z E U. Then the field Mu of complex functions in one complex variable z meromorphic in U may be considered as an ordinary difference field with a translation fj such that fjJ(z) = J(zoz) for any J(z) EMu. We shall denote this difference field by M U (zo ). It is easy to see that if z / Zo E U for any z E U, then the field Mu(zo) is inversive, (fj-l J(z) = J(z/zo) for any J(z) EMu).

Let Zo be a nonzero complex number and let U, V be two regions of the complex plain such that z + Zo E U for z E U, and z + Zo E V, for z E V (respectively, zZo E U for z E U and zZo E V for z E V). If U ~ V then, evidently, Mv(zo) is a difference subfield of the difference field Mu(zo) (respectively, Mv(zo) is a difference subfield of the difference field Mu(zo)),

8. Let A be a ring of functions of n real variables that are continuous on the n-dimensional real space ~n. Let us fix some real numbers hI, ... , hn and consider a set of inj ecti ve endomorphisms u = {a I, ... , an} of A such that

Then A becomes a difference ring with the basic set u, we shall denote this difference ring by Ao(hl , ... ,hn). Similarly, a ring (()'(~n) of all functions that are continuous on ~n together with all their partial derivatives up to the order P (p E N or P = +00), is a u-ring. Such difference ring will be denoted by Ap(hl, ... ,hn). Obviously, every u-ring Ap(hl, ... ,hn) is a difference subring of the u-ring Ap- l (hI, ... , hn) (p E N, P ~ 1), so we have the following chain of u-rings

It should be noted that difference rings Ap(hl , ... , hn) often arise in connection with equations in finite differences, when ith partial finite difference


of a function f(Xl, ... , xn) (1 :s; i :s; n) is written in the form t1fi = (ai - l)fi (translations ai are given by (3.3.1)).

9. An important example of a difference ring is the difference polynomial ring that will be defined below.

3.3.4. DEFINITION. Au-ideal J of au-ring R is called reflexive (or a u*-ideal) if for any a E u and a E R, the inclusion a(a) E J implies a E R.

Obviously, if R is a u* -ring i.e. al, ... , an are automorphisms of R, then any reflexive u-ideal is a u*-ideal in the sense of Section 3.1 (for u* = {al, .. . ,an ,

all, ... ,a;;-l}). Let J be a u-ideal of au-ring Rand J = {a E R I r(a) E J for some r E Ta}.

It is easy to show, that J is a u* -ideal, which is contained in any u* -ideal of R, containing J. The u* -ideal J is called the reflexive closure of J.

In what follows, the u-ideal of au-ring R generated by a set B S; R will be denoted by [Bla, and the reflexive u-ideal generated by B (i.e. the smallest u*ideal of R containing this set) will be denoted by [Bl.

3.3.5. DEFINITION. If a prime ideal of au-ring R is a u-ideal, then it is called a prime difference ideal of R. If a prime u-ideal P is reflexive, then it is called a prime u* -ideal.

3.3.6. Examples and Exercises. 1. The nilradical H and the Jacobson's radical J of any inversive difference

ring R are reflexive difference ideals, because a(H) = Hand a(J) = J for any automorphism a of R. In particular, the maximal ideal of a local u*-ring is a prime u* -ideal.

2. Let R = K[xl be the polynomial ring in one variable x over a field K of zero characteristic that is considered as a difference ring with a translation a such that (af)(x) = f(x + 1) for any polynomial f(x) E R. Show that R does not contain proper difference ideals.

3. Let S = K[xl be the polynomial ring in one variable x over a field K of zero characteristic that is considered as a difference ring with a translation a such that (af)(x) = f(2x) for any polynomial f(x) E R. Show that (x) is the unique prime difference ideal of S.

If f : R -+ R' is a u-homomorphism of u-rings, then it is easy to see that the kernel of f is a u* -ideal of R. Conversely, let 9 be a surjective homomorphism of au-ring R onto a ring S such that Ker 9 is a u*-ideal of R. Then there exists (obviously, unique) structure of u-ring on S such that 9 is a u-homomorphism of R onto S. In particular, if S = RIl, where I is a u*-ideal of R, then there exists a unique structure of u-ring on S, with respect to which the canonical surjection R -+ RI I is a u-homomorphism. The u-ring RI I is called a u-quotient ring of the u-ring R by the u* -ideal I.

Let R be a u-ring and let {Vi liE I} be a family of elements of a u-overring of R. As in general theory of rings with operators, the family {Vi liE I} is said to be difference (u-) algebraically dependent over R, if the family {r(vi) liE l,r ETa} is algebraically dependent over R; otherwise, the family {Vi liE I} is said to be difference (u-) algebraically independent over R or a family of difference (u-) indeterminates over R; the u-ring R{(Vi)iE/}a is called an algebra of difference (u-)


polynomials over R in u-variables (Vi )iEI. If a family consisting of one element v is u-algebraically dependent (independent) over au-ring R, then v is called u-algebraic (u-tmnscendental) over R.

3.3.7. PROPOSITION. Let R be a u-ring with a basic set u = {alJ .. . a n } and let 1 be an arbitrary set. Then the following statements hold:

(1) there exists an algebra of u-polynomials over R in the family of u-indeterminates with indices from the set 1 (See Definition 3.1.1.);

(2) if Sand S' are two such algebras of u-polynomials, then there exists a u-isomorphism between Sand S' whose restriction on R is an identical au tomorphism;

(3) if R is an integral domain, then any algebra of u-polynomials over R is also an integral domain.

PROOF. Let T = Tq and let S = R[(Yi,r)iEl,rET] be the R-algebraofpolynomials in variables (Yi,r )iEl,rET with indices from the set 1 x T. For any PES, a E u we shall denote by a(P) the u-polynomialobtained by replacing every coefficient a of P by a(a), and every variable Yi,r presented in P by Yi,ar. We obtain an injective homomorphism S -t S (denoted by the same letter a). For any i E 1 let us denote Yi,l by Yi (here 1 is the identity of the group T). Since the elements Yi,r = r(Yi) (i E 1, rET) are algebraically independent over R, the u-ring S is an algebra of u-polynomials over R.

Let S' be an algebra of u-polynomials over R in u-variables (Zi )iEI. Consider a R-homomorphism cp : R[(Yi,r )iEl,rET] -t R[(Zi,r )iEI,rET] that does not change elements of R and maps every Yi,r onto Zi,r (i E 1, rET). It is easy to see, that cp is a u-isomorphism. The last assertion of the proposition is obvious. 0

The ring of u-polynomials, whose existence was proved in Proposition 3.3.7, will be denoted by R{(Yi)iEl }q.

3.3.8. DEFINITION. Let R be a u-ring with a basic set u = {alJ ... , an}. A u-overring S ;2 R is called an inversive closure of R, if elements of u act as pairwise commuting automorphisms of S (they are denoted by the same symbols a1, ... ,an), and for any a E S, there exists an automorphism r E Tq of the ring S such that r(a) E R.

3.3.9. PROPOSITION.

(1) Every u-ring has an inversive closure. (2) If S1 and S2 are two inversive closures of au-ring R, then there exists a

u-isomorphism cp : S1 -t S2 whose restriction on R is an identity automorphism of R.

(3) If U is a u* -ring, containing R as a u-subring, then U contains the inversive closure of R.

(4) If au-ring R is an integral domain (field), then its inversive closure is also an integral domain (field).

PROOF. (1) First of all, let us construct the inversive closure it of an ordinary difference ring R with the basic set u = {a}. Let R' = a(R) and let R* be a ring that is isomorphic to R and such that R n R* = 12'. If f3 : R -t R* is the


corresponding ring isomorphism, then we set (R')* = (J(R'), so that (JO'.(R) = (R')*. Let us replace the elements of (R')* by the corresponding elements of R. After this replacement we transfer a ring R* into a ring R1 that may be considered as an overring of R. The mapping (p(J)-l, where p : R* -+ R1 denotes this replacement, is an isomorphism of R1 onto its subring R that extends 0'.. This isomorphism will be also denoted by 0'., and R1 will be treated as a cr-extension of the cr-ring R.

By induction we can determine a sequence of cr-overrings Ra, R1, R2, ... of R: Ra = R, Rn = (Rn- 1h for n = 1,2, .... Set R = UnEN Rn and define the mapping R -+ R that extends 0'. (it is denoted by the same letter) as follows. If a E R, then a E Rn for some n E N and there is an element O'.(a) in the cr-ring Rn, which we consider as the image of a under the mapping 0'. : R -+ R (the correctness of this definition of 0'. (a) , i.e. its independence of the choice of n EN such that a ERn, is obvious). It is easy to see that the cr-ring R is the inversive closure of R.

Let us consider now the case when a cr-ring has the basic set cr = {a 1 , ... , an}, (n> 1).

First of all, let us construct an inversive closure R1 of the cr1-ring R with the basic set cr1 = {0'.1}. We can consider R1 as a O"-ring, where the elements 0'.1, ... , an act in the following way: let a E R1 and r be the least nonnegative integer such that aHa) E R. Let O'.j(a) = O'.l r O'.jO'.Ha) for any i = 2, ... , n (recall, that the basic endomorphisms of R1, and also of the rings R2, ... ,Rn constructed below, are denoted by the same symbols as elements of the set cr). Then R1 becomes a cr-ring, and if a E Rl and aHa) E R for some SEN, then

O'.lsO'.iO'.~(a) = O'.lsO'.jO'.~-r O'.~(a) = O'. l sO'.f- r O'.iO'.~(a) = O'.l r O'.iO'.~(a) = O'.i(a)

(here r is the least nonnegative integer such that aHa) E R). In the same way we can construct an inversive closure R2 ofthe 0"2-ring R1 with the basic set 0"2 = {0'.2} and, as before, extend the action of all elements of the set cr on R2. Continuing this process for cr3-ring R3 with basic set 0"3 = {0'.3} and so on, we obtain on the nth step a difference ring S = Rn with the basic set cr = {0'.1,"" an}, which is, obviously, an inversive closure of the O"-ring R.

(2) Let Sl and S2 be two inversive closures of a O"-ring R. Let us define the mapping t.p : S1 -+ S2 in the following way: if a E S1 and T is an element of To such that T( a) E R, then t.p( a) = T- 1 (T( a)). Here T is treated as an automorphism of the ring Sl, and T- 1 is treated as the automorphism of S2 that is inverse to T (T is also treated as an automorphism of S2)' It is easy to check, that t.p is well-defined (i.e., the value t.p(a) does not depend on the choice of T E T such that T(a) E R), and t.p is a O"*-isomorphism such that t.p(a) = a for any a E R.

(3) Let U be a cr* -ring containing R as a cr-subring. let S denote the set of all a E U such that T( a) E R for some T E To. It is easy to check, that S is an inversive closure of the cr-ring R.

(4) If a O"-ring R is an integral domain (field) and S is its inversive closure, then for any 0 -:j:. a E S there exists a mapping T E To such that T(a) E R. Since T(a) is not a zero divisor, a is not a zero divisor too. If T(a) is invertible, then a is also invertible, and a-I = T-I(T(a)-I). The proposition is proved. 0

The following proposition shows that if an integral domain R is a cr-ring, then the quotient field Q(R) of the ring R can be considered as a cr-extension of R (in this case we say that Q(R) is a cr-quotient field of R).


3.3.10. PROPOSITION. Let R be a u-ring without zero divisors. Then the following assertions hold:

(1) there exists a u-quotient field of R; (2) if:F1 and :F2 are two u-quotient fields of R, then there exists au-isomorphism

<p : :F1 --+ :F2 whose restriction on R is an identical mapping; (3) if S is a u-field containing R as a u-subring, then S contains au-quotient

field of R.

We leave the proof of the proposition to the reader. (Actually, the proof for a partial u-ring is the same as in the ordinary case, see [CoR65, Ch. II, Th. 3].)

3.3.11. DEFINITION. Let R be a u-ring and let S = R{(Y;}iEl}a be the ring of u-polynomials in u-indeterminates variables (Y;)iEl' Then u-quotient field of S is called a u-field of rational fractions in u-indeterminates (Yi)iEI, it is denoted by R((Yi)iEl)a.

3.3.12. EXAMPLE. Let U be a ring of functions in n real variables Xl,"" x n, that are defined on Rn and vanish at every point of the cube

Consider the set of pairwise commuting injective endomorphisms u = {a1' ... , an} of U such that a;f(x1,"" xn) = f(x1,'" ,Xi-1, T,Xi+1, ... ,xn) (i = 1, ... , n) for any f(x1, ... , x,.) E U. Then U can be treated as a u-ring, which, however, is not a u*-ring (the mappings 01, ... , an are not surjective). It is easy t.o check that the inversive closure of U is the u* -ring of functions f(x1, ... , xn) that vanish on some parallelepiped

Sk" .. ,k n = {(Xl, ... , Xn) IIx11 < 2- k" ... , IXnl < Tkn}

(k1 = k1(f) EN, .. . ,kn = kn(f) EN).

3.3.13. EXERCISE. Let R = K[x] be the polynomial ring over a field K of zero characteristic treated as an ordinary difference field with the translation a such that a(k) = k for any k E K and a(x) = X3 (see Example 3.3.3.2). Describe the inversive closure of R.

3.3.14. EXERCISE. Let S be a multiplicatively closed subset of au-ring R such that a(S) ~ S for any a E u. Then the quotient ring S-l R can be uniquely supplied with a structure of a u-ring, so that the natural homomorphism -+ ~) is a u-homomorphism. Show that this statement is not true if S is an arbitrary multiplicatively closed subset of R (give a counterexample).

[Hint: The polynomial ring R = Q[x] over the field of rationals Q may be considered as an ordinary u-ring with the basic set u = {a}, where af(x) = f(x+l). Let S = {I, x, x2, ... , x n , ... } E R. Show, that there is no injective endomorphism a ofthe ring S-l R, such that for any f(x) E Q[x] the equality <p(af(x)) = a<p(f(x)) holds. (<p denotes the natural mapping f(x) --+ ¥ E S-l R.)]

The following example shows that the ring of u-polynomials over a u-ring with the maximality condition for u-ideals does not necessary satisfy the same condition.


3.3.15. EXAMPLE. Let:F be a O'-field with a basic set 0' = {Ol"'" on} and let :F{Y}a be the ring of O'-polynomials in one O'-indeterminate Y over:F. For every k = 1,2, ... , let~" denote the O'-ideal [yody),yoHy), ... ,yo~(Y)]a. Obviously, El ~ E2 ~ ... ~ E" ~ ... and, moreover, every inclusion E" ~ E"+l in this chain of 0'-ideals is proper. Indeed, any element of~" has the form E~=l E~"=l Qijrij(yoi (y)), where Qij E :F{Y}a, rij E Ta (1 ~ i ~ k, 1 ~ j ~ I,,), and therefore contains at least two elements ol(y), oi(y) such that Ir - sl ~ k. Hence, yo~+l (y) E E"+l \ E" so that ~" ~ ~"+1 (k = 1,2, ... ,). It is easy to see, that ~ = Uk=l ~k is a O'-ideal of :F {Y}a wIthout a finite set of O'-generators.

Algebraic difference equations and their solutions. Let R{(Yi )iE!} be an algebra of difference (0'-) polynomials in O'-indeterminates

(Yi)iE! over au-ring R and let 'f/ = ('f/i)iEI be a family of elements of a O'-overring of R. Since the elements (r(Yi))iEI,TET .. are algebraically independent over R, there exists a unique ring homomorphism 1fJ,., : R[r(Yi)iEI,TET] -+ R[r('f/i)iEI,TET .. ] that is identical on R and maps r(Yi) onto r('f/i) (i E I, r E Ta). Obviously, for U = r(Yi) (i E I, r ETa), as well as for U E R, we have 1fJ,.,(o(u)) = o(IfJ,.,(u)) for any 0 E 0'. Therefore, 1fJ,., is a O'*-homomorphism of R{(Yi)iEda onto R{('f/i)iEda; it is called a substitution of ('f/i)iEI instead of (Yi)iEI. It is easy to see that 1fJ,., is a O'-isomorphism iff the family (r/i)iEI is O'-algebraically independent over R.

If 9 E R{(Yi)iEIl, then the substitution of (r/i)iE! instead of (Yi)iEI maps 9 onto an element of the ring R{( 'f/i )iEI }a; this element is called the value of the difference polynomial 9 at the family ('f/i)iEI and is denoted by g(('f/i)iEI). Obviously, Ker 1fJ,., is a reflexive difference ideal of R{(Y;}iEI }a. This ideal is called a defining difference (0'-) ideal of the family (r/i)iEI over R.

3.3.16. REMARK. Let :F be a difference (0'-) subfield of a O'-field g and g = :F('f/l, ... , 'f/.) for some elements 'f/l,"" 'f/. E g. Consider the corresponding alge-bra of O'-polynomials :F {Yl. ... , Y.} in O'-indeterminates Yl, ... , Y. and substitution 1fJ,., : :F {Yl, ... , Ys} -+ :F {'f/1, ... , 'f/s}. Since:F {'1l, ... , '1.} is an integral domain (:F{'f/l, ... ,'f/.} is contained in the field g), P = KerlfJ,., is a prime O'*-ideal of :F {Yl, ... , Ys}, hence g may be treated as the difference quotient field of the differ-ence ring :F{yl, ... ,Y.}a/P'

3.3.17. DEFINITION. Let :F{Yl, ... , Y.}a be an algebra of O'-polynomials in 0'-indeterminates Yl. ... , Y. over a O'-field :F and let cf1 = {f>.(Yl, ... , y,)I.A E A} be a set of O'-polynomials from this algebra. Suppose, that 'TI = ('TIl, ... , '1.) is as-tuple with entries from some O'-extension of :F (it will be called (s-dimensional) point over :F). The point 'TI is called the solution of the set II> or of the system of algebraic difference (0'-) equations

J>.(Yl,"" y.) = 0 (.A E A), (3.3.2)

if cf1 ~ Ker 1fJ,." where 1fJ,., : :F {Yl, ... , Y.} -+ :F {'TIl, ... , 'TI.} is the substitution of ('TIl, ... , 'TI.) instead of (Yl, ... , y.).

Below, while considering algebraic difference (0'-) equations, we shall often omit the word" algebraic" and call a system of the form (3.3.2) a system of difference (or 0'-) equations.


3.3.18. DEFINITION. Two points 11 = (111. ... ,11.) and (= «(1, ... ,(.) over a u-field:F are called u-equivalent ifthere exists au-isomorphism cP : :F(111, ... , 11.}u -?

:F((l,"" (.}u such that cp(a) = a for any a E:F and cp(11i) = (i (i = 1, ... , s).

If :F {Y1, ... , Y.}u is the ring of u-polynomials in u-indeterminates Y1, ... , Y. over au-field:F, then, as it was shown above, for any s-dimensional point 11 = (111, ... , "18) over :F we can define a substitution of this point, i.e. au-homomorphism cpq : :F{Y1, ... ,Y.}u -? :F{"11,.··,"1.}u such that cpq(a) = a for any a E:F and CPf'/(Yi) = "1i (i = 1, ... ,s). If A E :F{Yl,""Y'}u, then cpq(A) will be denoted by A( TJl, ... , 11.) and called the result of substitution of the point 11 into the spolynomial A. In accordance with this terminology, we shall say that a point "1 is a solution of the u-polynomial A E :F{Yl, ... , Y.}u (or the solution of the algebraic u-equation A(Y1, ... , y.) = 0), if A( 111, ... , 11.) = 0, i.e., A E Ker cpq.

As Remark 3.3.16 shows, the set of all u-polynomials ofthe ring :F{Y1, ... , Y.}u for which a fixed s-dimensional point 11 = (111, ... , TJ.) is a solution over :F, forms a prime u· -ideal Pq = Ker CPr) of :F {Y1, ... , Y.}u; this ideal is called a defining ideal of the point 11.

3.3.19. DEFINITION. With the above notation, let ~ be a set of u-polynomials from the u-ring :F{Y1, ... ,Y.}u. A s-dimensional point "1 = {111, ... ,11.} over the u-field :F is called a generic zero of~, iffor any u-polynomial A E :F{Yl, ... , Y. }u, the inclusion A E ~ holds if and only if A(11l, ... ,11.) = O.

3.3.20. PROPOSITION. Let:F {Y1, ... , Y.}u be the ring of u-polynomials in uindeterminates Y1, ... , Y. over a u-field:F. Then:

(1) a set ~ ~ :F{Yl,"" Y.}u has a generic zero over:F iff~ is a prime u*-ideal of :F {Y1, ... , Y.}u different from (1);

(2) any s-dimensional point over :F is a generic zero of a prime u* -ideal of :F{Y1,···,Y.}u;

(3) if two s-dimensional points over :F are generic zeros of the same prime u·ideal of F {Y1, ... , Y.}u, then these points are u-equivalent.

PROOF. (1) It has been already proved that the set ~ ~ F{Y1, ... ,Y.}u with the generic zero TJ = (111, ... , 11.) is a prime u* -ideal (in this case ~ = Ker cpq, where Cpf'/ : :F {Yl, ... , Y. } u -? :F {TJ1, ... , TJ.} u is a u-homomorphism such that cpq (Yi) = TJ; for i = 1, ... , s and the restriction of cPq on :F is an identity mapping). Conversely, let ~ be a prime u* -ideal of the ring S = :F {Yl , ... , Y. } u different from (1). Let iii be the image of the u-indeterminate Yi (1 :s i :s s) under the canonical surjective u-homomorphism S -? S/~ (iii is considered as an element of the quotient field of the u-ring S/~). Thenl s-dimensional point ii = (iiI, ... , ii.) is the generic zero of the u-ideal ~.

(2) If 11 = (TJl, ... , 11.) is a s-dimensional point over :F, then ~ = Ker cPq is a prime u· -ideal of :F {Y1, ... , Y.}u with the generic zero TJ.

Let TJ = (TJl, ... , TJ.) and ( = «(1, ... , (.) be two s-dimensional points over :F, which are generic zeros of the same prime u*-ideal of F{Yl, ... , Y.}u. Let cPq and cp( be the u-homomorphisms of the ring F{Yl"'" Y.}u onto F{TJ1, ... , TJ.}u and :F {(l, ... , (.}u, respectively, that were described above. Then Ker cPq = Ker cp( = 0, hence the mappings cpq and cp( induce u-isomorphisms rpf'/ and rp( of the quotient field of the u-ring :F{Y1""'Y'}u/~ onto the u-fields :F(111, ... ,11.}u and


F((l, ... ,(.)", respectively. It follows that tf'(tf';;-1 : F(TJ1, ... ,TJ.)" -t F((l, ... , (.)" is a a-isomorphism such that tf'(tf';;-l(TJ;) = (; (i = 1, ... , s) and tf'Etf';;-l(a) = a for any a E F. Thus, the points TJ and (, are a-equivalent.

3.3.21. EXAMPLE. (See [CoR65, Ch. 2, § 10]). Let us consider the field of the complex numbers <C as an ordinary difference field whose basic set a consists of the identity automorphism a. Let <C{y}" be the ring of a-polynomials in one a-indeterminate Y over C. While considering this ring, we shall write Y(k) instead of ak (y) (k = 1,2, ... ). Let M be the field of functions in one complex variable z, meromorphic on all complex plane. We shall consider M as the a-extension of the a-field <C such that aJ(z) = J(z + 1) for any J(z) EM.

The a-polynomial A = Y(l) - Y - 1 E <C{y}" has, obviously, no solutions in <C, but in M this a-polynomial has solution z + c(z), where c(z) is a function from the field M such that c(z + 1) = c(z).

Let B = (Y(1) - y)2 - 2(Y(I) + y) + 1 E <C{y}". This a-polynomial is irreducible in the ring <C{y}" (that is considered as the polynomial ring in the countable set of indeterminates y, Y(l), ... , Y(n), ... over q, and ~ = (z + C(Z))2, TJ = (c(z)e i7rz + 1/2? E M are solutions of B. In this case, ~ is a solution of the system of a

polynomials Band B' = Y(2) - 2Y(1) + Y - 2, and TJ is a solution of the system {B, B" = Y(2) - y}. Note, that B' B" = a(B) - B.

This example shows, that an algebraical irreducible difference (a-) polynomial in one a-indeterminate Y may have two different sets of solutions, each of which depends on an arbitrary periodic function. It is known, that such situation is impossible for algebraic differential equations.

3.3.22. DEFINITION. Let R be a difference ring with a basic set a. A R-algebra U is called a difference (or a-) algebra over R if elements of a act on U in such a way that the algebra U is a a-ring and a(au) = a(a)a(u) for any a E R, u E U, a E a.

It is easy to see, that the algebra of difference polynomials over a difference ring R is a difference R-algebra in the sense of this definition. If F is a difference subfield of a difference field g, then 9 may be naturally treated as a difference algebra over F.

Perfect ideals of difference rings. Ritt's rings.

3.3.23. DEFINITION. Let R be a a-ring with a basic set a = {al,"" an}. A a-ideal J of R is called perfect if for any a E R, T1, ... , Tr E T", and k1, ... ,kr EN, the inclusion T1(a)kl ... Tr(a)kr E J implies a E J.

Obviously, every perfect a-ideal is reflexive and every prime a* -ideal is perfect. Let F be a difference (a-) field, F {YI, ... , Y.}" the ring of a-polynomials over F

in a-indeterminates Y1, ... , y., and A a set of s-dimensional points over F. Then the set (A) of all a-polynomials of the ring F {Y1, ... ,Y.}" that are annihilated by each point a E A, is a perfect a-ideal of F{Y1, ... , Y.}".

If B is a subset of a a-ring R, then the intersection of all perfect a-ideals of R containing B is the smallest perfect a-ideal containing B. This ideal is denoted by {B},,; it is called the perfect closure of B.


3.3.24. EXERCISE. (Construction of the perfect closure). Show that the perfect closure of a subset B of a difference (u-) ring R can be obtained by the following procedure. For any subset M ~ R, we denote by M' the set of all a E R, for which there exist elements Tl, ... , Tr E T" (r E N) and numbers k1 , ... , kr E N (depending on a) such that Tt{a)kl, ... , Tr(a)kr E M. Let Bo = B, Bl = [Bo]~, ... , Bm = [Bm-d~/", Then {B}" = UmEN Bm.

3.3.25. EXERCISE. Let A and B be two subsets of a difference (u-) ring R. Then (with the notation of the preceding exercise):

(1) AkBk ~ (AB)k+l for any kEN (if U ~ R and V ~ R, then UV denotes the subset {uv I u E U,v E V} of R);

(2) {A},,· {B}" ~ {AB},,; (3) (AB)k ~ Ak nBk for any kEN, k ~ 1; (4) Ak n Bk ~ (ABh+l for any kEN; (5) {A}" n {B}" = {AB}".

3.3.26. EXERCISE. Show, that the set of all perfect u-ideals of au-ring R is a perfect conservative system (see Definition 1.4.8).

3.3.27. DEFINITION. A basis of a subset M of au-ring R is a finite subset B ~ M such that {B}" = {M}". If {M}" = Bm for some mEN, then the set B is called m-basis of M.

A difference (u-) ring R, every subset of which has a basis, is called a difference (u-) Ritt's ring.

One can find the proves of the following Lemmas 3.3.28 and 3.3.29, as well as the proves of Theorems 3.3.30 and 3.3.31, in [CoR65, Ch. iii, §§ 4,5].

3.3.28. LEMMA. Suppose that a subset M of au-ring R has no basis (has no m-basis for some mEN). Then the u-ideal [M]", the inversive closure [M] and the perfect closure {M}" have no bases (m-bases, respectively).

3.3.29. LEMMA. If au-ring R contains a proper subset that does not have a basis (that does not have a m-basis for some mEN), then R contains a maximal proper subset that does not have a basis (respectively, m-basis).

3.3.30. THEOREM.

(1) If every perfect u-ideal of au-ring R has a basis, then R is a Ritt's u-ring. (2) If every perfect ideal J of au-ring R has a m-basis for some m = m( J) EN,

then every subset M of R has a m'-basis for some m' = m'(M) EN. (3) If au-ring R contains a set that does not have a basis (that does not have

a m-basis for some mEN), then there exists a prime u* -ideal in R that does not have a basis (respectively, m-basis) and that is maximal (with respect to inclusion) in the family of all proper subsets of R without basis (respectively, in the set of all proper subsets of R without m-basis).

3.3.31. THEOREM. Let R be a u-ring. Then the following conditions are equivalent:

(1) R is a Ritt's u-ring;


(2) the ring R satisfies the maximality condition for perfect CT-ideals, i.e. every set of perfect CT-ideals of R contains a maximal (with respect to inclusion) perfect CT-ideal;

(3) every strictly increasing sequence of perfect CT-ideals of R is finite.

This theorem and Exercise 3.3.26 show, that the set of all perfect CT-ideals of R is a Noetherian conservative system.

Let R = F{YI,"" Ys}" be the ring of difference (CT-) polynomials over a difference field F with a basic set CT = {D:l, ... , D:n }. Then R may be also treated as a polynomial ring over F in indeterminates T(Yi) (T E T", i = 1, ... , s), which we shall denote by TYi (without parentheses). By analogy with differential polynomial rings, we define the order of an element T = Q~l, ... , Q~n E T = Ta (k I , ... , kn E l'iJ) as the number ord T = L:7=1 ki. By a ranking of the family of CT-indeterminates Yl, ... , Ys we shall mean a well-ordering of the set TY = {TYi IT E Ta, 1 'S i 'S s} that satisfies the following conditions:

(1) 'U 'S T'U for any 'U E TY (we denote the order on the set TY by the same symbol'S);

(2) the inequality 'U 'S v ('U, V E TY) implies T'U 'S TV for any T E Ta.

Elements of TY will be called terms, and a ranking of a family Yl, ... , Ys of CTindeterminates will be also called a ranking of the set TY. In accordance with the similar notions of differential algebra, a ranking of TY will be said to be integrated if for any 'U, v E TY, there exists T E T such that v < T'U (i.e. v 'S T'U and v =1= T'U).

A ranking on TY will be said to be orderly, if the condition ord Tl < ord T2 (Tl, T2 E T = Ta) implies TIYi < T2Yj for any 1 'S i, j 'S s.

An important example of an integrated orderly ranking is a standard ranking: 'U = Q~', ... , Q~n • Yi 'S v = Q~l, ..• , Q~n • Yj (1 'S i, j 'S s; kn , 1m E l'iJ for 1 'S n, m 'S n) iff (n + 2)-tuple (L:~=1 kv, i, k1, . .. , kn ) is less than or equal to (L:~=1 Iv, j, 11, ... , In) with respect to the lexicographic order on l'iJn +2 .

Below we assume that some integrated orderly ranking of TY is fixed. Let A E F {Yl, ... , Ys }. The highest (with respect to a given ranking) element

of TY that appears in the CT-polynomial A is called the leader of A; it is denoted by UA. For example, if we consider the standard ranking of TY, then the leader of the CT-polynomial

IS UA = D:2D:3Y3· Obviously, every CT-polynomial A E F {Yl, ... , Ys}" can be uniquely written in

the form A = L:~=1 IiU~, where It, 12, .. . , Id are CT-polynomials that do not contain UA (therefore, if a term v appears in any of the polynomials Ii (0 'S i 'S d), then v < UA). The CT-polynomial Id is called the initial of A and is denoted by IA .

3.3.32. DEFINITION. Let A, B E F {YI, ... , Y.}a. We shall say that the CT-polynomial A has lower rank than the CT-polynomial B (and write rk A < rk B or simply A < B), if either A E F, B ft F, or UA < UB, or UA = UB, but deguA A < deguA B. In the case when UA = UB and deguA A = deguA B, CT-polynomials A and B will be said to be of the same rank (in this case we shall write rk A = rk B).


Obviously, the equality rk A = rk B can hold for different u-polynomials A, B E F{Yl, ... , Ys}a, and rk IA < rkA for any u-polynomial A.

3.3.33. DEFINITION. Let A, B E F{Yl, ... , Ys}a. The u-polynomial A is said to be reduced with respect to the u-polynomial B, if A does not contain any power of rUB (r E Ta) whose exponent is greater than deguB B.

Let ~ ~ F{Yl, ... ,Ys}a. A u-polynomial A E F{Yl, ... ,Ys}a is said to be reduced with respect to ~, if A is reduced with respect to every u-polynomial of the set ~.

3.3.34. DEFINITION. A subset E of a u-polynomial ring F {Yl, ... , Y.}a is called autoreduced, if no element of ~ belongs to F and each u-polynomial of ~ is reduced with respect to the set of all the other u-polynomials from ~.

Obviously, leaders of any two different elements of an autoreduced set are different. Therefore (see Lemma 2.2.1), every autoreduced set is finite. The following result is usually called the reduction theorem.

3.3.35. THEOREM. Let A = {A l , ... , Ap} be an autoreduced set in a u-polynomial ring F {Yl, ... , Ys}a over au-field F. Let I(A) denote the set of all u-polynomials of F{Yl, ... , Y.}a, each of which is equal either to 1, or to a product of a finite number of u-polynomials of the form r(IA), where r ETa, A EA. Then, for any u-polynomial B E F {Yl, ... , Y.}a, there exist au-polynomial J E I(A) and a u-polynomialC E F{Yl, ... ,Y.}a, such that C reduced with respect toA and JB == C (mod [Aja).

PROOF. If au-polynomial B is reduced with respect to A, then it is sufficient to take C = B, J = 1. If B is not reduced with respect to A, then B contains a power (ruA.)k of some term rUA; (r ETa, 1 :S i :S p) such that k 2: d;, where d; = deguA . A;. Such term rUA; of the highest possible rank will be called a A-leader orthe u-polynomial B.

Let E denote the set of all u-polynomials, for which the assertion of the theorem is false. Suppose, that ~ =1= 0. Let P be a u-polynomial from E, whose A-leader v has the lowest possible rank and whose degree d = degv P is the lowest one among all u-polynomials from ~ with the A-leader v.

Obviously, P can be expressed in the form P = P l vd+ P2 , where the u-polynomial P l does not contain v, and degv P2 < d. Since P is not reduced with respect to A, v = rUA; for some r ETa, Aj E A (1 :S j :S pl. In this case v is the leader of rAJ and degv(rAj) = deguA Aj = dj , IrA; = rIA;.

J

Consider the u-polynomial Q = (rIA;)P - vd - dj (rAj)Pl . If Q contains a term w such that v < w, then w appears in P. Furthermore, in this case degw P 2: degw Q. Therefore, Q f/:. ~ (because degv Q < d), i.e. there exist au-polynomial E and au-polynomial It E I(A) such that E is reduced with respect to A and It Q == E (mod [Aja). Thus, It((r IA; )p_vd - dj (rAJ )Pl ) == E (mod [Aja), so that J P == E (mod [Aja), where J = It(rIA;) E I(A). Thus, ~ = 0, so the theorem is proved. 0

Let U denote the set of all au tor educed subsets of F{Yl, ... , Y.}a. In what follows we shall write elements of each set A E U in the order of increasing rank (so that, if A = {Al , ... ,Ap} is an autoreduced set, then rkA l < ... < rkAp).


3.3.36. DEFINITION. An autoreduced set A = {AI, ... Ap} from the ring of G"-polynomials F {Yl, ... , Ys}" is said to have lower rank than an autoreduced set B = {B1 , ... , Bq} from F {Yl, ... , Ys}" (we write rk A < rk B), if one ofthe following conditions holds:

(1) there exists kEN, 1 :::; k :::; min(p, q), such that rk Ai = rk Bi for 1 :::; i :::; k - 1 and rk Ak < rk Bk;

(2) p> q and rkA; = rkBi for i = 1, .. . ,q.

If p = q and rk Ai = rk Bi for i = 1, ... , p, then we say that autoreduced sets A and B have the same rank (and write rkA = rkB).

3.3.37. PROPOSITION. Any nonemptyset of auto reduced sets ofF{Yl, ... , Ys}" contains an autoreduced set of the lowest rank.

PROOF. Let !m be a non-empty set of autoreduced subsets of F {Yl, ... , Ys}". Let !mo = !m and for any i EN, i > 0, let !m; denote the set of all auto reduced sets A E !mi-l such that CardA ~ i and the ith lowest elemant of A is of the lowest possible rank. Then we obtain a sequence of sets !mo 2 !ml 2 !m2 2 ... with the following property: in all elements of!mi (i = 1, 2, ... ), the ith lowest 17-

polynomials have the same leader Vi. If!mi i= 0 for i = 1, 2, ... , then {Vi = Ti Yj (i)}

is an infinite sequence such that no Vi is equal to TVj for some T E T, j i= i. The existence of such a sequence contradicts Lemma 2.2.1, whence there is a smallest i such that !m; = 0 (clearly, i > 0). It is easy to see, that any element of !mi-l is an autoreduced subset of!m of lowest rank. 0

3.3.38. DEFINITION. Let J be an ideal of a G"-polynomial algebra F{Y1, ... , Ys}" over a G"-field F. An autoreduced subset of J of the lowest rank is called a characteristic set of J. (Since for any G"-polynomial A E J the set {A} is autoreduced, Proposition 3.3.37 implies that each ideal of F {Yl, ... , Ys}" has a characteristic set.)

3.3.39. EXERCISE. Let Ll, ... , Lk be linear G"-polynomials from the G"-ring R = F{Yl, ... , Ys}" (i.e. the total degree of L j (1 :::; j ~ k) as a polynomial in indeterminates TYi (T E T", 1 :::; i ~ s) is equal to 1). Show, that if ULj i= UL,

for i i= j, (1 :::; i, j :::; k), then .c = {L 1 , ... , Ld is a characteristic set of the G"-ideal [L 1 , ... , Lk]" of R.

3.3.40. PROPOSITION. Let J be an ideal of F {Yl, ... , Y.}" and let A be its characteristic set. Then J does not contain nonzero G"-polynomials reduced with respect to A. In particular, if A E A, then fA fI. J.

PROOF. Suppose, that there exists a nonzero G"-polynomial B E J reduced with respect to A. Then B and the set {A E A I rkuA < rkuB} form an autoreduced set of J, whose rank is lower than the rank of A. This contradiction proves the proposition. 0

3.3.41. PROPOSITION. Let cf> be a prime G"*-ideal ofF{Yl, ... , y.}" and let A be its characteristic set. Suppose, that f = TIAEA fA and r(A) is a free commutative semigroup, generated by elements of the form T( 1), where T E T". Then

cf> = [A]" : f(A).


PROOF. Let BE [A]/7 : r(A). Then there exists au-polynomial C E r(A) such that CB E [AJ/7 ~ ~. Since the ideal ~ is prime and C fI. ~ (otherwise, IA E ~ that contradicts Proposition 3.3.40), we have B E ~, hence, [AJ/7 : r(A) ~ ~. Conversely, if B E ~, then, by Theorem 3.3.35, there exist au-polynomial L E r(A) and au-polynomial E such that LB == E (mod [A](7). Therefore, E E ~, hence E = 0 (see Proposition 3.3.39), so that E is reduced with respect to A and BE [AJ/7 : r(A). The proposition is proved. 0

The following theorem is an analogue of the Ritt-Raudenbush basis theorem for u-rings.

3.3.42. THEOREM. Let Ro be a Ritt's u-ring and R = RO{'11, ... , '1n}/7 be a u-overring of Ro generated by a finite family of elements '1 = {'11, ... , '1n}. Then R is also a Ritt's u-ring.

PROOF. Obviously, it is sufficient to prove the theorem for the ring of u-polynomials R = Ro{Yl, ... , Yn}" in u-indeterminates Yl,··., Yn over Ro. Assume that the conclusion is not true. Then the set of perfect ideals of R without basis contains a maximal element P, which is a prime u-ideal (see Theorem 3.3.30(3)). Let Po = P () Ro. Since Po is a perfect u-ideal of Ro, it has a basis C: Po = {C}/7. Let S be the set of all u-polynomials in P which have no coefficients from Po. It is easy to see that S ~ R \ Ro and S f. 0 (if S = 0, then each u-polynomial I E P has a coefficient from Po. If we subtract the corresponding monomial from I, then the remainder I' E P also has a coefficient from Po and so on. We obtain, that all coefficients of I belong to Po, hence P = {C} /7, that contradicts to the choice of P). Let us construct an autoreduced set F with elements from S as follows.

Let II (the first element of F) be a polynomial of the lowest rank in S. Suppose, that II, ... ,Ii (i 2: 1) have been constructed and they form an autoreduced set. If S contains no polynomials reduced with respect to II, ... , Ii, then we set F = {II, ... , Ii}. Otherwise, we choose au-polynomial Ii+l of the lowest rank reduced with respect to II, ... , Ii. Obviously, the set {II, ... , Ii, Ii+l} is autoreduced. After a finite number of steps this process will be completed and we shall obtain an autoreduced set F = {II, ... , Ir} ~ S such that S \ F contains no u-polynomial reduced with respect to F. Note, that if a u-polynomial A E P is reduced with respect to F, then all its coefficients belong to Po. Indeed, let A' be the sum of all monomials in A whose coefficients do not belong to Po. Then A' E P and A' is reduced with respect to F, therefore A' = O.

Now, let us show that if I j is the initial of a polynomial Ii (j = 1, ... , r) then Ij fI. P. Indeed, if Ij E P for some j (1 ~ j ~ r) then Ij, as a u-polynomial that is reduced with respect to F, has a coefficient from Po, hence Ii also has a coefficient from Po, that contradicts the inclusion Ii E S.

Let I = It h, ... Ir . Since P is a prime ideal, I fI. P. Furthermore, since P is a maximal perfect u-ideal without u-basis, {P, I}/7 = {B}/7 for a finite set B ~ R. Obviously, a set {gl, ... , g., I}, where gl, ... , g. E P, can be chosen as such set B. Let 9 E P. By the Reduction Theorem (see Theorem 3.3.35) there exist a u-polynomial h E [II, ... , Ir J/7 and elements Tl, ... , Tg E T/7; k1, ... , kg E N such that the u-polynomial 9 = j 9 - h, where j = Tl (I)k' ... Tq(I)kq, is reduced with respect to F. Since 9 E P, all coefficients of 9 belong to Po, so that 9 E {C} /7.


Therefore, jg E {G,h, ... ,fr}", hence [r1(Ig)]kl ... [rq(Ig)]k. E {G,fI, ... ,fr}", so that Ig E {G,h,···, fr}"

The last inclusion implies IP = {Ig I 9 E P} ~ {G, h, ... , fr },,' so we can use Exercise 3.3.25(5) to write the following sequence of inclusions:

P = P n {P, I}" ~ P n {I,gl, ... ,g.}" = {IP,glP, ... ,g.P}"

~ {JP,gl, ... ,g.}" ~ {G,h, ... ,fr,gl, ... ,g.}" ~ P.

Thus, G, h, ... , fr, gl, ... ,g. is a basis of P, that contradicts the choice of P. 0

A representation J = PI n ... n p. of an ideal J of a ring R as an intersection of a finite number of prime ideals PI, ... ,p. is called irreducible if Pi rt. Pj for i "f; j. Prime ideals Pi from this representation, are called essential prime divisors of J.

3.3.43. THEOREM. Let R be a Ritt's u-ring and let J be a perfect u-idealof R. Then:

(1) there exists an irreducible representation of J as an intersection of a finite number of prime u-ideals:

J = PI n·· ·nP.,

where Pi ~ ni;i:j Pj for any i = 1, ... , s; (2) u-ideals PI' ... ' p. are refiexive and uniquely determined by J.

(3.3.3)

PROOF. Let us prove, firstly, the existence of the representation (3.3.3). Assume that there exist perfect u-ideals of R which cannot be represented as an intersection of a finite number of prime u-ideals. Since R is a Ritt's u-ring, the set of all such ideals contains a maximal element I. Obviously, the u-ideal I is not prime, hence there exist a, bE R such that ab E I, and a f/; I, b f/; I. Applying Exercise 3.3.25(5), we obtain I ~ {I,a}" n {I,b}" = {I2 ,Ib,Ia,ab} = I, i.e. 1= {I,a}" n {J,b}". By the choice of I, each of the perfect u-ideals {I, a}", {I, b}" can be represented as an intersection of a finite number of prime u-ideals, therefore, J also has such a representation. This contradiction proves the existence of the representation (3.3.3) for any perfect u-ideal J. Now, let J = Q1 n· . ·nQr, where Q1, ... , Qr are prime uideals of R, and let Qi l , ... , Qip be the minimal (with respect to inclusion) elements of {Ql, ... , Qr}. Then J = Qil n ... n Qip ' and this representation is obviously irreducible.

Let us prove the uniqueness of the representation (3.3.3). Let PI n ... n p. = Ql n ... n Qr, where Pi (1 ~ i ~ s) and Qj (1 ~ j ~ r) are prime u-ideals such that Pi 2 Pj only for i = j (1 ~ i,j ~ s) and Qk 2 Q/ only for k = I (1 ~ k,l ~ r). Then the prime ideal PI must contain one of the ideals Qj (1 ~ j ~ r). Without loss of generality, we can assume, that PI ~ Q1. Similarly, Q1 contains some Pi (1 ~ i ~ s). Since PI ~ Pi for i"f; 1 we hav; i = 1, i.e. PI = Q1. Similarly, (may be after renumbering of Q2, ... , Qr) we have r = sand P2 = Q2, ... , p. = Q •.

Let us show that all prime u-ideals in (3.3.3) are reflexive. Denote by Pi the reflexive closure of u-ideal Pi (1 ~ i ~ s). Since u-ideal J is reflexive, J = n:=lPi, therefore, the set of essential prime divisors of J may be taken from the set {PI, ... , P.}. By uniqueness of the irreducible representation (3.3.3), we have Pi = Pi for all i = 1, ... ,s. The theorem is proved. 0


3.3.44. EXERCISE. Show, that every perfect oo-ideal in arbitrary oo-ring may be represented as an intersection of some (may be infinite) number of prime 00* -ideals.

3.3.45. EXAMPLE. (see [CoR65, Ch. 3, § 11]). Let Q{y}u be the ring of 00-

polynomials over rationals Q, that is considered as an ordinary oo-field whose basic set consists of the identity automorphism. Exersice 3.3.25(5) shows that

{y2 _ l}u = {y - l}u n {y + l}u. (3.3.4)

Let us set z = y -1 and order the terms of a oo-polynomial A E {y -l}u by the powers of r(z) (r E Tu). Obviously, A does not contain a constant term. Therefore, {y - l}u is a prime oo*-ideal of Q{y}u and similar conclusion can be drawn about {y+1}u. Besides, this property ofelementsof{y-1}u shows that y+1 fi {y-1}u, {y + l}u ct {y - l}u, and {y - l}u ct {y + l}u, hence {y - l}u and {y + l}u are essential prime divisors of {y2 -l}u. Thus, (3.3.4) is the irreducible representation of the oo-ideal {y2 - l}u as an intersection of prime oo-ideals.

Remark, that oo-ideal I = [y2 - l]u cannot be represented as an intersection of two oo-ideals hand 12 with the perfect closures {y-1}u and {y+ l}u, respectively, such that I ~ h, I ~ 12 (see [CoR65, Ch. 3, § 15]).

3.3.46. DEFINITION. Two oo-ideals J1 and h of au-ring R are called disjoint (or coprime) if {It,J2}u = R.

3.3.47. THEOREM. Let R be a Ritt's oo-ring. Then every perfect oo-ideal J:f (1) of R can be represented as a finite intersection of uniquely defined pairwise disjoint perfect oo-ideals J1, ... , Jk, different from (1) and such that none of them can be represented as an intersection of two disjoint perfect oo-ideals different from (1).

PROOF. See [CoR65, Ch. 3 § 20]*. 0

Perfect oo-ideals It, . .. , Jk, whose existence is proved in the last theorem, are called essential disjoint divisors of the perfect oo-ideal J.

&-varieties.

3.3.48. DEFINITION. Let F be au-field, F{Y1, ... , Y.}u the ring of oo-polynomials over F in oo-indeterminates Y1, ... , y., and & an arbitrary family of oo-overfields of :F. The set Me(~) of all s-tuples a = (at, ... , a.), with coordinates from au-field of the family &, that are solutions of every oo-polynomial of ~, is called a & -variety defined by the subset ~ ~ :F{Y1' ... ' Y.}u.

In this case the oo-field :F is also called a basic oo-field, and the set Me(~) is called the &-variety of the set ~ over:F or over :F{Y1, ... ,Y.}u. Now, let A be a set of s-dimensional points, whose coordinates belong to a oo-subfield of F from the family &. We shall write that A is an &-variety over F {Y1, ... , Y.}u, if there exists (may be not uniquely determined) a set ~ ~ :F{Y1, ... , Y.}u such that A = Me(~).

Below, while considering &-varieties, we suppose that the ring of oo-polynomials :F {Y1, ... , Y.}u is fixed.

* All statements in [CoR65) are formulated and proved for ordinary difference rings. However. in this case the generalization for partial difference rings is trivial.


3.3.49. EXAMPLE. Let M be the field of functions of one complex variable meromorphic on the whole complex plane, that is considered as an ordinary u-field with a basic endomorphism a such that aJ(z) = J(z + 1) for any J(z) E M. If cI> ~ M{Y1, ... ,Ys}" and & = {M}, then the &-variety defined by cI> consists of all s-tuples (h(z), ... , J.(z)) EM' that are solutions of the system of functional equations P(Y1,"" Ys) = 0 (P E cI».

3.3.50. DEFINITION. Let F be au-field, 9 = F(X1, X2, ... ) the field of rational fractions in countable set of indeterminates Xl, X2, ... , and Q the algebraic closure of 9. The family U of all u-overfields of F defined on different subfields of Q is called the universal system of u-overfields of F. If cI> ~ F {Y1, ... , y.}", then U -variety Mu(cI» (denoted below simply by M(cI>)) is said to be the variety defined by the set cI> over F (or over F {Y1 , ... , Ys}" ). Below, if we say that a set A is a variety we shall mean that there exists a set cI> ~ F{Y1,"" Y.}" such that A = M(cI» = Mu(cI».

3.3.51. PROPOSITION. Let F be a u-field and "'I = ("'11, ... , "'I.) an arbitrary s-tuple whose coordinates belong to a u-overfield of F. Then there exists as-tuple ( = ((1. ... , (.) u-equivalent to "'I, whose coordinates belong to a u-field contained in an universal system U of u-overfields of F.

PROOF. Denote by J the set {T("'Id IT E T", i = 1, ... , s} and choose a transcendence basis J' of the extension F(J) :2 F such that J' ~ J. Since the set J' is finite or countable, the field F(J') is isomorphic to a subfield 91 of the rational fraction field 9 = F(X1, X2, ... ) in a countable set of indeterminates Xl. X2, .... Moreover, there exists an isomorphism cp : F(J') ~ 91 whose restriction on F is the identity mapping. Since the field F(J) is algebraic over F(J'), the algebraic closure Q of 9 contains a subfield Q1 such that 91 ~ Q1 and there exists an isomorphism if; : F(J) -t Q1, whose restriction on F(J') coincides with cpo As a field, F("'I1,"" "'I.)" coincides with F(J), therefore, the isomorphism if; induces a structure of u-field on Ql, for which if; : F ("'11, ... , "'I.)" -t Q1 is a u-isomorphism identical on :F. Let ( = ((1. ... , (.)", where (i = if; ( "'I;) (1 :5 i :5 s). Then the points "'I and ( are u-equivalent and all entries of ( belong to au-field Q1, contained in U. The proposition is proved. 0

Properties of &-varieties. Let & be an arbitrary family of u-overfields of au-field F and F {Y1, ... , y.}"

the ring of u-polynomials over:F. We shall use the above notation: if cI> ~ F {Y1, ... , y.}", then the &-variety defined by cI> will be denoted by Me( cI»; if A is a set of s-tuples, whose coordinates belong to a u-extension of F, then the set of all upolynomials {J E F {Y1, ... , Ys}" I J(a1, ... , a.) = 0 for any a = (a1, ... , as) E A} will be denoted by cI>(A).

The following proposition describes the properties of &-varieties, which are similar to the appropriate properties of algebraic varieties and can be easily checked (see, for example, [VdW67, § 126]).

3.3.52. PROPOSITION. With the preceding notation, the following assertions hold:

(1) ifcI>l and cI>2 are subsets of the ring F{Yl,""Y'}" and cI>1 ~ cI>2, then Me(cI>2) :2 ME(cI>I)i


(2) if Al and A2 are two sets of s-tuples over au-field :F and Al C A2, then (A2) ;2 (Ad ; -

(3) if A is an E-variety, then A = Md(A)); (4) if h, ... ,Jk are u-ideals of :F{Yl, ... ,Y.}" and J = Jl n ... n Jk, then

Me(J) = Me(Jd u··· u MdA); (5) if Al , ... , Ak is a finite family of E-varieties and A = Al U· .. U Ak, then A

is also an E-variety and (A) = (Ad n··· n (Ak); (6) the intersection of any family of E-varieties is an E-variety.

3.3.53. DEFINITION. An E-variety Al is called a subvariety (proper subvariety) of an E-variety A, if Al ~ A (respectively, Al ~ A).

3.3.54. DEFINITION. An E-variety A is called reducible if it may be represented as a union of two proper subvarieties. If there exists no such representation, the variety A is called irreducible.

Suppose that an E-variety A is represented as a union of a finite number of irreducible E-varieties A l , ... ,Ak:

(3.3.5)

Representation (3.3.5) is called irreducible if A; cf:. Aj for i =j:. j (1 ~ i,j ~ k). The proof of the following statement is similar to the proof of the corresponding

result for algebraic varieties (see [VdW67, § 126]).

3.3.55. PROPOSITION. E-variety A is irreducible iff (A) is a prime reflexive u-ideal of :F {Yl, ... ,Y.}".

PROOF. Suppose, firstly, that an E-variety A is reducible: A = Al U A2, where Al, A2 are proper subvarieties of A. By Proposition 3.3.52, (A) = (Adn(A2), where (A) ~ (A;) (i = 1,2). If / E (Ad \ (A), 9 E (A2) \ (A), then /g E (A), and the ideal (A) is not prime.

Now, suppose that an [-variety A is irreducible. If the u-ideal (A) is not prime, then there exist u-polynomials /,g ft (A) such that /g E (A). Let Al = Md(A)u{f}), A2 = Md(A) U{g}). By Proposition 3.3.52(4), Al UA2 = Md(A)) = A, and the irreducibility of A implies A = Al or A = A2. However, the first equality contradicts the condition / ft (A) , and the second one contradicts the condition 9 ft (A). Thus, (A) is a prime u-ideal and the proposition is proved. 0

3.3.56. THEOREM. Every E-variety has a unique representation as a union of finite number of irreducible E-varieties none of which contains another.

PROOF. Let A be an E-variety and let (A) = P l n ... n Pr be an irreducible representation of the perfect u-ideal (A) of the ring :F {Yl, ... , Y.}" as an intersection of prime u-ideals. By Proposition 3.3.52(3,4)' we have A = Me((A)) = MdPl) U··· U Me(Pr). Let us set Ai = Me(Pi) and show that Al , ... ,Ar are irreducible E-varieties. By Proposition 3.3.52(5), we have (A) = (Ad n ... n (Ar), and evidently, (Ai) ;2 Pi (i = 1, ... , r). Suppose that / E (Ad and 9 is a u-polynomial such that 9 E P2 n··· n Pr , 9 ft Pl· Then 9 E (A2) n··· n (Ar),


hence, f 9 E (A) ~ Pl. Since 9 rt. PI, we have f E Pl· Therefore, (AI ) :::: PI, that implies (see Proposition 3.3.55) the irreducibility of the t'-variety AI' The irreducibility of all t'-varieties A; (1 :S i :S s) can be proved in the same way.

If Ak ~ AI (1 :S k, I :S r), then Pk :::: (Ak) ;2 (At) :::: Pt, that can be only for k :::: I. Thus, A :::: Al U ... U Ar is an irreducible representation of A as a union of irreducible t'-varieties. To show the uniqueness of such a representation suppose that A :::: Al U ... U Ar :::: A~ U ... A~, where A;, Aj are irreducible t'-varieties (1 :S i :S r, 1 :S j :S s) and both representations are irreducible. Then Al :::: A n Al :::: (A~ n Ad u ... U (A~ n Ad. By Proposition 3.3.52(6), every intersection Aj n Al (1 :S j :S s) is an t'-variety. Since the t'-variety Al is irreducible, Al :::: Al n AI. for some k (1 :S k :S s), hence, Al ~ AI.. Similarly, AI. ~ AI for some I (1 :S I :S r), that implies Al ~ AI, hence, I :::: 1, Al :::: AI.. In the same way we can prove that each t'-variety Ai (1 :S i :S r) coincides with a (unique) t'-variety Aj(i). The theorem is proved. 0

Irreducible t'-varieties in the irreducible representation (3.3.5) of an t'-variety A are called irreducible components of A. Note, that if AI, ... , Ak are all irreducible components of an t'-variety A, then Ai 1;. Uj:;tiAj for each i:::: 1, ... , k. Indeed, if, for example, Al ~ A2U ... Ak , then we have two different irreducible representations A:::: Al U A2 U ... U Ak :::: A2 U ... U Ak of A that contradicts Theorem 3.3.56.

The proof of Theorem 3.3.56 implies the following result.

3.3.57. PROPOSITION. Let AI, ... ,Ak be irreducible components of an t'-variety A. Then the prime (T*-ideals (Ad, ... , (Ak) are essential prime divisors of the (T* -ideal (A) .

Varieties. As Proposition 3.3.52 shows, the mapping A -t (A) establishes a one-to-one

correspondence between the set of t'-varieties over :F {YI, ... , YS}C7 and a subset P of the set of all perfect (T-ideals of :F {YI, ... , Ys} C7' Denote by P* the image of the set of all irreducible t'-varieties under this mapping. Propositions 3.3.52 and 3.3.57 imply that P consists exactly of those perfect (T-ideals whose essential prime divisors belong to P*. Let us show that if t' is the universal system U of (T-overfields of :F then P* contains all prime (T* -ideals of :F {YI, ... , Ys} C7, so that P consists of all perfect (T-ideals of this ring. Indeed, let P be a prime (T* -ideal of :F {YI, ... , YS}C7' Let us determine a variety (i.e., a U-variety) A such that p:::: (A). If P :::: (1), then we set A :::: 0, otherwise we take as A the set A(P) consisting of all solutions A:::: (AI, ... , As) of the system of all (T-polynomials f E P in a (T-extension of:F from the universal system U. In the case when P is a proper (T-ideal and 1/ :::: (1/1, ... , 1/s) is its generic zero, there exists (see Proposition 3.3.41) a s-tuple ~ :::: (~l"'" ~s) with coordinates from some (T-field r;; E U that is equivalent to 1/. Since ~ is a generic zero of the (T-ideal P, we have p:::: (O ;2 (A). However, the definition of the variety A implies P ~ (A), therefore, P :::: (A).

Theorem 3.3.56 for t' :::: U implies the similar structural theorem for varieties. Irreducible components of A as an U-variety will be called irreducible components of the variety A.

As it has been shown, mapping 'P : A -t (A) is a one-to-one correspondence between the set of all varieties over :F {YI, ... , y.}C7 and the set of all perfect (T-ideals


of F{YI, ... , Y.}". The equality A = A(~(A)) shows that the mapping P -t A(P) (where P is a perfect O"-ideal of F {YI, ... , Y.},,) is inverse to <p, and Proposition 3.3.57 gives us the action of this correspondence on irreducible components of the variety. As a result, we have the following theorem.

3.3.58. THEOREM. Let F be a O"-field and F{YI, ... , y.}" the ring of O"-polynomials in O"-indeterminates Yl, ... , Y. over F. Then the correspondence A -t ~(A) considered above is a one-to-one mapping from the set of all varieties over F{Yl, ... , Y.}" onto the set of all perfect O"-ideals of F{YI, ... , Y.}". The inverse mapping is given by the formula P -t A(P), where P is an arbitrary perfect 0"

ideal of F {Yl, ... , Y.}". Moreover, the correspondence A -t ~(A) maps irreducible components of an arbitrary variety B onto essential divisors of ~(B). In particular, irreducible varieties over F {Yl, ... , Y.}" one-to-one correspond to the prime O"-ideals ofF{YI,.·.,Y.},,·

By a generic zero of an irreducible variety over au-field F we shall always mean the generic zero of the corresponding prime 0"* -ideal. If the contrary is not said explicitly, we shall assume that coordinates of the generic zero lie in a O"-field from the universal system of O"-overfields of F.

3.3.59. COROLLARY. Let J be a perfect O"-ideal of the ring of O"-polynomials F{Yl, ... , Y.}", over au-field F. Then the following statements hold:

(1) J = ~(A(J)); (2) A(J) = 0 iff J = (1).

3.3.60. DEFINITION. A set £ of O"-overfields of au-field F is called a complete system of O"-overfields, if different perfect O"-ideals of any ring of O"-polynomials over F correspond to different £-varieties.

Note, that Theorem 3.3.58 implies that an universal system U of O"-overfields of au-field F is complete.

The following theorem is an analogue of the Hilbert's Nullstellensatz for O"-fields.

3.3.61. THEOREM. Let F{Yl, ... ,Y.}" be the ring of O"-polynomials in 0"

indeterminates Yl,""Y' over au-field F. Suppose that f ~ F{YI, .. ·,Y.}", f E F {YI, ... ,Y.}" and A(f) is the variety defined by the set f. Then every point of A(r) is a solution of the O"-polynomial f iff f E {f}".

PROOF. If every point of A(f) is a solution of f then A(f) = A(f U {f}), therefore, A( {f},,) = A( {f U {f}},,) and, by Theorem 3.3.58, {f}" = {f U {J}}", hence, f E {f}". The sufficiency of the condition of the theorem is evident. 0

3.3.62. DEFINITION. Varieties Al and A2 over the ring F{Yl, ... ,Y.}" of O"-polynomials over a field F are called separated if Al n A2 = 0.

3.3.63. EXERCISE. Prove the following statements.

(1) Varieties Al and A2 over F{YI, ... ,Y.}" are separated if and only if {~(Al), ~(A2))" = (1), i.e., the ideals ~(Ad and ~(A2) are disjoint.

(2) Two perfect ideals Jl and J2 of F{Yl, ... , Y.}" are disjoint iff the varieties A(Jt} and A(h) are separated.


Suppose that A can be represented as a union of non-empty varieties AI, ... , Ar which are pairwise separated (i.e., A n Aj = 10 for i i- j). Suppose also that none of Ai can be represented as a union of non-empty varieties. Then the varieties AI, ... , Ar are called essential separated components of A.

3.3.64. THEOREM.

(1) Every non-empty variety A can be represented as a union of a uniquely determined family of its essential separated components. Essential separated components of A in this representation are unions of irreducible components of this variety.

(2) Essential disjoint divisors of the perfect ideal (A) corresponding to a variety A are the ideals determined by the essential separated components of A.

PROOF. (1). Obviously, there exists a representation A = Al U ... U Ar where AI, ... , Ar are essential pairwise separated components of A. If A = A~ U ... U A~ is another representation of A as a union of its essential pairwise separated components, then A~ ~ A 1 U·· ·UAr . Let B' be an irreducible component of A~ and let B1 , ... , Bq be all essential irreducible components of the varieties AI, ... , Ar . Then B' ~ Bl U ... U Bq , hence B' n Bi i- 10 for some index i (1 ::; i ::; q), i.e., B' ~ Bi. Therefore, every irreducible component of A~ is contained in an irreducible component of Aj for some index j, 1 ::; j ::; r. (If two irreducible components of A~ are contained in different varieties Ai, Aj (i i- j) then A~ may be represented as a union of two proper subvarieties, one of which is contained in Ai, and the other one is contained in Uj;ti A j . These subvarieties are separated in contrast to the supposition that A~ is an essential separated component.) We may assume that j = 1, i.e., A~ ~ AI. Similarly, Al ~ A~ for some k (1 ::; k ::; s) hence A~ = AI. In the same way we obtain that r = s and, after some renumeration of AI, ... , Ar, the equalities Ai = Ai for all i = 1, ... , r.

(2). Obviously, if AI, ... , Ar are essential separated components of A, then u-ideals (A) (1 ::; i ::; r) are pairwise disjoint. By Proposition 3.3.52(5), we have (A) = (Al) n ... n (Ar). In order to prove that (A) (1 ::; i ::; r) are essential disjoint divisors of the u-ideal (A) , it is sufficient to show that the ideal (A) (1 ::; i ::; r) cannot be represented as (A) = <1>: n <1>:', where <1>:, <1>:' are disjoint perfect u-ideals of a u-polynomial ring that are different from (1) and strictly contain (Ai). If such representation exists, then by Proposition 3.3.52, A = Mu((A)) = Mu(D U Mu(i)' therefore, the varieties Mu(D and Mu(i') are nonseparated, so that the ideals <1>:, <1>:' are not disjoint. The theorem is proved. 0

u-field extensions. Let g be a u-overfield of au-field :F, A ~ g and v E g. We say that v is

u-algebraically dependent on a set A over :F if v is u-algebraic over the u-field :F(A)". Obviously, in this case there is a finite family 711, ... , 71. E A such that v is u-algebraic over the field :F (711, ... , 71. )".

The four following lemmas can be proved in the same way as in the ordinary case (see [CoR65, Ch. 5 §§ 2-6]).


3.3.65. LEMMA. A family r of elements of a u-overfield of au-field F is u-algebraically dependent over F iff there exists an element v E r which is ualgebraically dependent on the set r \ {v} over :F.

3.3.66. LEMMA. Let r be a family of elements of a u-overfield of au-field :F. Then r contains a maximal subset u-algebraically independent over F, i.e., there exists a subset r 1 ~ r that is u-algebraically independent over F and that is not contained in another u-algebraically independent over F subset of r.

3.3.67. DEFINITION. Let r be a subset of au-extension g of a u-field F. Then a maximal u-algebraically independent over F subset B ~ r (whose existence is proved in Lemma 3.3.66) is called a u-transcendence basis of r over F. If r = g, then B is called a u-transcendence basis of the u-extension g of F.

3.3.68. LEMMA. Let g be a u-overfield of au-field F, let r, r' be two subsets of g, and let v, Ul, ... , U m be some elements of g. Then the following statements hold:

(1) if v E r, then v is u-algebraically dependent on rover F; (2) if v u-algebraica1ly depends on rover F and each element of r is u

algebraically dependent on a set r' over F, then v is u-algebraically dependent on r' over F;

(3) if v u-algebraically depends on a set {Ul, ... , urn} over F but v is not u-algebraica1ly dependent on the set {Ul, ... , urn-d over F, then Urn is (J'-algebraically dependent on {Ul, ... , Um-l, v} over F;

(4) if r' ~ r and {Vl' ... , v.} is a (J'-algebraica1ly independent over F subset of r such that each element Vi (1 SiS s) is u-algebraically dependent on r' over F, then there exist elements Wl, ... , w. E r' such that each Wi

(1 SiS s) is (J'-algebraically dependent on the set r" (over F), which can be obtained from r' by replacing Wj by Vj (j = 1, ... , s).

3.3.69. LEMMA. Let r be an arbitrary subset of a (J'-overfield of a (J'-field F and let B be a subset of r that is u-algebraically independent over F. Then the set B is a (J'-transcendence basis of rover F iff every element of r is u-algebraically dependent on B over F.

3.3.70. THEOREM. Let r be an arbitrary subset ofau-overfield g ofau-field F. Then all u-transcendence bases of rover F either contain the same finite number of elements or are infinite.

PROOF. See [CoR65, Ch. 5, § 7, Theorem 2]*. 0

This theorem allows to give the following definition.

3.3.71. DEFINITION. Let a (J'-field g be a u-extension of au-field F and r ~ g. A (J'-transcendence degree of rover F (denoted (J'-trdeg.1" g) is the number of elements in any u-transcendence basis of rover F, if this number is finite, or infinity in the contrary case.

* All statements in [CoR65] are formulated and proved for ordinary difference rings, but in this case the generalization to partial difference rings is trivial.


Let :F be au-field, g = :F(f)" , and B a O'-transcendence basis of f over F. By Lemma 3.3.68(2), B is also a O'-transcendence basis of g over F, so we have the following proposition.

3.3.72. PROPOSITION. A O'-transcendence basis of au-field g over a O'-subfield :F can be chosen from any system of O'-generators of g over :F.

3.3.73. COROLLARY. Suppose that :F is a O'-field and Til, ... , 1'1m a.re elements of a O'-overfield of :F. Then O'-trdeg,T :F(1'11 , ... , 1'1m}" ~ m. If :F(1'11, ... ,1'1m}" = :F ((1, ... , (.}", where {(I, ... , (.} is a family of O'-algebraica.lly independent over :F elements of :F(1'11, ... , 1'1m}" , then m = s.

3.3.74. COROLLARY. Suppose that :F is a O'-field and S = F{Y1, ... , Y.}" is the ring of O'-polynomia.ls over :F in O'-indeterminates Y1, ... ,Y •. If kEN, k f. s, then S cannot be a ring of O'-polynomia.ls over :F in any O'-indeterminates Zl, .•. , Zk.

3.3.75. DEFINITION. Au-extension g of au-field :F is called O'-algebraic if every element of g is u-algebraic over :F.

By Lemma 3.3.68, au-extension g = :F(1'11 , ... , 1'1m}" of au-field :F is O'-algebraic iff the elements 1'11, ... , 1'1m are u-algebraic over:F. Note also that g is u-algebraic over :F iff O'-trdeg,T g = O.

3.3.76. THEOREM. Let :F,g, and H be u-fields such that:F ~ g ~ H. Ifone of the numbers u-trdeg,T H, u-trdeg,T g + u-trdegg H is finite, then the other one is also finite and O'-trdeg,T H = O'-trdeg,T g + O'-trdegg H.

PROOF. See [CoR65, Ch. 5, § 7, Theorem 3]*. 0

The following theorem has been proved by P. Cohn for ordinary case (see [CoR65, Ch. 5, Th. 18]) and was generalized by P. Evanovich for partial u-fields (see [Ev84]).

3.3.77. THEOREM. Let:F, g, and H be O'-fields such that F ~ g ~ H. If H is generated over :F by a finite system of O'-generators, then g has a finite system of generators as a O'-extension of :1".

Rings of difference operators. Difference modules. Let R be a difference ring with a basic set 0' = {0'1, ... , O'n} and let T = T" be

a free commutative semigroup generated by 0'1, ... , O'n. We shall call the number ord T = E~l k; the order of an element T = O'~l ... O'~n E T (k1, ... , kn EN). The set of elements T E T such that ord T = r (r E N) will be denoted by Tr , and the set {T E T I ordT ~ r} will be denoted by T(r).

3.3.78. DEFINITION. Expressions of the form ETeT aTT, where aT E R for any T E T and only finite number of elements aT are different from zero, are called difference (or 0'-) operators over a difference ring R. Two O'-operators ETeT aTT and ETeT bTT are defined to be equal iff aT = bT for any T E T.

* All statements in [CoR65] are formulated and proved for ordinary difference rings, but in this case the generalization to partial difference rings is trivial.


The set of all u-operators over au-ring R will be denoted by U. The set U may be considered as a ring in which

LaTT+ LbT'T= L(aT+bT)T, TeT TeT TeT

a L aT'T = L (aaT )'T, TeT TeT

( L aT'T) 'T1 = L aT ('T'T1), TeT TeT

( L aT'T) a = L(aT'T(a))'T TeT TeT

for any u-operators ETeT aT'T, ETeT bT'T E U and for any a E R, 'T1 E T. This ring will be called the ring of difference (or u-) operators over the difference (or u-) ring R.

The order ord w of au-operator w = ETeT aT T E U is defined as max{ ord TlaT =F O}.

3.3.79. DEFINITION. Let R be a difference ring with a basic set u = {al, ... , an} and let U be a ring of u-operators over R. Then any left U-module will be called a difference (or u-) R-module. In other words, a R-module M will be called a difference R-module (or a u-R-module) if elements of u act on M in such a way that the following conditions hold:

(1) a(x + y) = ax + ay; (2) a({3x) = (3(ax); (3) a(ax) = a(a)ax

for any x, y EM; a, (3 E u; a E R.

If R is a difference (u-) field, then a u-R-module M is also called a difference vector R-space (or a vector u-R-space).

The ring of u-operators U over a difference ring R with a basic set u = {a1, ... ,an} may be considered as a graded ring (with positive grading): U = EBqez U(q), where U(q) = {w = Ei=l aTj'Tj lord 'Tj = q (j = 1, ... , m)} for any q E Nand U(q) = 0 for q < O. Below, while considering U as a graded ring, we shall suppose that it is supplied with this grading.

The following proposition can be proved in the same way as the similar proposition in commutative algebra (see [AM69, Ch. 11, p. 143]).

3.3.80. PROPOSITION. Let M = EBqez M(q) be a graded module over the ring of u-operators U over au-ring R. Then every Abelian group M(q) (q E Il) is a R-module. If M is a finitely generated U -module then M(q) is a finitely generated R-module for any q E Il.

3.4. Inversive Difference Rings and Modules

Let R be an inversive difference ring with a basic set u = {a1, ... , an} of mutually commuting automorphisms.

3.4. INVERSIVE DIFFERENCE RINGS AND MODULES 159

As in the previous section, such ring will be called a u' -ring and the set 0'1, ... , an, all, ... ,0';;1 of ring automorphisms of R will be denoted by u*. Furthermore, r CI

will denote the free commutative group generated by aI, ... ,an, and the number ord"l = 2:7=1 Ikil will be called the order of an element "I = a~l, ... , a~n E r CI

(k1' ... ,kn E f"J). (We use multiplicative notation for the operation in r a, the identity element of this group will be denoted by 1. The fact that the identity of the ring R is also denoted by 1, will not lead to any confusion.)

Since any u* -ring is a difference (u-) ring, all definitions and results of the previous section may be applied to u* -rings. In this case we use the same notation for the u* -structures as for the corresponding u-structures (with respect to prefix u*instead of u-). Thus, if B is a subset of a u* -ring R, then the u* -ideal generated by B will be denoted by [B] ([B] is the smallest u* -ideal of R containing B; this ideal coincides with the ideal of R generated by the set h(b) I "I Era, b E B}). If Ro is a u*-subring of a u*-ring R (i.e. alRo is an automorphism of Ro for any a E u*) and B S; R, then the smallest u* -subring of R containing Ro and B will be denoted by Ro{B} and will be called the u*-overring of Ro generated by the set of (u*-) generators B (it is clear, that Ro{B} coincides with Ro[h(b) 1"1 Era, bE B}]). If Ro and Rare u*-fields (where Ro is a u*-subfield of R), the smallest u*-subfield of R containing Ro and B (B S; R) will be called the u* -overfield (or u* -extension) of R o, generated by the set of (u* -) generators B. This u* -overfield coincides with the field Ro( h(b) 1"1 Era, bE B}) and will be denoted by Ro(B). If f : R -+ S is a uhomomorphism of u* -rings then we shall also say that f is a u* -homomorphism (it is clear, that the equality fa (a) = af(a) (a E R) is valid not only for any a E u but also for any a E u*). It is easy to check that the kernel of any u* -homomorphism f is a u* -ideal and vice versa, if g is an epimorphism of a u* -ring R onto a ring S whose kernel Ker 9 is a u* -ideal of R, then S has a unique structure of a u* -ring with respect to which 9 is a u*-epimorphism. In particular, if I is a u*-ideal of R, then the factor-ring R/ I has a unique structure of u* -ring with respect to which the canonical epimorphism R -+ R/ I is a u* -epimorphism. This u* -ring will be called the u* -factor-ring of R by the u* -ideal I.

Several examples of u* -rings were given in the previous section (see Examples 3.3.3 and 3.3.6). The following exercise allows to determine if the polynomial ring over an ordinary u* -ring without zero divisors is a u* -ring. (u* -ring is said to be ordinary if Card u = 1 and partial if Card u > 1.)

3.4.l. EXERCISE. Let R be an ordinary u*-ring without zero divisors with a basic set u = {a}. Example 3.3.3.2 shows that the univariate polynomial ring R[x] will be a u-overring of R, if we extend a on R[x] defining the value a(x) E R[x] in an arbitrary way. Prove that R[x] is a u*-overring of R iff a(x) = ax + b (a, bE R) where a is an invertible element of R.

3.4.2. DEFINITION. Let R be an inversive difference ring with a basic set u = {a1,"" an}. A family (Vi)iEl from some u*-overring of R is called u*-algebraically dependent over R, if the family {"I ( Vi) 1"1 E r CI, i E I} is algebraically dependent over R; in the contrary case the family (Vi )iE! is called u* -algebraically independent over R or a family of u* -indeterminates over R. If the family has only one element, then this element is called u* -algebraic or u* -transcendental over R, respectively.

It is easy to see that a family (Vi )iE! of elements of a u* -overring S of a u* -ring


R is a* -algebraically dependent over R iff it is a-algebraically dependent over R.

3.4.3. DEFINITION. If R is a a* -ring and (Yi)iEl is a family of a* -indeterminates over R (from some a'-extension of R), then the R-algebra R{(Yi)iEI} is called the algebra of inversive difference (or a* -) polynomials in a* -indeterminates (Yi )iEI. Elements of R{(Yi )iEl} are called inversive difference (or a* -) polynomials.

3.4.4. PROPOSITION. Let R be an inversive difference ring with a basic set a = {a!, ... an} and let I be an arbitrary set. Then:

(1) there exists an algebra of a* -polynomials over R in a* -indeterminates with indices from I;

(2) if Sand S' are any two such algebras, then there exists a a* -isomorphism 'P : S -t S' whose restriction on R is the identity mapping;

(3) if R is an integral domain, then any algebra of a* -polynomials over R also is an integral domain.

PROOF. Let r = r u and let S = R[(Yi,-y)iEI,-yEr) be the R-algebraofpolynomials in indeterminates (Yi,-y)iEl,-yEr with indices from the set I x r. Let a(P) (P E S, a E a*) denote a polynomial in S, that is obtained by replacing each coefficient a

of P by a(a) and each indeterminate Yi,-y that appears in P by Yi,o"Y. Then, for any a E a, we obtain a mapping S -t S (it will be denoted by the same symbol a) which is an automorphism of S extending the automorphism a of R.

Thus, S is the a* -overring of R. Set Yi = Yi,l for any i E I. Since (Yi,-y )iEl,-YEr

are algebraically independent over Rand I'(Yd = Yi,-y for any I' E r, i E I, it follows, that (Yi )iEl are a* -algebraically independent over R, hence the inversive difference ring S = R[(Yi,-y)iEI,-YEr) = R{(Yi,-y)iEl} is the algebra of a*-polynomials in a* -indeterminates (Y;)iEl over R.

Suppose that S' is another algebra of a* -polynomials over R in a* -indeterminates (Z;)iEI, with indices from I. Consider the mapping 'P : R[(Yi,-y)iEI,-YEr) -t R[(Zi,-y)iEI,-YErl which keeps fixed any a E R and translates each I'(yd (i E 1" E r) into I'(Zi). It is clear, that 'P is a a*-isomorphism whose restriction on R is an identity mapping. The last assertion of the proposition is obvious. 0

3.4.5. DEFINITION. Let a a*-ring R be an integral domain and let Q(R) be its quotient field. If Q(R) has the structure of a*-overring of R, then we shall say that Q(R) is the a* -quotient field of the a* -ring R.

The following proposition can be proved in the same way as the similar result for difference rings (see Proposition 3.3.10).

3.4.6. PROPOSITION. Let R be an inversive difference integral domain with a basic set a. Then:

(1) there exists a a* -quotient field of R; (2) if Fl and F2 are two a* -quotient fields of R, then there exists a a* -isomor

phism t.p : Fl -t F2 that keeps fixed any element of R; (3) if a field S is a a* -overring of R then S contains a a* -quotient field of R.

3.4.7. DEFINITION. Let F be a a*-field and let S = F{(Yi)iEI} be the algebra of a* -polynomials in a* -indeterminates (Y;)iEI over:F. The quotient field of S


will be denoted by F((Y;}iEI) and will be called a (T*-field of rational fractions in (T* -indeterminates (Y;}iEI over F (by Proposition 3.4.6, S is an integral domain).

3.4.8. REMARK. The maximality condition for (T* -ideals in a (T* -ring R does not imply the same one in the ring R{Y1, ... ,Ys} of (J"* -polynomials in a finite number (J"* -indeterminates Y1, ... ,Ys' Indeed, if F {y} is the ring of (T* -polynomials in (J"*indeterminate Y over an ordinary inversive difference field F with a basic set (T = {o:} and ~k = [yo:(y) , y0:2 (y), ... ,yo:k (y)] (k = 1,2 ... ), then ~1 ~ ~2 ~ ... ~ ~k ~ ... (the proof is similar to the proof in Example 3.3.15).

Let R{(Y;}iEI} be the algebra of (J"* -polynomials in (J"* -indeterminates (Yi)iEl over an inversive difference ring R with a basic set (J". If (1JdiEI is a family of elements of a (T* -overring of R, then the algebraic independence of {,(Yi) I, E r", i E I} over R implies the existence of a unique ring homomorphism 1/!" : R[(b(Y;}),Er~, iET)] ~ R( b( 1J;) )'YEr ~, iEI), identical on R and mapping ,(y;) to ,( 1J;) b E r", i E 1). It is clear, that 1/!" is a (T*-homomorphism of R{(Y;)iEI} onto R{(1Ji)iEI} , which is a (T* -isomorphism iff the family (1J;}iEI is (J"* -algebraically independent over R. The (T*-homomorphism 'l/J" is called the substitution of (1J;}iEI for (or "instead of") the indeterminates (Y;)iEI' If 9 E R{(Y;)iEI} , then 1/!" (g) is called the result of substitution of (1Ji)iEI in the (T*-polynomial 9 or the value of 9 on (1Ji)iEI; it is denoted by g((1Ji)iEI)' It is easy to see that Ker 1/!" is a prime (J"*-ideal of R{(Yi)iEI}' This ideal is called a defining (J"* -ideal of the family (1Ji)iEI over R.

As in the difference case, we shall usually deal with the rings of (J"* -polynomials in a finite set of (J"* -indeterminates Y1, ... ,Y. over a (T* -field F and with substitutions of fini te sets 1J = (1J1, ... , 1Js); such ordered sets of elements from some (J"* -extension of F we shall call s-tuples or s-dimensional points over F.

If 1J = (1J1,"" 1J.) is a s-tuple over a (T*-field F and A E F{Y1, ... , Ys} is a (T* -polynomial such that A(1J1 , ... , 1Js) = 0 (i.e. A E Ker 1/!,,), then we say that 1J is a solution of the (J"-polynomial A or of the algebraic (J"* -equation A(Y1' ... , Ys) = O.

If 0 = F(1J1,"" 1J.) is a (J"*-extension of a (J"*-field F generated by elements 1J1, ... , 1Js, then the kernel P of the substitution 1/!" : F {Y1, ... , Ys} ~ F {1J1, ... , 1Js} is a prime (T* -ideal of F {Y1, ... , y.} (since F {Y1, ... , Ys} / P == F {1J1, ... , 1J.} and the last ring is contained in the field 0, it has no zero divisors). In this case 0 can be considered as the (J"* -quotient field of the (J"* -ring F {Y1, ... , y.} / P.

3.4.9. DEFINITION. Let F be an inversive difference field with a basic set (T. Two s-dimensional points (1J1, ... , 1Js) and (1, ... , (.) over F are called (T* -equivalent if there exists a (T* -isomorphism ip : F (1J1, ... , 1J.) ~ F (1, ... , (s) such that ip( 1Ji) = (i (i = 1, .. . ,s) and ip(a) = a for any a E F.

3.4.10. REMARK. Let F be a difference field with a basic set (J" and let F* be the inversive closure of F. Then any finitely generated (J"* -extension 0 = F* (1J1 , ... , 1Js) of F* coincides with the inversive closure of the (T-field 0" = F(1J1 , ... , 1J.)". Indeed, any element of 0 has the form f "1,""'" ,where f(Y1, ... , Y.), g(Y1, ... ,Ys) E

9 fJl,···,118

F* {Y1, ... , Ys }. Therefore, for some T E T", the polynomials T f(Y1 , ... , Ys) and T f(Yl, ... , y.) belong to the ring F{Y1, ... , Ys}", hence T(~) EO", so that 0 is the inversive closure of the (T-field 0". In particular, if F is a (J"* -field, then F(1J1, ... , 1Js) is the inversive closure of F(1J1,"" 1J.)o, and the ring of (T* -polynomials F {Y1, ... , Ys} in (J"* -indeterminates Y1, ... , Y. is the inversive closure of


F{Yl,"" Y.},,· This note shows that two s-dimensional points .,., = (""1, ... ,""') and ( =

((1, ... ,(.) are 0'* -equivalent over a 0'* -field F iff they are O'-equivalent over F.

3.4.11. DEFINITION. A s-dimensional point (""1, ... ,""') over a O'*-field F is called a generic zero of the set ~ ~ F{Y1' ... , Y.}, if for any A E F{y1' ... , Y.} the inclusion A E ~ is equivalent to the equality A(""l, ... , .,.,.) = O.

Let F be a 0'* -field, P a prime 0'* -ideal of R = F {Y1, ... ,Y.}, G the 0'*quotient field of the 0'* -ring R/ P, and iii the image of the 0'* -indeterminate Yi under the canonical epimorphism R -t R/ P (i = 1, ... , s). It is clear, that the point (fit, ... , f)s) is a generic zero of P. As in the case of difference polynomials (see Proposition 3.3.20), we have the following result.

3.4.12. PROPOSITION.

(1) A set ~ ~ F {Y1, ... , Y.} has a generic zero over a 0'* -field F iff ~ is a prime O'*-ideal of F{Y1,"" y.}.

(2) Any s-dimensional point over a 0'* -field F is a generic zero of some prime O'*-ideal of F{Y1, ... , y.}.

(3) Any two generic zeros of a prime 0'* -ideal of F {Y1, ... , Y.} are 0'* -equivalent.

3.4 .13. DEFINITION. Let R be an inversive difference ring with a basic set 0'. A R-algebra U is called a O'*-algebm over R (or a O'*-R-algebm), if 0' acts on U in such a way that U is a O'*-ring in which the relation a(au) = a(a)a(u) holds for any a EO', a E R, u E U.

A 0'* -ideal J of an inversive difference ring R with a basic set 0' is called perfect if J is a perfect O'-ideal of R in the sense of Definition 3.3.23. It is clear that a 0'* -ideal J is perfect iff the inclusion ,/t( a)kl ... '/r (a )kr E J (a E R; '/1, ... ,'/r E f,,; k1 , ... , kr EN) implies a E J.

If B ~ R, then the smallest 0" -ideal of R containing B will be denoted by {B} and will be called a perfect closure of B (if R is treated as a O'-ring, then {B} = {B},,). Furthermore, for any set M ~ R, we shall denote by M' the set {a E RI'/t(a)kl .. ''/r(a)kr E M for some (depending on a) elements '/1,.· .,'/r E fa; k1 , ... ,kr EN}. It is clear, that if M is a O'"-ideal of R then M' = {a E R I 71 (a)kl ... 7r(a)kr EM for some (depending on a) 71, ... , 7r E T,,; k1, ... , kr E N}. As in Section 3.3 (see Exercise 3.3.24), we obtain the following construction of the perfect ideal {B} (B ~ R): {B} = U~=oBk' where Bo = B, B1 = [Bo]', ... , Bk = [Bk-1]', ....

If B is a finite set and J = {B}, then we shall say that B is a basis of the perfect O'"-ideal J. In this case, if there exists mEN such that J = Bm , then B is called a m-basis of J. More generally, a finite subset B of a set M ~ R is called a basis (m-basis) of M if {B} = {M} (respectively, Bm = {M}).

3.4.14. DEFINITION. An inversive difference ring, every subset of which has a basis, is called a Ritt's inversive difference ring.

3.4.15. REMARK. The properties of perfect ideals of difference rings given in the previous section hold also for perfect 0'" -ideals of inversive difference (0'" -) rings.


In particular, if R is a u*-ring and A, B ~ R then

AkBk ~ (AB)k+1,

Ak () Bk ~ (AB)k+l (k EN),

{A}{B} ~ {AB},

(ABh~Ak()Bk (kEN,k~1),

{A} () {B} = {AB}

(see Proposition 3.3.25). Furthermore, the analogues of Lemmas 3.3.28, 3.3.29 and Theorems 3.3.30, 3.3.31 are also valid for u* -rings.

Let R be a Ritt's u*-ring and let R{Yl, ... , Y'}O" and R{Yl, ... , Y.} be the rings of u- and u* -polynomials, respectively, in u* -indeterminates Yl, ... , Y. over R. If Y is a perfect u* -ideal of R {Yl, ... , Y. }, then Yo = Y () R {Yl , ... , y. } 0" is a perfect u-ideal of R{Yl, ... ,Y.}O". By Theorem 3.3.42, there exist u-polynomials 1I,···,fr E Yo such that Yo = {II,· .. , fr } 0" • It is clear, that Y = {II, ... , fr}, so that we have the following result.

3.4.16. THEOREM. Let R be a Ritt's inversive difference ring with a basic set u and let R{ 111, ... , 11.} be a u* -overring of R, generated by a finite family 111, ... , 11 •. Then R{ 111, ... , 11.} is a Ritt's u* -ring.

3.4.17. REMARK. It is easy to prove the results on the representation of a u* -ideal of a u* -ring as an intersection of its prime divisors that are similar to Theorem 3.3.43 and Statement 3.3.44. Namely, every perfect u* -ideal of an inversive difference (u* -) ring R can be represented as an intersection of prime u* -ideals, and if R is a Ritt's u*-ring, then every its perfect u*-ideal Y can be represented as a finite irreducible intersection of prime u* -ideals uniquely defined by Y (these ideals are called essential prime divisors of Y). Furthermore, if Y -::j:. (1) is a perfect u*ideal of a Ritt's u*-ring R, then Y can be represented as an intersection of a finite set of pairwise disjoint proper perfect u* -ideals Y1 , ... , Yp of R, none of which is the intersection of two strictly containing it disjoint perfect u* -ideals of R different from (1) (u* -ideals It, h of R are called disjoint (or coprime) if{ It, h} = R). The ideals Y1 , ... , Yp are defined by Y uniquely, they are called essential prime divisors of the perfect u* -ideal Y.

3.4.18. EXAMPLE. Let F be an inversive difference ring with a basic set u and let S = F{Yl, Y2} be the ring of u*-polynomials over F in two u*-indeterminates YI,Y2· Then {YIY2} = {YI} () {Y2} (see Remark 3.4.15). Since {Yi} = [Yi] (i = 1,2) are prime u* -ideals of S (obviously, S/[Yi] :=:::! F {Y2-i}) and Yi ft [Y2-i], we see that {ytl, {Y2} are essential prime divisors of the perfect u* -ideal {YIY2}.

Ordering and reduction in rings of inversive difference polynomials. Let R be an inversive difference ring with a basic set u = {al, ... , an} and let

r = r C1 be the free commutative group generated by al, ... , an. Let us set

n

r. = h = a~' ... a~" E r lord 'Y = L Ikd ~ s} i=l


for any sEN. As in Chapter 2, ~_ denotes the set of nonpositive integers and /Zl, /Z2, ... ,/Z2" denote all different Cartesian products of n factors, each of which is equal to either N or ~_ (we suppose, that /Z1 = Nn). These sets will be called ortants of the set /Zn.

Furthermore, for any j = 1, ... ,2n we set

Let R{YI, ... ,Y.} be the algebra of (To -polynomials in (T* -indeterminates YI, ... ,Y. over a (TO -ring R. Then we set Y = bYi I 1 ~ i ~ s, "{ E r} (here and below we write "{Yi instead of "{(Yi)) and Yj = bYi 11 ~ i ~ S,,,{ E r j } for j = 1, ... ,2n . (It is clear that Y = UI<j<2"Yj.) In what follows, elements ofY will be called terms. - -

3.4.19. DEFINITION. A term v E Y is called a tmnsform of a term u E Y if u and v belong to the same set Y; (1 ~ i ~ 2n) and v = "{U for some "{ E r i. If "{ f:. 1 then the transform v of u is called proper.

Consider the following linear order on the set Y (we use the standard symbols ~, <, ~, and> for this order): if u = a~l, ... , a~"Yi' v = ail, ... , a~"Yj E Y, then u < v iff the vector

is less than the vector

with respect to the lexicographic order on the set /Zn+2. It is easy to check, that this order has the following properties:

(1) if "{,,,{' E rand ord"{ < ord"{', then "{Yi < "{'Yj for any (To-indeterminates Yi,Yj (1 ~ i,j ~ s);

(2) if v, wE Yj for some j = 1, ... , 2n and v < w then "{v < "{w for any "{ E rj.

3.4.20. REMARK. Sometimes we shall also consider some other oderings of Y satisfying conditions (1) and (2). Such orderings are called admissible. Note, that (1) implies v < "{v for all v E Yj, 1 =I "{ E rj (1 ~ j ~ 2n).

3.4.21. DEFINITION. Let A E R{YI, ... , Y.}. The greatest (with respect to the order ~) term of the (T* -polynomial A is called the leader of A and is denoted by UA·

If deguA A = d, then A can be written in the form A = Idu1 +Id_IU~-I+ .. . +10 ,

where Ij E R{YI, ... , Y.} and (To-polynomials Ij (0 ~ j ~ d) are free of UA. The (T* -polynomial Id is called the initial of A and is denoted by IA.

Now we are going to define an ordering on the algebra R{YI' ... , Y.} that extends the given ordering of Y (it will be denoted by the same symbol ~). Namely, we set a < A for any a E R, A E R{YI, ... , Y.} \ R, and if A, B E R{YI, ... , Y.} \ R, then A < B iff either UA < UB, or UA = UB and deguA A < deguB B. If A < B,


then we say that the rank of the u· -polynomial A is lower than the rank of the u· -polynomial Bj if UA = UB and deguA A = deguB B, then we say that A and B are u· -polynomials of the same rank (in this case we write rk A = rk B). In what follows, the algebra of u· -polynomials R{Yl, ... , Y.} will be always considered as a partially ordered set with respect to the order introduced above.

3.4.22. DEFINITION. It will be said that a u·-polynomial BE R{Yl, ... , Y.} is reduced with respect to a u· -polynomial A E R{Yl, ... , Y.}, if B does not contain any power of a transform of the leader UA whose exponent is greater than or equal to d = deguA A.

3.4.23. DEFINITION. The set A ~ R{Yl>"" Y.} is called autoreduced, if every u· -polynomial PEA is reduced with respect to every other u· -polynomial from A. In particular, any set consisting of a single u·-polynomial is autoreduced.

3.4.24. EXAMPLE. Let R be an inversive difference ring with a basic set u = {Q:l,Q:2} and let R{y} be the ring of u·-polynomials in one u"-indeterminate y. Consider u· -polynomials

A = al(Q:lY)(Q:1 1Q:2y)2 + bl(Q:11Q:2ly),

B = a2Y(Q:~y)2 + b2y2(Q:IY)'

C = a3(Q:2Y)(Q:1 l Q:22y)2 + b3(Q:1lQ:22y),

D = a4y2(Q:IQ:~Y) + b4(Q:1lY)(Q:12Q:22y)2,

where 0 =1= ai, bi E R for i = 1, ... , 4. We can construct autoreduced sets {A, B}, {A, C}, {A, D}, {B, C}, {B, D},

{A,B,C}, but, for example, sets {C,D} and {A,C,D} are not autoreduced. Indeed, the u·-polynomial D is not reduced with respect to C, since it contains the power of the term Q:12Q:22y = Q:1l(Q:1 l Q:22y) = Q:1lue with the exponent 2 = degueC (the element Q:1 l = Q:1 l Q:g corresponds to the vector (-1,0) E Z2, which belongs to the same ortant as the vector (-1, -2), corresponding to the term uc).

The definition of an autoreduced set implies that the leaders of all its elements are different. Furthermore, as in the case of u-polynomials, the following proposition can be proved.

3.4.25. PROPOSITION. Every autoreduced set A in the ring of u· -polynomials R{Yl, ... I Y.} is finite.

3.4.26. DEFINITION. It will be said that a u·-polynomial A E R{Yl, ... ,Y.} is reduced with respect to a set A ~ R{Yl, ... , Y.}, if A is reduced with respect to every element of A.

The following reduction theorem can be proved similarly to the analogous result for the rings of difference polynomials (see Theorem 3.3.35).

3.4.27. THEOREM. Let R{Yl, . .. , Y.} be the algebra of u· -polynomials in u· -indeterminates Yl, ... , Y. over an inversive difference ring R with a basic set u. Let A = {AI"", Ar} be an autoreduced subset of R{Yl, ... , y.} and let B E


R{YI, ... , Ys}. Then there exists a (J'*-polynomial C ER{YI, ... , Y.} reduced with respect to A such that

Y B == C (mod [AD, (3.4.1)

where the (J'* -polynomial Y is equal to either 1, or a product of a finite number of (J'-polynomials of the form if Ai ({ E f, 1 ~ i ~ r).

The reduction process, i.e. the transition from a given (J'* -polynomial B to a (J'*-polynomial C reduced with respect to A and satisfying (3.4.1), may be performed in different ways. Let us consider one of them.

First of all, we exclude from B all powers of transforms of UA, whose exponents are greater than or equal to deguA , AI. It can be done as follows. Suppose that v = iUA, is the greatest (with respect to our fixed order <) transform OfUA, such that B contains powers of v whose exponents are greater than or equal to d l = degu AI. A, Let m be the greatest exponent of such power of v. Then the (J'* -polynomial B can be written as B = Yvm + YI vm - l + ... + Y m, where (J'* -polynomials Y, YI, ... , Y m

do not contain v and, moreover, if u is any transforms of UA, that is greater than v, then Y, YI, ... , Ym do not contain any power of u whose exponent is greater than or equal to d l . Then the (J'*-polynomial B' = i(fA,)B - Yvm-d'iAI can contain only those powers of v whose exponents do not exceed m - 1. It is also clear that if u is any transform of UA, and u > v, then B' does not contain any power of u whose exponent is greater than or equal to d l . Furthermore, B' == B (mod [AD. Continuing in the same way we obtain a (J'*-polynomial iJ such that iJ == B (mod [AD, deg" iJ < d l and if u is any transform of UA, that is greater than v, then iJ does not contain any power of u whose exponent is greater than or equal to d l .

Let w be the greatest transform of UA, in iJ such that degw iJ ~ d l . It is clear that w < v. Repeating the procedure described, we obtain a (J'* -polynomial BI such that BI == B (mod [AD and if u is any transform of UA" then BI does not contain any power of u whose exponent is greater than or equal to d l .

On the next step we exclude from BI all powers of transforms of UA. whose exponents are greater than or equal to d2 = deguA2 Al (repeating the procedure described above). Since the (J'*-polynomials Al and A2 are mutually reduced, the (J'* -polynomial B2 (B2 == B (mod [A])) obtained on this step contains no powers of transforms of UA, whose exponents are greater than or equal to dl .

Subsequently applying this procedure to the polynomials A3 , . .. , Ar , we obtain a (J'* -polynomial C that satisfies the condition of Theorem 3.4.27.

3.4.28. EXAMPLE. Let F{YI' Y2} be the ring of (J'*-polynomials in (J'* -indeterminates YI,Y2 over an inversive difference field F with a basic set (J' = {al,a2}. Consider in F{yl, Y2} the autoreduced set

A = {AI = a(ala2"lyt}(aIY2)2 + b(ala2yt}(aIY2) + c(a12Y2),

A2 = p(a2yt}(aIa2Y2) + q(aIYt}},

where a, b, c, p, q are some nonzero elements of F. It is clear, that the (J'*-polynomial


where I, g, h, k are nonzero elements of F, is not reduced with respect to A, because it contains a power of the transform a 21uA, of the leader UA, = a~Y2 whose exponent is equal to deguA1 Al = 2, and besides, it contains a power of the leader UA, = ala2Y2 whose exponent is equal to deguA2 A 2. In order to fulfill the process of reduction, we subsequently find

Bl = a 21(a)(ala2 2yt}B - (fyt}a2 1(Al)

= [ga21(a)(ala2Yl)(ala22yt} - la21(b)ydalYl)](a~a21Y2)

+ ha21(a)(alYl)(ala2Y2)(ala22Yt}

+ ka21(a)Yl(a2Y2)(ala22Yt} - la21(c)Yl(a12a21Y2);

B2 = p(a2Yl)B1 - ha21(a)(alyt}(ala22yt}A2

= p(a2yt}[ga21(a)(ala2yt}(ala22Yl) - la21(b)Yl(alyt}](a~a21Y2)

+ pka21(a)Yl(a2Yt}(a2Y2)(ala 22Yl) - gha21(a)(alyt}2(ala22yt}

- pla21 (C)Yl (a2Yl)(a 12a 21Y2).

The u*-polynomial B2 is reduced with respect to A and Y B == B2 (mod [A]), where Y = pa21(a)(a2Yl)(ala22Yt} = lA, . a 21(IA,).

3.4.29. DEFINITION. Let A = {A1, ... ,Ar } and B = {B1, ... ,Bd be two autoreduced sets in the algebra of u* -polynomials R{Yl, ... , Y.} over a u* -ring R. Suppose that elements of these two sets are arranged in such a way that Al < ... < Ar and Bl < ... < B t . The set A is said to have lower mnk than B (that is denoted as A < B), if one of the following conditions holds:

(1) there exists an integer kEN, 1 ~ k ~ min{r,t}, such that rkAi = rkB; for i = 1, ... , k - 1, and Ak < Bk;

(2) r > t and rk Ai = rk Bi for all i = 1, ... , t. If r = t and rkAi = rkBi for all i = 1, .. . ,r, then A is said to have the same

rank as B. (In this case we write rkA = rkB.)

The following proposition can be proved similarly to the analogous result for autoreduced sets in rings of difference polynomials (see Proposition 3.3.37).

3.4.30. PROPOSITION. Any nonempty set of autoreduced subsets of the ring R{Yl, ... , y.} contains an autoreduced subset of the lowest rank.

3.4.31. DEFINITION. An autoreduced set of the lowest rank of a u*-ideal I of R{Yl, ... , y.} is called a chamcteristic set of I (with respect to the given ranking).

Using Theorem 3.4.27, the following proposition can be proved similarly to the corresponding statement for difference polynomial rings.

3.4.32. PROPOSITION. Let S = R{Yl, ... , Y.} be the ring of u*-polynomials in u* -indeterminates Yl, ... , Y. over a u* -ring R, P a prime u* -ideal of S, and A a characteristic set of P. Then the following statements hold:

(1) P does not contain nonzero u*-polynomials reduced with respect to A (in particular, lA (/. P for every A E A);

(2) if YA is the set of all finite products of elements of the form -ylA, where -y E r (1, A E A, then P = [A] : YA


E-varieties and varieties over the ring of IT*-polynomials. E-varieties .and varieties of IT* -polynomials can be defined in the same way as the

corresponding notions for IT-polynomials. Namely, let F {Yl, ... , Y.} be the ring of IT* -polynomials in IT* -indeterminates Yl, ... , Y. over a IT* -field F, ~ F {Yl, ... , Y.} and E a system of IT* -overfields of F (in this case F is called the basic IT* -field of the system). Then the set Me ( <I» consisting of all s-dimensional points II = (111, ... , II.) with coordinates from a IT* -field of E, that are solutions of every IT* -polynomial from , is called a IT* -E -variety or simply E -variety defined by the set . It is also called a E-variety of the set over F (or over F {Yl, ... , Y.}).

Let A be a set, whose elements are s-tuples with entries in a field from a system E of IT* -overfields of a IT* -field F. Then the set A is called an E -variety over F (or over F{Yl, ... , Y.}), if there exists a set ~ F{y1, ... , Y.} such that A = Me(<I».

3.4.33. EXAMPLE. Let M be the field of meromorphic on the whole complex plane functions considered as an ordinary inversive difference field with the basic set IT = {a}, where af(z) = f(z + 1) for any f(z) EM (compare with Example 3.3.49, where M is considered as a IT-field). If ~ F{y1, ... , Y.} and E = {M}, then the E-variety defined by consists of all s-tuples (!t(z), ... ,!.(z)) EM', which are solutions of the system of functional equations P(!t(z), ... , f.(z)) = 0, (P E <I». For example, if s = 1, and consists of a single IT*-polynomial o:y - y, then E-variety Me(<I» consists of all periodic functions with period 1 (in particular, Me(<I» contains e2lriz , sin 211"z, cos 211"z).

Let F be a IT* -field, G = F (Xl, X2, ... ) the field of rational fractions in a de-numerable set of indeterminates Xl, X2, ... over F, and G the algebraic closure of G. Let U denote the family of all IT* -overfields of F, which are defined on different subfields of G. Such family U is called the universal system of IT* -over fields of F. If ~ F {Y1, ... , Y.}, then the U -variety Mu (<I» (it will also be denoted M( <I») is called the variety defined by the set over F (or over the ring of IT* -polynomials F {Y1, ... , Y.} ). A set of s-tuples A is called a variety over a IT* -field F, if A is a U -variety over F.

The formulation and proof of the following proposition are similar to the formulation and proof of the corresponding result for varieties of IT-polynomials.

3.4.34. PROPOSITION. Let F be a IT*-field and let "1 = ("11, ... ,"1.) be an arbitrary s-tuple, whose coordinates lie in a IT*-overfield of F. Then there exists a s-tuple ( = ((1, ... ,(.) that is equivalent to "1 and all coordinates of which lie in a IT* -field from the universal system U of IT* -overfields of F.

3.4.35. DEFINITION. An E-variety (variety) M1 is called a E-subvariety (respectively, a subvariety) of an E-variety (variety) M2, if M1 ~ M2. If the inclusion is proper then we say that M1 is a proper E-subvariety (respectively, subvariet ) of M 2 . E-variety (variety) M is calle reducible, if it can be represented as an union of two proper E-subvarieties (respectively, subvarieties). Otherwise, M is called irreducible E-variety (respectively, irreducible variety).

Below we shall use the following notation: F {Y1, ... , Y.} denotes the ring of IT*polynomials over a IT* -field F, E denotes a system of IT* -overfields of F, Me( <I» and M (<I» denote E-variety and variety, respectively, defined by a set ~ F {Y1, ... , Y.}


over F. If A is a set of s-tuples over F, then ~(A) denotes the set of all (J'*polynomials in F {YI, ... , Y.} that vanish on A.

The main properties of E-varieties and varieties over an inversive difference field are listed in the following proposition. The properties are similar to the corresponding properties of E-varieties and varieties over a (J'-field; they can be proved analogically (see Proposition 3.3.52).

3.4.36. PROPOSITION. With the preceding notation the following statements hold.

(1) If A is an E-variety, then ~(A) is a perfect (J'*-ideal of F{YI,"" Y.} and A = Me(~(A)).

(2) If I!, ... , hare (J'*-idealsofF{YI, ... , Y.} then Me(n~=1 Ij) = U~=I Me (Ij). (3) The intersection of any set of E-varieties is an E-variety. (4) The union of any finite family of E-varieties AI, ... , Ale is an E-variety, and

~(U7=1 Ai) = n7=1 ~(Ai)' (5) An E-variety A is irreducible iff~(A) is a prime (J'*-ideal of F{YI, ... ,y.}. (6) Any E-variety A can be uniquely represented as an irreducible union of

a finite set of irreducible E-varieties AI, ... , Ale (recall, that irreducibility

of the representation A = U~=I Ai means that Ai cf:. Uj;ti Aj for any i = 1, ... , k). These E-varieties are called irreducible components of A.

(7) If A = Al U ... U Ale is an irreducible representation of an E-variety A as the union of irreducible subvarieties, then the (J'* -ideals ~(Ad, ... , ~(AIe) of F{YI,"" Ys} are essential prime divisors of the perfect (J'*-ideal ~(A) (see Remark 3.4.17).

(8) The correspondence A ~ ~(A) is a one-to-one mapping from the set of all varieties over F {Yl, ... , Ys} onto the set of all perfect (J'* -ideals of F {YI, ... , Ys}. It maps the irreducible components of A onto the essential prime divisors of ~(A), and irreducible varieties over F {YI, ... , y.} correspond to prime (J'* -ideals of F {YI, ... , Ys}.

(9) If Y is a perfect (J'*-ideal of F{YI,"" Y.}, then ~(M(Y)) = Y. Furthermore, M(Y) = 0 iffY = (1).

(10) Let 1: ~ F{y1, ... , Y.} and let f be a (J'*-polynomial in F{YI, ... , y.}. Then f E ~(M('E)) iff f E {1:} (an analogue of the Hilbert Nullstellensatz).

By a generic zero of an irreducible variety over a (J'* -field F we shall mean the generic zero of the corresponding prime (J'* -ideal of a (J'* -polynomial ring F {YI, ... , y.}. We may assume that the coordinates of the generic zero belong to a (J'*-field from an universal system of (J'*-overfields of F.

The notions of complete system of (J'* -overfields of F, of separability of two varieties over F {YI, ... , Y.}, and of separable components of varieties may be introduced analogically to the case of (J'-fields and rings of (J'-polynomials (see Definitions 3.3.60 and 3.3.62, Proposition 3.3.63, and Theorem 3.3.64).

Extensions of (J'* -fields. The theory of extensions of (J'*-fields is developed in the same way as the corre

sponding theory for differential and difference cases. Namely, if G is a (J'* -overfield of a (J'* -field F and 'E ~ G, then an element v EGis called (J'* -algebraically dependent on the set 1: over F, if v is (J'* -algebraic over the (J'* -field F('E). (Obviously,


in this case there exist V1, ... , v. E E such that v is u* -algebraic over the u* -field F(V1' ... ' v.)). As in the case of difference extensions, the following two facts can be established.

(1) A family B of elements of a u*-extension of a u*-field F is u*-algebraically dependent over F iff some v E B is u* -algebraically dependent on the set B \ {v} over F.

(2) Let B be a family of elements in a u*-overfield G of a u*-field F. Then B contains a subset B1 ~ B which is u* -algebraically independent over F and is not contained in any other u* -algebraically independent over F subset of B. The set B1 is called a u* -trancendence basis of B over F. If B = G, then B1 is called a u*-trancendence basis of the u*-field Gover F (or a u* -trancendence basis of the u* -extension G ;2 F).

The main results about the u-algebraic dependence (see Lemmas 3.3.68, 3.3.69 and Theorem 3.3.70) may be transmitted to u* -algebraic dependence. In particular, if B is a subset of a u* -extension G of a u* -field F, then either all u* -trancendence bases of B over F contain the same number of elements, or they are all infinite.

3.4.37. DEFINITION. Let G be a u*-extension of a u*-field F. Then the u*trancendence degree of Gover F is the number of elements in some u* -trancendence basis of Gover F, if such bases are finite. If a u*-trancendence basis of Gover F is infinite, then we say that G is a u*-extension of F with infinite u*-trancendence degree.

The u* -trancendence degree of a u* -field G over its u* -subfield F will be denoted by u* -trdegp G (if G is a u* -extension of F with infinite u* -trancendence degree, then we shall write u*-trdegp G = 00).

3.4.38. REMARK. The condition u*-trdegp G = 0 means that every element of the u* -extension G of the u* -field F is u* -algebraic over F. In this case we say that G is a u* -algebraic extension of F. The analogue of Lemma 3.3.68 for u* -extensions shows that a u· -field G = F(Tl1, ... , TIm) is u* -algebraic over its u* -subfield F iff the elements TIl, ... , TIm are u* -algebraic over F.

The following proposition gives us some properties of the u* -trancendence degree that are similar to the properties of difference trancendence degree (see Proposition 3.3.72, Corollary 3.3.73, and Theorem 3.3.76).

3.4.39. PROPOSITION.

(1) A u*-trancendence basis of a u*-field G over its u*-subfield F can be chosen from any set of u* -generators of Gover F. In particular, if G = F(Tl1, ... , TIm), then u*-trdegp G ~ m.

(2) Let F be a u* -field and TIl, ... , TIm elements of a u* -overfield G of F that are u*-algebraically independent over F. Then u*-trdegp F(Tl1, ... , TIm} = m. In particular, if F(Tl1, ... , 17m} = F((l, ... , (.), where elements (1, ... , (. are u*-algebraically independent over F, then s = m.

(3) Let R = F {Y1, ... , Y.} be the ring of u* -polynomials in u* -indeterminates Y1, ... , Ys over a u*-field F. If k =F s (k E 'Z., k ~ 1) then R cannot be represented as a ring of u* -polynomials over F in some u* -indeterminates Zl,···, Zk.


(4) Let F, G, and H be u* -fields such that F ~ G ~ H. If one of the quantities u* -trdegF H or u* -trdegF G + u* -trdegG H is finite, then the other is also finite and

u* - trdegF H = u* - trdegF G + u* - trdegG H.

The proof is analogous to the proof of the corresponding results for difference field extensions. 0

The result by P. Evanovich for finitely generated difference extensions (see Theorem 3.3.77), as it is shown in [Ev84, § 1], may be adjusted to the case of u*-field extensions.

3.4.40. THEOREM. Let F, G, and H be u*-fields such that F ~ G ~ Hand H a finitely generated u*-extension of F. Then G is also a finitely generated u*extension of F.

Ring of inversive difference operators. Inversive difference modules. Let R be an inversive difference ring with a basic set of automorphisms u =

{aI, ... , an} and let r = r (7 be the free commutative group generated by aI, ... ,an' As before, we say that R is a u* -ring and denote the set of automorphisms aI, ... ,an, a 11, ... ,a;;-l by u*. If I = a~l ... a~l E r then the number ord I = L7=1 I k; I will be called the order of I' The set b E r lord I ~ r} (r EN) will be denoted by r(r).

3.4.41. DEFINITION. An inversive difference (or u*-) operator over a u*-ring R is an expression of the form L-YEr a-YI' where a-y E R for any I E r and only finitely many elements a-y are different from zero. Two u*-operators L-YEr a-YI and L-YEr b-YI are equal iff a-y = b-y for any I E r.

Let £ denote the set of all inversive difference operators over a u* -ring R. The set £ may be equipped with a ring structure if we set

a L a-YI = L(aa-yh, -YEr -yEr

(L a-Ylhl = L a-Y('''f'YI), -yEr -yEr

Il a = Idahl

for any L-YEr a-YI, L-YEr b-YI E £; a E R"l E r and extend multiplication by distributivity. The ring obtained is called the ring of inversive difference operators (or the ring of u* -operators) over R. It is clear, that the ring U of u-operators over R introduced in Section 3.3 (see Definition 3.3.78) is a subring of £.

For every u* -operator w = L-YEr a-YI E £, we define its order ord w as the greatest of the orders of elements I E r for which a-y -:f. O. The ring £ may be considered as a filtered ring with the increasing filtration (£r)rEZ, where £r = {w E £ I ordw ~ r} for r E Nand £r = 0 for r < 0 (it is clear, that (£r)rEZ is an exhaustive and separable filtration on £). Below, if a ring of inversive difference operators £ is considered as a filtered ring, we assume that its filtration is (£r)rEZ,


3.4.42. DEFINITION. Let R be an inversive difference ring with a basic set (1' = {a1, ... , an} and let £ be the ring of (1'* -operators over R. Then every left £-module is called an inversive difference R-module (or (1'*-R-module). In other words, a R-module M is called a (1'* -R-module, if elements of (1'* act on M in such way that the following conditions hold:

(1) a(x + y) = ax + ay; (2) a({3x) = (3(ax); (3) a(ax) = a(a)ax; (4) a(a-lx) = x

for any a,{3 E (1'*; x,y E M;a E R. If R is a (1'*-field, then a (1'·-R-module M is also called a vector (1'*-R-space (or an inversive difference vector space over R).

It is clear, that every (1'*-R-module is a (1'-R-module (see Definition 3.3.79) when R is considered as a (1'-ring.

3.4.43. DEFINITION. Let R be a (1'*-ring and let M,N be (1'*-R-modules. A R-module homomorphism f : M ~ N is called difference or (1'·-homomorphism if f(ax) = af(x) for any a E (1', X EM (it is clear, that if f is a (1'*-homomorphism then the last equality holds for all a E (1'* and x E M). The notions of (1'. -epimorphism, (1'* -monomorphism, and (1'* -isomorphism of (1'* -R-modules are defined in the natural way.

3.4.44. PROPOSITION. Let R be an inversive difference ring with a basic set (1' = {al, ... , an} and let £ be the ring of (1'* -operators over R. Let M and N be arbitrary (1'*-R-modules. Then each of the R-modules HomR(M, N) and M ®R N can be supplied with a structure of a (1'* -R-module, so that the natural R-module isomorphism TJ : HomR(P ®R M, N) ~ HomR(P, HomR(M, N)) (where P is a (1'·-R-module) is a (1'*-isomorphism.

PROOF. Let us set (aip)(x) = aip(a-lx) for any ip E HomR(M,N), a E (1'., and x E M. It is easy to check that aip E HomR(M, N) and that the conditions (1)-(4) of Definition 3.4.42 hold. (We leave the proof to the reader as an exercise). Therefore, the action of (1'* on the R-module HomR(M, N) induces a (1'*-R-module structure. In order to define a (1'*-R-module structure on the tensor product M ®R N, we set a(E~=l Xi ® Yi) = E~=l aXi ® aYi for any E~=l Xi ® Yi E M ®R Nand a E (1'*. Since a(a E~=l Xi ® y;) = a(E~=l aXi ® Yi) = E~=l a(a)axi ® aYi = a(a)a(E7=l Xi ® y;) for a E (1'., a E R, E7=l Xi ® Yi E M ®R N, and, obviously, a(a-lz) = z, a(zl + Z2) = aZl + aZ2, a({3z) = (3(az) for any a,{3 E (1'*, z, Zl, Z2 E M ®R N, the conditions (1)-(4) of Definition 3.4.42 hold. Therefore, the action of (1'* induces a structure of (1'*-R-module on M ®R N.

The natural R-module isomorphism

TJ : HomR(P ®R M, N) ~ HomR(P, HomR(M, N))

is given by the rule [(TJJ)x](y) = f(x®y) for any f E HomR(P®RM, N), x E P, Y E M. Taking into account the above defined action of (1'* on R-modules HomR(M, N) and M ®R N (M and N are arbitrary (1'* -R-modules), it is easy to check (this is left to the reader as an exercise) that TJ(aJ) = a(TJJ) for any f E HomR(P ®R M, N), a E (1'*, i.e. TJ is a (1'*-isomorphism. 0


Below, while considering R-modules HomR(M, N) and M ®R N (M, N are (1·-R-modules) as inversive difference R-modules, we shall assume that (1" acts of (1* on these modules in the same way as in the proof of Proposition 3.4.44.

3.4.45. Exercises. 1. Show that if an inversive difference ring R with a basic set (1 is an integral

domain, then the ring of (1. -operators over R has no zero divisors. 2. Let R be a (1. -ring and £ the ring of (1. -operators over R. For any (1. -R

module M, let C(M) be the set of all x E M such that ox = x for any 0 E (1. Then C(M) is an Abelian subgroup of the additive group of M, and the mapping C is a functor from the category of (1. -R-modules to the category of Abelian groups. Prove that C has the following properties:

(1) C(HomR(M, N)) = Home(M, N); (2) the functors C and Home(R,·) are naturally isomorphic; (3) the functor C is left exact and for any p E IE, p > 0, its pth right derivative

functor is naturally isomorphic to the functor ExtHR, .).

3. Using Proposition 3.4.44 and the properties of C given in the previous exercise, show that for any (1"-R-modules M and N, the functors Home(· ®R M, N) and Home (., HomR (M, N)) are naturally isomorphic, and the similar assertion is valid for the functors Home(M ®R ., N) and Home(M, HomR(·, N)).

4. Let R be a (1* -ring, £ the ring of (10 -operators over R, and M, N arbitrary (1°-R-modules. Then for any positive integers p and q there exists a spectral sequence converging to Exti+q (M, N) whose second term is equal to E~,q = (RPC)(Ext'h(M, N)).

[Hint: Use the results of the two previous exercises and the following theorem by A. Grothendick on the spectral sequences (see [Gr57, Ch. 2, Th. 2.4.1]): Let A, B, C be rings and kA , kB, kc be the categories of the left modules over A, B, and C, respectively. Let F : kA -* kB and G : kB -* kc be covariant functors, for which the following conditions hold:

(1) the functor G is left exact; (2) if M is an injective A-module, then the B-module F(M) is annihilated by

any right derivative functor RqG (q > 0) of the functor G.

Then for any left A-module N there exists a spectral sequence in the category kc that converges to Rp+q(GF)(N) (p, q E N) and whose second term has the form E~,q(N) = RPG(RqF(N)).]

Let G be an inversive difference field with a basic set of automorphisms (1 = {01, ... ,on}, f the free commutative group generated by the elements 01, ... , on and (1* = {01, ... ,On,011, ... ,0;;1} ~ f. Let F be a (1°-subfield of G and let DerF G be the vector G-space of all F-derivations of the field G into itself (see Section 1.5). In the following proposition we use the notation that was introduced after Proposition 1.5.7: (DerF G)* denotes the dual vector G-space HomG(DerF G, G) for DerF G and OF(G) denotes the appropriate module of differentials (OF (G) is a vector G-subspace of (DerF G)* generated by the set {d1J 11] E G} where elements d1J (1] E G) act on the derivations D E DerF G as follows: d1](D) = D(1])).

3.4.46. PROPOSITION. VectorG-spacesDerFG, (DerFG)", andOF(G) can be provided with structures of vector (1-G-spaces such that OF (G) becomes a vector


UO -subspace of (DerF G)* and, dTJ = d,(TJ) for any , E r, TJ E G.

PROOF. Let us set o:(D) -o:oDoo:- 1 for every 0: E uO, D E DerF G. It is easy to check that o:(D) is a F-derivation of the field G into itself and that all conditions of Definition 3.4.42 are satisfied, so that DerF G becomes a vector uO-G-space. Now we can define actions of elements of the set u· on (DerF G)· as in the proof of Proposition 3.4.44: (o:(tp))(D) = o:(tp(o:-l(D))) for any 0: E u·, tp E (DerF G)·, D E DerF G. It can be easily verified that all conditions of Definition 3.4.42 hold, so (DerF G)· becomes a u·-G-module with respect to the introduced action of elements 0: E u· (we leave the verifications of properties (1)-(4) in Definition 3.4.42 to the reader as an exercise).

It remains to show that our definition of the action of the set u· on (DerF G)* implies the equality 0: dTJ = dO:(TJ) for any 0: E u", TJ E G. (This equality, as it is easy to see, implies that ,dTJ = d,(TJ) for any, E r, TJ E G, DE DerFG, TJ E G, and 0: E u·. Then

(o:(dTJ))(D) = o:((dTJ)(o:-l(D))) = 0:((0:- 1 (D))(TJ)) = 0:(0:- 1 (D(O:(TJ))))

= D(O:(TJ)) = (dO:(TJ))(D),

whence o:(dTJ) = d(O:(TJ)). This completes the proof. 0

3.5. Differential-Difference Structures

3.5.1. DEFINITION. Let R be a commutative ring considered with a finite set Ll = {d1, ... ,dm } of derivations of this ring and finite sets u = {O:l, ... ,O:n}, c = {,81, ... , ,8q} of injective endomorphisms and automorphisms of R, respectively. Suppose that the elements of the set Ll U u U c are pairwise commuting. Then R is called a Ll-u-c· -ring or a differential-difference ring with respect to the basic set LlUuUc.

With the above notation, the set of automorphisms {,81, ... ,,8q, ,811, ... , ,8;;1} will be denoted by by c·. If Ll = u = 0, then we shall consider R as an inversive difference ring with the basic set c. Similarly, if u = c = 0 then R will be treated as a differential (Ll-) ring, and if Ll = c = 0 then R will be treated as a difference (u-) . ring. Also, if u = 0, then we will say that R is a Ll-c· -ring, if c = 0 then R will be said to be a Ll-u-ring, and if Ll = 0 then R will be called a Ll-c·-ring.

Let A denote the commutative semigroup of elements of the form

(3.5.1)

where k1 , ... , km , 11, ... , In EN; U1, ... , uq E Z. The free commutative semigroups S (or Sa) and T (or Tq) introduced in Sections 3.2-3.4 may be considered as subsemigroups of A, generated by all elements of the sets Ll and u, respectively. The free commutative group r (or r £), generated by the elements of the set c may be also considered as a subsemigroup of A. By the order of A in (3.5.1) we shall mean the number

m n q

ordA = L ki + Llj + L lulIl· i=l j=l

3.5. DIFFERENTIAL-DIFFERENCE STRUCTURES 175

For every r EN, the set {>. E A lord>. :s r} will be denoted by A( r). If a Ll-O'-c* -ring R is a field then we will say that R is a Ll-O'-c' -field. In

accordance with the general concepts for rings with operators (see Section 3.1), we may consider Ll-O'-c' -homomorphisms of Ll-O'-c' -rings, Ll-O'-c' -subrings and LlO'-c* -overrings, Ll-O'-c* -ideals, Ll-O'-c* -subfields and Ll-O'-c' -overfields (or Ll-O'-c·extensions), etc. If R is a Ll-O'-c* -ring then the set

{a E R 18(a) = 0, o:(a) = (J(a) = a for every 8 E Ll, 0: E O',(J E c}

is, obviously, a subring of R. This subring is called a ring of constants of the Ll-O'-c* -ring R. If Ro is a Ll-O'-c* -subring of the Ll-O'-c* -ring Rand B ~ R, then Ro{B} denotes the Ll-O'-c'-subring of R generated by the set B over Ro, i.e. the minimal Ll-O'-c'-subring of R containing Ro and B (evidently, Ro{B} coincides with Ro[{>.(b) I >. E A, b E Bm. If Ro is a Ll-O'-c'-subfield of a Ll-O'-c*-field R and B ~ R, then the Ll-O'-c*-field, generated by B over Ro will be denoted by Ro(B) (this field coincides with the field Ro({>'(b) I >. E A, b E B})). If a LlO'-c'-ring (Ll-O'-c*-field) R is generated over a Ll-O'-c*-subring (respectively, over a Ll-O'-c· -subfield) Ro by a finite family of elements TIl, .. . ,TI. then we shall write R = Ro{ TIl, ... , TIs} (respectively, R = RO(Tl1, ... , TI.))·

Let R be a Ll-O'-c· -ring and B ~ R. Then the minimal Ll-O'-c· -ideal of R containing B will be denoted by [B] (evidently, [B] coincides with the ideal ({>.(b) I>' E A,bEB})ofR).

3.5.2. Examples. 1. Any differential, difference, or inversive difference ring is a Ll-O'-c* -ring in the

sense of Definition 3.5.1. Therefore, all rings in Examples 3.2.4-3.2.7 and 3.3.3 are Ll-O'-c* -rings.

2. Let R = Q[x] be the ring of polynomials with rational coefficients in one indeterminate x. Consider the usual derivation 8 = d~ on R and automorphism (J : R -+ R such that (Jf(x) = f(x+1) for any f(x) E R. Then R may be considered as all-c' -ring, where Ll = {8}, c = {(J}.

3. Let R be the ring of functions of m real variables Xl, ... ,Xm that are infinitely differentiable on ]Rm. Fix some real numbers h1 , ... ,hm and consider the set of automorphisms c = {(J1, ... ,(Jm} such that

Consider also the set of derivations Ll = {81 , ... , 8m } of R, where 8i = -ao .. Then R x, can be treated as a Ll-c· -ring. If S is the overring of R, consisting of all functions of Xl, ... , xm infinitely differentiable in the domain Xl > 0, ... , xk > 0 (1 :s k < m) and hi > 0, ... , hk > 0, then S is a Ll-O'-ci-ring, where Ll = {81 , ... , 8m }, 0' = {(J1, ... , (Jk}, c = {(Jk+l, ... , (Jm}.

4. Let zo be a complex number. Suppose that U is a domain of the complex plane such that z E U always implies z + Zo E U. The field Mu of all functions meromorphic in U can be regarded as a Ll-O'-field, where the set Ll consists of one derivation 1., and 0' = {o:}, where 0: is the injective endomorphism of Mu such that o:f(z) = fez + zo) for any fez) EMu.


5. Let R = K {Y1, ... , Y2.} be the ring of differential polynomials in differential indeterminates Y1, ... , Y2. over a differential field K with a basic set ~ = {1I"1, ... ,lI"m}. Let OJ (1 ~ j ~ s) be the mapping from R into itself, which replaces any ()Yj (() E e, 1 ~ j ~ s) by ()YHIJ and ()YH. by ()Yj. Evidently, U = {01"'" as} is a set of ring automorphisms of R, which do not change elements of K. The elements of u commute with each other and with derivations from ~. Therefore, R may be considered as a ~-u* -ring generated by Y1, ... , Y. over the ~-u* -field K.

3.5.3. DEFINITION. A ~-u-e*-ideal I of a ~-u-e*-ring R is called a reflexive ~-u-e*-ideal iffor any a E u, a E R, the inclusion o(a) E I implies a E I.

It is easy to see, that the kernel of any ~-u-e* -homomorphism of ~-u-e* -rings f : R --t S is a ~-u-e* -ideal of R and conversely, if g is a ring epimorphism of a ~u-e* -ring R onto a ring S such that Ker g is a reflexive ~-u-e* -ideal of R, then there exists a unique structure of ~-u-e* -ring on S such that g is a ~-u-e* -epimorphism with respect to this structure. In particular, if I is a ~-u-e* -ideal of a ~-u-e*ring R , then there exists a unique structure of ~-u-e* -ring on the factor-ring R/ I such that the natural epimorphism of R onto R/ I is a ~-u-e* -epimorphism. The ~-u-e* -ring obtained is called a ~-U-e* -factor-ring of R by the ~-u-e* -ideal I.

As in the difference case, we may construct the reflexive closure j of a ~-u-e*ideal J of a ~-u-e*-ring R. Namely, j = {a E R I r(a) E J for some r ETa}.

3.5.4. EXAMPLE. Let R be a ~-u-e*-ring containing the field ofrationals. Then the nilradical N of R is a reflexive ~-u-e* -ideal.

Indeed, if a E N, II" E ~ and a E u U e, then lI"(a) EN by Theorem 3.2.12 and, obviously, o(a) E N. Besides, if a E u and o(a) EN, then o(a)" = o(a") = 0 for some kEN, whence a E N.

3.5.5. DEFINITION. A family V = {Vi liE I} of elements of some ~-u-e*overring of a ~-u-e* -ring R is called ~-u-e* -algebraically dependent over R, if the family {>.( v;) I >. E A, i E I} is algebraically dependent over R; otherwise, the family V is called ~-u-e* -independent over R or a family of ~-u-e* -indeterminates over R. If V consists of a single element v, then v is called ~-u-e* -algebraic or ~-u-e* -transcendental over R, respectively.

Let R be a ~-u-e* -ring and let Y = {Yi liE I} be a family of elements of some ~-u-e* -overfield of R that is ~-u-e* -algebraically independent over R. Then R{Y} is called an algebra of ~-U-e* -polynomials over R, and elements of this algebra are called ~-u-e* -polynomials over R. If the family Y is finite, Y = {ylJ ... , Ys}, then the algebra of ~-u-e*-polynomials in ~-u-e*-indeterminates Y1, ... , Ys over a ~-u-e* -ring R will be denoted by R{YI, ... , y.}.

The algebra of ~-u-e*-polynomials in ~-u-e*-indeterminates {Yi liE I} over a ~-u-e*-ring R may be constructed by introducing a ~-u-e*-ring structure on the ring of polynomials R[{Yi,A liE I, >. E A}], where the derivations from ~ and mappings from u U e* are extended in the same way as in the proof of Propositions 3.3.7 and 3.4.4 (in this case J1.Yi,A = Yi,/JA for every i E I, >',J1. E A). As in the difference case, we obtain the following result.


3.5.6. PROPOSITION. Let R be a Ll-lJ"-c*-ring and let I be a set. Then the following statements hold:

(1) there exists an algebra of Ll-lJ"-c* -polynomials over R in Ll-lJ"-c* -indeterminates with indices from I;

(2) there exists a Ll-lJ"-c* -isomorphism between any two such algebras that is identical on R;

(3) if R is an integral domain then any algebra of Ll-lJ"-c* -polynomials over R is also an integral domain.

Let a Ll-lJ"-c* -ring R be an integral domain and let Q(R) be its quotient field. If Q(R) has a structure of a Ll-lJ"-c*-overring of R, then Q(R) will be said to be a Ll-lJ"-c* -field of quotients of R. The following proposition can be proved in the same way as the corresponding propositions for differential and difference rings.

3.5.7. PROPOSITION. Let R be a Ll-lJ"-c* -integral domain. Then the following statements hold:

(1) there exists a Ll-lJ"-c* -field of quotients of R; (2) there exists a Ll-lJ"-c' -isomorphism between any two Ll-lJ"-c* -fields of quo

tients of R that is identical on R; (3) if Ll-lJ"-c' -field S is a Ll-lJ"-c' -overring of R, then S contains a Ll-lJ"-c' -field

of quotients of R.

Note, that in conditions of Proposition 3.5.7 the structure of Ll-lJ"-c'-ring on the field of quotients Q(R) is such that 8( ~) = o(a)b;,ao(b) and ,( ~) = ~ for every

elements ~ E Q(R), 8 E Ll, , E lJ" U E* (the correctness of these actions can be verified in the same way as in the differential and difference cases).

3.5.8. DEFINITION. Let R be a Ll-CT-c' -ring and S = R{(Yi )iEf} the ring of Ll-lJ"-c* -polynomials in Ll-CT-c* -indeterminates (Yi)iEI over R. Then Ll-lJ"-c* -field of quotients of S will be called a Ll-lJ"-c* -field of rational fractions in Ll-CT-c*indeterminates (Yi )iEf and will be denoted by R( (Y;)i Ef).

3.5.9. EXERCISE. Let R be a Ll-lJ"-c'-ring and La multiplicatively closed subset of R such that ,(L) ~ L for any, E lJ" U c*. Then there is a unique structure of Ll-lJ"-c* -ring on the ring of quotients L- l R such that the canonical homomorphism R -T L- I R (a -T if) is a Ll-lJ"-c*-homomorphism (compare with Exercise 3.3.14).

A Ll-lJ"-c*-ring R is said to be inversive, if the elements of lJ" are automorphisms of R (thus, R may be considered as Ll-w*-ring, where w = lJ" U c). In this case the set {al,"" an, all, ... , a;;-l} is denoted by lJ"* and R is called a Ll-lJ"* -c* -ring.

3.5.10. DEFINITION. Let R be a Ll-CT-c' -ring. Its Ll-CT-c* -over ring S is called an inversive closure of R, if S is inversive and if for any a E S there exists an element r E Ta such that r(a) E R.

The construction of the inversive closure of a Ll-lJ"-E* -ring is the same as in the difference case, so the proof of the following proposition is similar to the proof of Proposition 3.3.9.


3.5.11. PROPOSITION. Let R be a Ll-u-g* -ring. Then the following statements hold:

(1) R has an inversive closure; (2) if SI and S2 are two inversive closures of R, then there exists a Ll-u-g*

isomorphism 'fJ: SI -+ S2 identical on R; (3) if U is an inversive Ll-u-g* -ring containing R as a Ll-u-g* -subring, then U

contains an inversive closure of R; (4) if R is an integral domain (a field), then its inversive closure is also an

integral domain (respectively, a field).

Algebraic Ll-u-g* -equations and their solutions. Let S = R{(Yi)iE/} be an algebra of Ll-u-g* -polynomials over a Ll-u-g* -ring

Rand 1/ = (1/i)iEI a family of elements of some Ll-u-g*-overring of R. Let 'fJf/ : S -+ R{(1/i)'EI} be a Ll-u-g*-epimorphism that is identical on R and maps '\(Yi) (,\ E A, i E I) to '\(1/.) (obviously, 'fJf/ is an isomorphism iff the family 1/ is Ll-u-g*algebraically independent over R). This epimorphism will be called the substitution of elements (1/;)'EI instead of Ll-u-g* -indeterminates (Y')'EI; if 9 E R{(Y')'E/ }, then we shall say that the element 'fJf/(g) of R{(1/,)iEJ} is the result of this substitution (or the value of the Ll-u-g*-polynomial 9 on 1/). We shall usually denote 'fJf/(g) by g( 1/) (if 1/ = {1/1, ... ,1/.} (s EN, s ~ 1), then we shall write g(1/1 , ... ,1/.) instead of g(1/)). It is easy to see, that Ker'fJf/ is a prime reflexive Ll-u-g*-ideal of R{(Y,)iEJ}. This ideal is called the defining Ll-u-g* -ideal of the family 1/ over the ring R.

If {1/1, ... , 1/.} is a finite family of elements of a Ll-u-g* -overring of R and A E R{Y1' ... , Y.}, then we shall say that as-tuple 1/ = (171, ... , 17.) is a solution of a Ll-u-g*-polynomial A, if A E Ker 'fJf/' i.e. if A(1/1, ... , 1/.) = o.

Similarly, if {II-' I J1. E M} is a family of elements from the kernel of the mapping 'fJf/' then one says that the point 1/ is a solution of this family or a solution of the system of algebraic Ll-u-g* -equations

(J1. EM).

Let F be a Ll-u-g* -field and G = F(1/1 , ... ,1/.) be its Ll-u-g* -extension generated by a finite family 1/ = {1/1, ... , 1/.}. Then G may be considered as the Ll-u-g*quotient field of the factor-ring of the ring of Ll-u-g* -polynomials G{Yl, ... , Y.} by its prime Ll-u-g* -ideal P = Ker( 'fJf/ : F {Y1, ... , Y.} -+ F {1/1, ... , 1/.}). In this case the images of Ll-u-g*-indeterminates fI, = Yi + P (1 :5 i :5 s) correspond to the elements 1/i.

Let 1/ = (1/1, ... , 1/.) and ( = ((1, ... , (.) be two s-tuples over a Ll-u-g* -field F (the elements 1/1, ... , 1/., (1, ... ,(. lie in some Ll-u-g* -extension of F). We shall say that 1/ and ( are Ll-u-g* -equivalent over F, if there exists a Ll-u-g* -isomorphism 'fJ: F(1/l, ... , 1/.) -+ F((1, ... , (.) such that 'fJ(1/;) = (, for i = 1, ... , s, and 'fJ(a) = a for any a E F.

3.5.12. DEFINITION. As-tuple (1/1, ... ,1/.) over a Ll-u-g*-field F is called a generic zero of a set q> ~ F {Yl, ... , y.} if for any Ll-u-g* -polynomial A E F {Yl, ... , y.} the inclusion A E q> is equivalent to A( 1/1, ... , 1/.) = O.


3.5.13. PROPOSITION. Let F{Y1, ... , Y.} be the ring of 6.-u-c*-polynomials in 6.-u-c· -indeterminates Y1, ... , Y. over a 6.-u-c· -field F. Then the following statements hold:

(1) a set cl> ~ F{Y1, ... , Y.} has the generic zero over F iffcl> is a prime reflexive 6.-u-c*-ideal of F{Y1, ... ,Y.};

(2) any s-tuple over the 6.-u-c* -field F is a generic zero of some prime 6.-u-c*ideal of F {Y1, ... , Y.};

(3) any two generic zeros of the same prime 6.-u-c* -ideal of F {Y1, ... , Y.} are 6.-u-c* -equivalent.

The proof of this statement is similar to the proof of Proposition 3.3.20. In this case the generic zero of a prime 6.-u-c* -ideal cl> of F {Y1, ... , Y.} has the form (:ih, ... , Y.), where Yi = Yi + cl> are canonical images of the 6.-u-c* -indeterminates Yi (1 ~ i ~ s) in the quotient field of the 6.-u-c· -ring F {yl, ... , y.} / cl>, and the prime 6.-u-c*-ideal of F{Y1, ... , y.}, whose generic zero is a given s-tuple 17 (171, ... , 17.) over F, coincides with the kernel of the substitution

cp" : F {Y1, ... , y.} -t F {171, ... , 17.}. 0

3.5.14. DEFINITION. Let R be a 6.-u-c*-ring. A R-algebra S is called a 6.-uc* -R-algebm (or a 6.-u-c* -algebra over R), if elements of the set 6. U u U c* act on S in such a way that S becomes a 6.-u-c*-ring, and c5(au) = c5(a)u + ac5(u), -y(au) = -y(a)-y(u) for any elements c5 E 6., -y E u U c*, a E R, u E S.

It is easy to see that the algebra of 6.-u-c* -polynomials over a 6.-u-c* -ring R is a 6.-u-c*-R-algebra in the sense of Definition 3.5.14. Obviously, any 6.-u-c*-field may be regarded as a 6.-u-c* -algebra over any its 6.-u-c* -subfield.

Let R be a 6.-u-c* -ring. In addition to the commutative semigroups of operators 8, T = Tu, A and commutative group f = fe introduced in the beginning of this section, let us consider the commutative subsemigroup Tf of A, consisting of all elements of the form a~l ... a~ .. ~l •• • ri/ (k1' ... ' kn E Nj h, ... , lq E Z). The following definition generalizes the notion of a perfect ideal of a difference ring to the case of 6.-u-c* -rings.

3.5.15. DEFINITION. A 6.-u-c*-ideal J of a 6.-u-c*-ring R is called perfect, if for any a E R; J.l1, ... ,J.lr E Tf (r E N, r 2: 1) and for all k1, ... ,kr EN the inclusion J.l1(a)kl .. . J.lr(a)kr E J implies a E J.

It is easy to see, that if R is considered as a difference ring with the basic set u U c (or with the basic set u, or as an inversive difference ring with the basic set c), then any perfect 6.-u-c*-ideal of R is a perfect ideal in the sense of Definition 3.3.23. If R is considered as a differential ring with the basic set 6., then any perfect 6.-u-c* -ideal of this ring is its differential perfect ideal. Obviously, the intersection of any family of perfect 6.-u-c* -ideals is also a perfect 6.-u-c* -ideal. Therefore, if B is a subset of a 6.-u-c* -ring R, then the intersection of all perfect 6.-u-c* -ideals of this ring containing B is the minimal perfect 6.-u-c*-ideal containing B. This ideal will be denoted by {B}; it will be called the perfect closure of B. Some properties of perfect 6.-u-c*-ideals are formulated in the following exercise (clearly, they are similar to the properties of perfect ideals of difference rings).


3.5.16. EXERCISE. Prove the following properties of perfect ~-u-c'-ideals of arbitrary ~-u-c' -ring.

(1) Every perfect ~-u-c' -ideal is reflexive, and every prime reflexive ~-u-c'ideal is perfect.

(2) Let R be a ~-u-c' -field, let R{Yl, ... , Y.} be the ring of ~-u-c' -polynomials in ~-u-c* -indeterminates Yl, ... , Y. over R, and let A be a set of s-tuples over R. Then for any (al, ... ,a.) E A the set <JI(A) = {f(Yl, ... ,Ys) E R{Yl,"" Ys} I f(al, ... , a.) = O} is a perfect ~-u-c'-ideal of R{Yl, ... , Ys}.

(3) The construction of the perfect closure for a subset B of a ~-u-c* -ring R is similar to the construction of the perfect closure for a subset of a difference ring R with a basic set u U c. Namely, for any M ~ R, let M' be the set of all elements a E R for which there exist elements J.ll,.'" J.lr E Tf (r E N, r ~ 1) and integers kl, ... ,kr EN such that J.lda)kl"'J.lr(a)kr E M. Let Bo = B, Bl = [B]', ... ,Bk = [Bk-l]', .... Then {B} = UkENBk.

(4) For any two subsets A and B of a ~-u-c' -ring R, let AB be the set {ab I a E A, bE B}. Then the following statements hold:

(a) AkBk ~ (AB)k+l for every kEN; (b) {A}· {B} ~ {AB}; (c) (AB)k ~ Ak n Bk for every kEN, k ~ 1; (d) Ak n Bk ~ (AB)k+1 for all kEN; (e) {A} n {B} = {AB}.

(5) The set of all perfect ~-u-c'-ideals of a ~-u-c*-ring R is a perfect conservative system.

3.5.17. DEFINITION. A set B in a ~-u-c'-ring R is called a basis of a set L ~ R if B is finite and {B} = {L}. If p:} = Bm for some mEN, B ~ L, and Card B < 00, then the set B is called a m-basis of the set L.

3.5.18. DEFINITION. A ~-u-c*-ring R is called Ritt's ~-u-c*-ring if it contains the field of rationals Q and if every subset of R has a basis.

3.5.19. EXERCISE. Show, that every perfect ~-u-c*-ideal in a ~-u-c*-ring R may be represented as an intersection of prime ~-u-c' -ideals, and if R is a Ritt's ~-u-c' -ring then for every its perfect ~-u-c' -ideal J there exists a representation J = PI n· . ·npr, where PI, ... , Pr are prime ~-u' -c' -ideals of R such that Pi i. Pi for i #- j (1 ::; i, j ::; r). In the last case the prime ~-u' -c' -ideals PI, ... , Pr are uniquely defined by J; they are called essential prime divisors of J and the representation J = PI n ... n Pr is called irreducible.

3.5.20. REMARK. It follows from the definition of Ritt's ~-u-c'-ring that the set of all perfect ~-u-c' -ideals of such a ring is a Noetherian perfect conservative system (see Definition 1.4.16). Hence, if R is a ~-u-c* -ring containing the field of rationals, then the following conditions are equivalent:

(1) R is a Ritt's ~-u-c*-ring; (2) R satisfies the maximality condition for perfect ~-u-c' -ideals; (3) any ascending chain of perfect ~-u-c' -ideals of R becomes stable on a finite

step.


The ordering and reduction in the ring of Ll-u-c*-polynomials. Let R be a Ll-u-c* -ring, where Ll = {01' ... ' Om} is a set of derivation op

erators of R, u = {a1, ... , an} is a set of injective endomorph isms of R, and c = {.B1, ... , .Bq } ~ Aut R. Let 8 and T be the free commutative semigroups introduced above (they are generated by the elements of Ll and u, respectively), f the free commutative group generated by the elements of c, and Tf, A the commutative

. . t· fIt 1<, I< all nI. d semlgroups consls mg 0 e emen s a 1 .. . a nnp1 .. . Pq an

\ _ N, Um 1<, I<n aI, al. A- 0 1 ... om a 1 ... a n 1-'1 ... I-'q

respectively. Elements of 8 will be called derivations, and elements of Tf will be called translations. Remind, that the order of an element A E A is defined as ord A = E~l Si + E.i=1 kj + E~=l Ilvl and that the orders of elements of 8, T, f and Tf are defined by the restriction of this definition.

Let R{Y1, ... , y.} be the algebra of Ll-u-c* -polynomials in Ll-u-c* -indeterminates Y1, ... , Y. over a ring R and let Y = {AYj I A E A, 1 $ j $ s} (elements of the set Y will be called terms). Consider the following two orders on the set Y:

1. The differential order. This order is denoted by the symbol < (or by the symbol $, if we permit the coincidence of terms compared). A term AYj = (JIlYj is less than the term A'Yi = (J' Il'Yi (1 $ i, j $ s; (J = o~' ... O:,.m, (J' = o~' .. . O;;"m E 8;

1<, I<n alt a'. '- U, Un aVl aV• E Tf) ·th t t thO Il = a 1 ... an 1-'1 ... I-'q , Il - a 1 ... an 1-'1 ... I-'q WI respec 0 IS

order iff the vector (ord (J, ord Il, Sl, ... , Sm, kl , ... , kn , It, ... , lq, j) is less than the vector (ord (J' , ord Il', r1, ... , rm , U1, ... , Un, VI , .•. , Vq , i) with respect to the lexico-graphic order on the set zm+n+q+3.

2. The regular order. This order is denoted by the symbols ~ (or ~, if we permit the coincidence of elements compared). In this case the inequality >'Yj ~ A' Yi holds for the elements AYj and A' Yi considered above iff the vector (ord A, j, Sl, ... , Sm, k1, ... , kn, 11, ... , lq) is less than (ord A', i, r1, ... , rm , Ut, ... , Un, VI, ... , vq) with respect to the lexicographic order on the set zm+n+q+2.

Let ZI, ... , Z2. be all ortants of the set zq, i.e. all different Cartesian products of q multipliers, each of which is equal to N or Z_ = {a E Z I a $ O}. Set

fj = iT' = f1il .. . .B~. E f I (11, ... lq) E Zj},

(Tf)j = {Il = TI IT E T'I E fj},

Aj = {(JTI I (J E 8, T E T, I E f j }

Y; = {Ay; I 1 $ i $ s, A E Aj}

for any j = 1, ... , 2q . It is easy to see that A = U1~j~2.Aj and Y = UI~j9.Y;.

3.5.21. DEFINITION. A term V = (JTIYj is called a multiple of a term U = (J'T'I'Yj ((J, (J' E 8; T, T' E T; I' I' E f; 1 $ i, j $ s), if U and v lie in the same set Yp (1 $ p $ 2q) and v = AU for some A E Ap. If (J = (J' and v is a multiple of u, then we shall say, that v is a transform of u. If v = (Ju ((J E 8), then v will be called a derivative of u.

Below, if the opposite is not said explicitly, we shall suppose that the set of terms Y of the ring of Ll-u-c* -polynomials R{Y1, ... , Y.} is ordered with respect to the differential order.


3.5.22. DEFINITION. Let A E R{Yl, ... ,Y.}. The maximal term from the set Y present in A with a nonzero coefficient is called a leader of A and is denoted by UA·

Obviously, any /j.-U-6* -polynomial A can be represented as a polynomial of degree d = deguA A of its leader UA: A = Iau1 +Id_1U~-1+ .. . +10 . The coefficients Ij (0 ~ j ~ d) are /j.-u-6*-polynomials free of UA (all terms of Ij are less than UA). The /j.-U-6* -polynomial Id is called the initial, and the /j.-U-6* -polynomial SA = :UAA is called the separant of A.

Consider now the following ordering of the /j.-U-6* -algebra R{Yl, ... , Y.} that extends the differential ordering of the set Y. This ordering will be also called differential and will be denoted by the same symbol < (or ~ if we permit the coincidence of the terms compared). We set a < A for any elements a E R, A E R{Yl, ... , Ys} \ R, and A < B for A, B E R{Yl, ... , Y.} \ R iff either UA < UB, or UA = UB and deguA A < deguB B. If A < B, we say that A has a lower rank than B; if UA = UB and deguA = deguB B, we say that A and B are of the same rank and write rk A = rk B. Below the algebra R{Yl, ... , Y.} will be treated as a set with this differential ordering.

3.5.23. DEFINITION. A /j.-u-6*-polynomial B E R{Yl,".,y.} is said to be reduced with respect to a /j.-U-6* -polynomial A E R{Yl"'" Y.}, if B does not contain any transform of a derivative of the leader UA and if the degree of B with respect to any transform of UA is less than d = deguA A.

3.5.24. DEFINITION. A set A ~ R{Yl, ... , Y.} is called autoreduced if every /j.U-6* -polynomial A E A is reduced with respect to every other /j.-U-6* -polynomial from A.

3.5.25. REMARK. The definition of an autoreduced set implies that the leaders of all its elements are different. Furthermore, as in the case of autoreduced sets of differential, difference and inversive difference polynomials, it can be proved that any autoreduced set of /j.-U-6* -polynomial is finite.

3.5.26. DEFINITION. A /j.-u-6*-polynomial A E R{Yl, ... ,Y.} is said to be reduced with respect to a set A ~ R{Yl, ... , y.}, if A is reduced with respect to each element of A (if A = 0, then every /j.-u-6*-polynomial is considered as a reduced one with respect to A).

The following reduction theorem generalizes the analogical results for difference, inversive difference and differential rings (see Theorems 3.3.35, 3.4.27 and Theorem 5.3.7 below).

3.5.27. THEOREM. Let R be a /j.-u-6*-ring and let R' = R{Yl, ... ,Y.} be the ring of /j.-U-6* -polynomials in /j.-U-6* -indeterminates Yl, ... , Y. over R. Let A = {F1 , ... , Fr } be an autoreduced subset of R' and G a /j.-U-6* -polynomial from R'. Then there exist /j.-U-6* -polynomials H, J, L E R' such that the following properties hold:

(1) H is reduced with respect to A; (2) J is equal to a product of finite powers of translations of separants and

initials of elements of A;


(3) L E [A] and H= JG-L, (3.5.2)

where A-u-g* -polynomials J and L do not contain terms that are greater than the leader of G.

If R is a U-g* -ring (i.e., A = 0), then J can be taken as a product of translations of the initials of F1 , ... , Fr.

PROOF. If A = 0, it is sufficient to take H = G, J = 1, and L = O. Let A =1= 0

and let Uj, Ij and Sj be the leaders, initials and separants of A-u-g* -polynomials Fj (1 ~ j ~ r), respectively. Without loss of generality we may assume that Sj =1= 0 for all j = 1, ... , r (otherwise, the statement of the theorem is trivial, because we may take J = 0). If A E R' and A is not reduced with respect to A, then A contains terms which are either transforms of derivatives of some leaders Uj (1 ~ j ~ r), or transforms v = J1.Uj (J1. E Tf, 1 ~ j ~ r), such that degv A (the degree of A with respect to v) is greater than or equal to dj = deguj Fj. The highest term of this type will be called the A-leader of the A-u-g* -polynomial A.

Now, suppose that the statement of the theorem does not hold. Let M be the nonempty set of all A-u-g*-polynomials G E R' such that there are no A-ug*-polynomials H, J, L satisfying conditions (1)-(3). It is clear, that every A-ug* -polynomial from the set M is not reduced with respect to A, hence it has a A-leader. Let us choose the A-u-g*-polynomial A E M with lowest A-leader v and such that A has the lowest degree d = degv A among all polynomials from M with A-leader v. Let v = AUj (A E A, 1 ~ j ~ r) be a multiple of Uj in the sense of Definition 3.5.21 and let E be the initial of AFj. If A E Tf (in this case v = AUj

is a transform of Uj), then E = >.Ij , deg>,uj AFj = dj and d ~ dj . If A = 9J1. f!. Tf (9 E e, J1. E Tf), then E = J1.Sj and A-u-g* -polynomial AFj is linear with respect to AUj. In the both cases there exist MER' and pEN such that the degree of the polynomial

(3.5.3)

with respect to v is less than dj (if A f!. Tf, then Al is free of v). Besides, M and AFj contain no terms which are greater than UA. Obviously, Al contains only the terms that appear either in A or in AFj, and the terms of A, that are not contained in AFj, are present in A and Al in the same powers.

Let w be a term present in Al in some power e ~ 1 and let w > v. Then w does not appear in AFj and the above considerations imply that degw A ~ e. Furthermore, if w is a multiple of some leader Uj (1 ~ j ~ r), then (since v is a A-leader of A) the term w is some transform J1.Uj of Uj (J1. E Tf), and e < dj . Therefore, w is not the A-leader of AI, hence either Al has no A-leader, or the A-leader of Al is less than v, or v is the A-leader of Al and degv Al < degv A. In all these cases we have Al f!. M, i.e., there exist A-u-g*-polynomials HI, Jl, Ll E R' with the following properties: HI is reduced with respect to A; Jr is equal to the product of powers of translations of separants and the initials of A-u-g* -polynomials F l , ... , Fr; Ll E [A]; HI = J1A1 - L 1 , and the representations of J and L contain no terms which are greater than the leader of AI. Taking into account (3.5.3), we obtain that HI = (J1EP)A - (Ll + M(AFj)). Therefore, A E M that contradicts the choice of A. Thus, M = 0 and the theorem is proved (the statement on the


rings of U-e* -polynomials follows from the fact that the proof can be repeated for such rings without mentioning separants). 0

The following theorem generalizes Theorems 3.3.42 and 3.4.16 to the case of arbitrary conservative systems of perfect A-u-e* -ideals. .

3.5.28. THEOREM. Let Ro be a u-e*-ring whose basic set consists of a set of injective endomorphisms u = {Q1' ... ' Qn} and a set of automorphisms e = {/h, ... ,,Bq} of the ring Ro· Let R = RO{1]l, ... ,1].} be a u-e*-overring of Ro generated by a finite family of elements 1] = {1]b ... , 1].} and let E be a conserva-tive system of perfect U-e* -ideals of R such that the conservative system EIRo is Noetherian. Then the system E is also Noetherian.

The proof of this theorem can be received similarly to the proof of Theorem 3.3.42 (using Theorem 3.5.27 instead of Theorem 3.3.35 and changing all references to Theorem 3.3.30 and Proposition 3.3.25 by references to the corresponding results for perfect conservative systems given in Lemma 1.4.22 and Proposition 1.4.23). We leave details to the reader as an exercise.

3.5.29. THEOREM. Let R = RO{1]b ... ,1].} be a A-u-e*-overring ofa Ritt's A-u-e* -ring Ro generated by a finite family of elements 1] = (1]1, ... ,1].). Then R is also a Ritt's A-u-e* -ring. In particular, the ring of A-u-e* -polynomials in a finite number of A-u-e* -indeterminates over a Ritt's A-u-e* -ring is a Ritt's A-u-e* -ring.

PROOF. As in the proof of Theorem 3.4.16, we may reduce the case of A-u-e*rings ("inversive" with respect to the set of automorphisms e) to the case, when Ro is a A-(u U e)-ring, and R is its finitely generated A-(u U e)-overring. Since u U e may be considered as a set of injective endomorphisms commuting with derivation operators from A, it is sufficient to prove the theorem for the case of A-u-rings (when e = 0), i.e. we can suppose, that Ro is a A-u-ring, and R is its finitely generated A-u-overring. Furthermore, it is sufficient to prove the theorem in the case, when R = RO{Y1, ... , Y.} is the ring of A-u-polynomials in A-u-indeterminates Y1, ... , Y. over a Ritt's A-u-ring Ro.

Suppose, that the theorem does not hold, i.e. R is not a Ritt's A-u-ring. Since the set of all perfect A-u-ideals of R is a perfect conservative system, we can apply Lemma 1.4.22 to this system and concludu that there exists a maximal perfect A-u-ideal P of R which has no basis. It is also known that such ideal is prime. Let Po = P n Ro and let U be the set of all A-u-polynomials in P, that do not belong to Ro and whose separants do not belong to P (it is possible, that U = 0). By induction we construct an autoreduced set A, contained in U and such that no element from U \ A is reduced with respect to A. If U = 0, then A = 0. Suppose that U =F 0. Let F1 be a A-u-e*-polynomial in U, minimal with respect to the differential ordering on the set Y = {AYj I A E A, 1 ~ j ~ s}. Suppose, that elements F1 , ... ,Fi of the autoreduced set A have been already constructed. If U contains no A-u-polynomials, reduced with respect to any of the polynomials F1, ... , Fi, then set A = {F1' ... , Fi}. If U contains A-u-polynomials reduced with respect to {F1, ... , Fi}, then choose a minimal such A-u-polynomial and denote it by Fi+1. It is easy to see, that {F1 , . .• , Fi, Fi+d is an autoreduced set. After a finite number of steps this process, obviously, stops and we receive the desired autoreduced set A = {F1 , ••• , Fr } ~ U.


Let Uj be the leader, Sj the separant, and I j the initial of Fj , respectively, and let dj = deguj Fj (1 ::; j ::; r). Then Fj can be represented in the form

Fj = Iju~j + Aj , where Aj < Fj and deguj Aj ::; dj - 1. Since Aj is reduced with respect to A, its separant belongs to P, i.e., fJAj!fJuj E P. If Ij E P, then

Sj = djU~j-1Ij + fJA!fJuj E P, that contradicts the inclusion Fj E U. Therefore, I j tt P for all j = 1, ... , r. Moreover, J tt P for J = h ... Ir S 1 ... Sr, because P is a prime ideal.

Let {Po, A} R denote the perfect ~-u-ideal of R, generated by Po U A. Since Ro is a Ritt's ~-u-ring, there exists a finite set B ~ Po such that {Po, A} R = {B, A} n. We shall show, that J P ~ {B, A} n. First of all, we note that if a ~-u-polynomial A E P is reduced with respect to A, then A E RPo. Indeed, if this is not so, then let G tt RPo be a ~-u-polynomial of minimal rank reduced with respect to A. Then G tt U, so that the separant Sa of G belongs to P. Since Sa is reduced with respect to A and Sa < G, we have Sa E RPo. Now, the fact that Ro contains the field of rationals implies the inclusion G E RPo, which contradicts our assumption.

Let A E P. If A is reduced with respect to A, then A E RPo ~ {B,A}n. If A is not reduced with respect to A, then there exists a product J of powers of translations of J such that J A - L is reduced with respect to A for some L E [F1 , ... , Fr 1 (see Theorem 3.5.27). Therefore, JA E {B,A}n, so that Jp ~ {B,A}R.

Since J ~ P, we have the strict inclusion {P, J} R ~ P, hence the pe~fect ~u-ideal {P, J} R has a basis, which can be chosen in the form C1 , ... , Cq , J, where C1 , ... , Cq E P. Now, Proposition 1.4.9 implies that

i.e., the perfect ~-u-ideal P has a finite basis BuAu {C1 , .•. , Cq }. Thus, we obtain the contradiction with the choice of P, so the theorem is proved. 0

3.5.30. DEFINITION. Let R be a ~-u-c'-ring and let A = {Al, ... ,Ap} and B = {B1 , ... , Bq} be two autoreduced sets in R{Yl,"" y.}. Suppose that elements of A and B are ordered in such a way that Ai < ... < Ap and Bl < '" < Bq (below, if the contrary is not said explicitly, we shall always assume that elements of an autoreduced set are arranged in the increasing order). It will be said, that the autoreduced set A has lower rank than the autoreduced set B (and it will be written as rk A < rk B), if one of the following conditions holds:

(1) there exists an integer k E ~, 1 ::; k ::; min(p, q), such that rk A; = rk Bi for i = 1, ... , k - 1 and Ak < Bk;

(2) p > q and rk A; = rk Bi for i = 1, ... , q.

If p = q and rk Ai = rk Bi for all i = 1, ... , p, then we shall say that A and B are auto reduced sets of the same rank; it will be written as rk A = rk B.

The formulation and the proof of the following proposition are similar to those in differential and difference cases.

3.5.31. PROPOSITION. Any nonempty family I; of autoreduced sets of the ring of ~-u-e' -polynomials R{Yl, ... , Y.} contains an autoreduced set of lowest rank.


3.5.32. DEFINITION. Let R be a Ll-O'-c*-ring and I a Ll-O'-c*-ideal of the ring R{Yl, ... , y.}. Denote by U the set of all Ll-O'-c* -polynomials in I, whose separants do not belong to I. An autoreduced subset of U of the lowest possible rank is called a characteristic set of I.

3.5.33. PROPOSITION. Let Ro be a Ll-O'-c* -ring, R = Ro{y!' ... , Y.} the ring of Ll-O'-c* -polynomials in Ll-O'-c* -indeterminates Yl, ... , Ys over R o, and J a Ll-O'-c*ideal of R. Let A = {Fl , ... , Fr} be a characteristic set of J, Fl < ... < Fr, and let h, Sk denote the initial and separant of Fk (1 ~ k ~ r), respectively. Then the following statements hold:

(1) if a Ll-O'-c* -polynomial A E J ~ R is reduced with respect to A then the separant SA of A belongs to J;

(2) Ik ¢ J for all k = 1, ... , rj (3) if Ro is a field of zero characteristic, then J contains no nonzero Ll-O'-c*

polynomials reduced with respect to Aj in particular, Sk ¢ J for k = 1, ... ,r.

PROOF. (1) Suppose that a Ll-O'-c*-polynomial A E J \ Ro is reduced with respect to A. Let Fl , ... , F/ be elements of A whose leaders have lower rank than the leader of A. Then B = {Fl , ... , F/, A} is an autoreduced subset of J of lower rank than A that is possible only if SA E J.

(2) Suppose, that h E J for some kEN, 1 ~ k ~ r. If Uk is the leader of Fk and dk = deguk Fk, then F~ = Fk - Ik U~k E J and F~ is reduced with respect to A, so that the leader u~ of F~ either coincides with Uk, or is strictly less than Uk. In the both cases {)F~/{)Uk = Sk - dkIku~k-l E J (by the part (1) of the proposition), whence Sk E J. On the other hand, Sk is the separant of an element of a characteristic set of J, hence Sk ¢ J. This contradiction shows that h ¢ J for allk=I, ... ,r.

(3) If the last statement of our proposition is not true, then the set of all nonzero Ll-O'-c*-polynomials, that are reduced with respect to A, contains a Ll-O'c*-polynomial G of the lowest rank. Its separant SG does not belong to J because SG is reduced with respect to A and is oflower rank than G. Therefore, G and the elements of A that are lower than G form an autoreduced set whose rank is lower than the rank of A. This contradiction completes the proof. 0

3.5.34. EXERCISE. Let R = F{Yl, ... ,Ys} be the ring of Ll-O'-c*-polynomials in Ll-O'-c* -indeterminates Yl, ... , Y. over a Ll-O'-c* -field F of zero characteristic. Suppose that P is a prime Ll-O'-c*-ideal of R and A = {Fl , ... , Fr } is its characteristic set. Let J = It ... IrSl ... Sr, where h is the initial and Sk is the separant of Fk (1 ~ k ~ r), and let r(A) be the commutative semigroup, generated by the elements of the form jl(J), where jl E Tr. Prove, that P = UGEr(.A) ([Fl , ... , Frl : G).

Extensions of Ll-O'-c* -fields. Let F be a Ll-O'-c*-field (as usual, we suppose that Char F = 0), G a Ll-O'

c* -overfield of F, and ~ ~ G. An element v EGis called Ll-O'-c* -algebraically dependent of the set ~ over F if v is Ll-O'-c*-algebraic over the Ll-O'-c*-field F(~). Obviously, if an element v is Ll-O'-c* -algebraically dependent of ~ over F, then v is Ll-O'-c* -algebraic dependent of some finite set ~l ~ ~ over F.


The following proposition describes some properties of tl.-u-c· -algebraic dependence, which are analogical to the corresponding properties of differential and difference extensions.

3.5.35. PROPOSITION. Let F be a tl.-u-c·-field, G its tl.-u-c·-overfield, and B ~ G. Then the following statements hold:

(1) the set B is tl.-u-c·-algebraicallydependent over F iff there exists an element v E B that is tl.-u-c· -algebraically dependent of the set B \ {v} over F;

(2) if B is tl.-u-c* -algebraically dependent over F and B ~ B1 ~ G then B1 is also tl.-u-c· -algebraically dependent over F;

(3) if B is tl.-u-c* -algebraically dependent over F and F1 is a tl.-u-c· -extension of F contained in G then B is tl.-u-c· -algebraically dependent over F1 ;

(4) if v is tl.-u-c· -algebraically dependent of the set B over F, and every element of B is tl.-u-c· -algebraically dependent of the set B1 ~ Gover F, then v is tl.-u-c· -algebraically dependent of the set B1 over F;

(5) if an element v is tl.-u-c· -algebraically dependent of B over F and v is tl.-uc· -algebraically independent of the set B \ {u} over F, where u E B, then u is tl.-u-c·-algebraically dependent of the set (B U {v}) \ {u} over F;

(6) let B' ~ B and let {V1' ... , v.} be a subset of B that is tl.-u-c· -algebraically independent over F. If all elements V1, ... ,v. are tl.-u-c* -algebraically dependent of B' over F, then there exist elements W1, ... , w. E B' each of which is tl.-u-c* -algebraically dependent over F of the set B" that is obtained from B' by replacing Wj by Vj (1 ~ j ~ s).

The proof of these properties is similar to the proof of the corresponding properties of algebraic dependence; it and is left to the reader as an exercise. 0

The following statement is a consequence of Proposition 3.5.35.

3.5.36. COROLLARY. Let F be a tl.-u-c·-field, G its tl.-u-c·-extension, and B ~ G. Then the following conditions are equivalent:

(1) the set B is tl.-u-c*-algebraically independent over F and G is tl.-u-c·algebraic over F(B) (i.e., every element v EGis tl.-u-c*-algebraic over F(B));

(2) B is a minimal (with respect to inclusion) subset of the field G such that G is tl.-u-c·-algebraic over F(B);

(3) B is a maximal subset of the field G which is tl.-u-c* -algebraically independent over F.

3.5.37. DEFINITION. Let F be a tl.-u-c*-field and G a tl.-u-c*-extension of F. A set B ~ G that satisfies the equivalent conditions of Corollary 3.5.36 is called a tl.-u-c* -transcendence basis of Gover F.

The following theorem can be proved in the same way as the corresponding statements in differential and difference cases.

3.5.38. THEOREM. Let F be a tl.-u-c*-field and G a tl.-u-c*-extension of F. Then the following statements hold:

(1) if A ~ C ~ G, the set A is tl.-u-c· -independent over F, and the field G is tl.-u-c· -algebraic over F (C), then there exists a tl.-u-c* -transcendence basis B ofG over F such that A ~ B ~ C;


(2) all ~-O"-e*-transcendence bases ofG over F have the same cardinality; (3) if H is a ~-O"-e* -extension of G, B is a ~-O"-e* -transcendence basis of Gover

F, and B' is a ~-O"-e*-transcendence basis of Hover G, then B' n B = 0 and BuB' is a ~-O"-e* -transcendence basis of Hover F.

The second statement of Theorem 3.5.38 allows to introduce the following definition.

3.5.39. DEFINITION. Let F be a ~-O"-e*-field, and G a ~-O"-e*-extension of F. If the number of elements of any ~-O"-e* -transcendence basis of Gover F is finite, then this number is called ~-O"-e* -transcendence degree of Gover F. If any ~-O"-e* -transcendence basis of Gover F is infinite then we shall say that the ~-O"-e*-extension G of F has infinite ~-O"-e*-transcendence degree.

The ~-O"-e* -transcendence degree of a ~-O"-e* -field G over its ~-O"-e* -subfield F will be denoted by ~-O"-e* -tr degFG (if this degree is infinite, then we shall write ~-O"-e*-trdegFG = 00).

The following result is a consequence of Theorem 3.5.38.

3.5.40. COROLLARY.

(1) A ~-O"-e* -transcendence basis of a ~-O"-e* -field G over its ~-O"-e* -subfield F may be chosen from any system of ~-O"-e* -generators of Gover F. In particular, ifG = F('f/l, ... , 'f/,), then ~-O"-e*-trdegFG ~ s.

(2) Let F be a ~-O"-e*-field and G = F(TJl, ... , 'f/,), where 'f/l, ... , 'f/, are elements from some ~-O"-e* -overfield of F which are ~-O"-e* -algebraically independent over F. Then ~-O"-e*-trdegFG = s. In particular, if F('f/I. ... , 'f/,) = F«(I, ... ,(r), where {(I, ... , (r} is a family of ~-O"-e* -independent over F elements from some ~-O"-e*-extension of F, then r = s.

(3) The ring of ~-O"-e* -polynomials F {Yl, ... , y,} in ~-O"-e* -indeterminates Yl, ... ,Y. over a ~-O"-e* -field F is not isomorphic to any ring of ~-O"-e*polynomials F {ZI' ... , zr} in ~-O"-e* -indeterminates ZI, ... ,Zr over F, if r =p s.

(4) Let F be a ~-O"-e*-field, G its ~-O"-e*-extension, and H a ~-O"-e*-extension of G. If one of the quantities

~-O"-e* -tr degFG + ~-O"-e* -tr degGH,

is finite, then the other is also finite and

~-O"-e· -tr degFH

~-O"-e* -tr degFH = ~-(J'-e* -tr degFG + ~-O"-e* -tr degGH.

Differential-difference modules. Let R be a ~-O"-e* -ring, where ~ = {1I"1' ... ,lI"m} is a set of derivation operators

of the ring R, 0" = {al, ... , an} is a set of injective endomorphisms of R, and e = {,Bl, ... , ,Bq} ~ Aut R. Let A be the commutative semigroup consisting of the elements of the form

(S1' ... , Sm , k1' ... , kn EN; 11'···' lq E IE).

(As above, by the order of .A we mean the integer ord.A = 2:~1 Sj + 2:;=1 k j + 2:~=1 ILvl·)


3.5.41. DEFINITION. An expression of the form I:>'EA a>.A, where a>. E R for all A E A and only finitely many coefficients a>. are different from zero, is called a differential-difference operator over the A-17-e* -ring R (or a A-17-e* -operator over R). Two A-17-e*-operators I:>'EA a>.A and I:>'EA b>.A are considered to be equal iff a>. = b>, for all A E A.

The set of all A-17-e*-operators over a A-17-e*-ring R will be denoted by :F (or by :FR, if we want to mention the ring of coefficients). The set :F may be supplied with a structure of a ring, if we set

a E a>.A = E(aa>.)A; >'EA >'EA

(E a>'A) P. = E a>.A(Ap.); >'EA ),EA

p.a = p.(a)p.;

da = ad + d(a)

for all elements I:>'EA a>.A, I:>'EA b>.A E :F; a E R; p. E 17 U e U e*, d E A (e* = {,8I, ... , ,8q, ,81 1 , ... , ,8i I }) and require the validity of the distributive laws. This ring will be called the ring of differential-difference operators over the A-17-e* -ring R.

It is clear that the ring of A-17-e* -operators :F contains (as subrings) the ring of differential (A-) operators over R, and the ring of difference (17-) operators over R.

Let us define the order of a A-17-e* -operator w = I:),EA a>.A E :F as the integer ordw = max{ordA I a), i= O} and let :Fr = {w E :F I ordw ~ r} for r E N and :Fr = 0 for r E IZ, r < O. Then the sequence (:Fr )rEZ can be considered as an exhaustive filtration of the ring :F. In what follows, while considering :F as a filtered ring we will suppose (if the contrary is not said explicitly) that :F is supplied with the filtration (:Fr)rEZ.

3.5.42. DEFINITION. Let R be a A-17-e*-ring and :F a ring of A-17-e* -operators over R. Then a left :F-module is called a differential-difference R-module (or A-17-e*-R-module). In other words, a R-module M is called A-17-e*-R-module, if elements of the set AU 17 U e* act on M in such a way that the following conditions hold:

A(X + y) = AX + AY; A(P.X) = P.(AX);

d(ax) = adx + d(a)x; r(ax) = r(a)rx; ,8(,8-1x) = x

for any A E A, d E A, r E 17 U e* , ,8 E e; x, y EM; a E R. If R is a field, then a A-17-e*-R-module M will be called a vector A-17-e*-space

(or a differential-difference vector space over R).

It is clear that every A-17-e* -R-module is at the same time a A-R-module, 17-Rmodule, (17Ue)-R-module and e*-R-module (see Definitions 3.3.79, 3.4.42, and the definition before Example 3.2.43), if we consider R as A-, 17-, (17Ue)-, or e*- ring, respectively.


3.5.43. DEFINITION. Let M and N be two modules over a .a.-oo-c*-ring R. A homomorphism of R-modules 1 : M -+ N will be called a .a.-oo-c· -homomorphism (or a differential-difference homomorphism), if I()..x) = )../(x) for any).. E A, x E M. A surjective (respectively, injective or bijective) .a.-oo-c·-homomorphism will be called a .a.-oo-c· -epimorphism (a .a.-oo-c· -monomorphism or a .a.-oo-c· -isomorphism, respectively) .

3.5.44. REMARK. Let R be a .a.-c*-ring (i.e., a .a.-oo-c*-ring with 00 = 0) and M, N two .a.-c*-R-modules. Then each of the R-modules HomR(M, N) and M®RN can be supplied with the structure of .a.-c·-module in such a way, that 6(ip) = 60 ip - ip 0 6, (3(ip) = {3 0 ip 0 {3-1 for any 6 E .a., (3 E c·, ip E HomR(M, N) and 6(x ® y) :::::: 6(x) ® y + x ® 6(y), (3(x ® y) = (3(x) ® (3(y) for any 6 E .a., {3 E c· and any generators x ® y of M ®R N (x E M, YEN). The proof of the correctness of these definitions, as well as the checking of the conditions (1)-(5) of Definition 3.5.42, are left to the reader as an exercise.

3.5.45. EXERCISES.

(1) Show, that if a .a.-oo-c·-ring R is an integral domain, then the ring of .a.-ooc· -operators over R has no zero divisors.

(2) Let P, M, N be three .a.-c·-modules over a .a.-c·-ring R. Show, that the canonical isomorphism of R-modules

is a .a.-c·-isomorphism (here and below P ®R M and HomR(M, N) are considered as .a.-c·-R-modules with respect to the action of .a.Uc· described in Remark 3.5.44).

(3) Let R be a .a.-c· -ring, :F a ring of .a.-c· -operators over R, and M a.a.-c· -Rmodule. Denote by C(M) the set of all constants of M, i.e.,

C(M) = {x EM I 6x = O&{3x = x for any 6 E .a.,{3 E c}.

Obviously, C(M) is an Abelian subgroup of the additive group of M and the mapping C is a functor from the category of .a.-c· -R-modules to the category of Abelian groups. Prove the following properties of this functor:

(a) C(HomR(M, N)) = Hom.1"(M, N); (b) the functors C and Hom.1"(R, .) are naturally isomorphic; (c) the functor C is left exact and for any p E IE, p> 0 its right derivative

functor is naturally isomorphic to the functor Extj.(R, .). (4) Let R be a .a.-c·-ring, :F a ring .a.-c·-operators over R, and M, N two

.a.-c·-R-modules. Prove the following statements: (a) the functors Hom.1"(· ®R M, N) and Hom.1"(·, HomR(M, N)) are natu

rally isomorphic and the similar statement is true for the functors Hom.1"(M ®R·, N) and Hom.1"(M, HomR(·, N));

(b) for any positive integers p and q there exists a spectral sequence converging to Extj.+q (M, N), the second term of which is equal to E~·q(RPC)(Exth(M, N)). (Use Hint to Exercise 3.4.45).

CHAPTER IV

GROBNER BASES

4.1. Grabner Bases for Polynomials, Differential and Difference Modules

Let X = {Xl, ... , Xm} be a finite system of elements. By T = T(X) we denote the free commutative semigroup with unity (written multiplicatively), generated by the elements of X. Elements of T will be called monomials. Let 0 E T, 0 = X~l .•. x:',.m. By the order of 0 we shall call the sum e1 + .. +em that will be denoted by ord O. Suppose, that the set of monomials is linearly ordered and for any (J E T the following conditions hold:

1 < 9· - , (4.1.1)

(4.1.2)

In this case we shall say, that a ranking is defined on the set of monomials T. The following examples show that for every finite set X there exist different rankings.

4.1.1. EXAMPLE. The lexicographic ordering of monomials. Let

Then 91 < 92 , if either e1 < i1 or ej = ij for j = I, ... , k and ek+1 < ik+1 for some k (1 < k < m).

4.1.2. EXAMPLE. The standard ranking. We shall assume that

if either ord 91 < ord 92 or ord 01 = ord 92 and 91 < (J2 in the sense of lexicographic ordering.

4.1.3. EXAMPLE. The total degree then inverse lexicographic ordering. Let 01 = X~l ••• x:',.m, O2 = X~l ••. X;.;:.. We set 91 < O2 if either e1 + e2 + ... + em < i1 +i2 + .. ·+im or ej = ij for j = 0, ... , k and ek+1 > ik+1 for some k (0 < k < m).

Let K be a field, P the vector K-space with the basis T = T(X). We define on P the function "taking the leader" in the following way: any 9 in P may be represented as a sum 9 = EeET aeO, where only a finite number of coefficients ae E K are distinct from zero (such representation is unique up to the order of the terms). Among all monomials, present in this expression with nonzero coefficients, we choose the maximal with respect to the order introduced on the set T. This monomial will be called the leader of 9 E P and will be denoted by u g . The correctness of this definition follows from the fact that the linear ordered set T is a basis of the vector space P.

191


192 IV. GROBNER BASES

4.1.4. DEFINITION. Let some ranking on the set of monomials T = T(X) be given and let P be the vector K-space with the basis T. Suppose that P is a K-algebra, and UAB = UAUB for all A, B E P. Furthermore, suppose that lilt . 102 = 10102 E P for any 01 , O2 E T; in particular, the generators Xl, ... , xm pairwise commute. Such ring we shall call the ring of generalized polynomials in the indeterminates X = {Xl, ... , x m }.

4.1.5. EXAMPLE. The ring of commutative polynomials over a field. Consider an arbitrary ranking on the set X = {Xl, ... , x m }. Let P be the algebra K[Xl' ... , xml of polynomials in the commutative indeterminates Xl, ... , Xm over a field K. It is easy to see, that the condition UAB = UA UB holds for all A, B E P and therefore we may treat K[Xl' ... , xml as a ring of generalized polynomials in the indeterminates Xl, ... , X m .

4.1.6. EXAMPLE. The ring of differential operators over a field. Let K be a differential field with the basic set ~ = {dl , ... , dm } of pairwise commuting derivation operators, and let an arbitrary ranking be fixed on the set T = T(~). Then the ring D = K[d l , ... , dml of linear differential operators over K (see Definition 3.2.38) is a ring of generalized polynomials in the indeterminates d l , ... , dm .

4.1. 7. EXAMPLE. The ring of differential operators over a ring of polynomials. Let K be a differential field with a basic set ~ = {d l , ... , dm } and R a ring of commutative polynomials in the indeterminates Yl, ... ,Yn over K. We define the derivation operators ~' = {d~, ... ,d~} on R in the following way. If 1::S i::S m, then d:(Yj) = 0 for all j = 1, ... , n. For any i E N m we fix a number j E N n and set dHk) = dj(k)Yj for all j = 1, ... , nand k E K. Then the ring DR of linear ~/-operators over R is a ring of generalized polynomials in the indeterminates X = {d~, ... , d~, Yl , ... , Yn}. Indeed, if we consider the ranking such that d: > Yj for all i = 1, ... , m, j = 1, ... , n, then the condition ufug = Ufg is fulfilled.

4.1.8. EXAMPLE. The ring of difference operators over a field. Let K be a difference field with basic set of mutually commuting automorphisms {O'l' ... , O'm}. Then the ring R = K[ 0'1, ... , O'ml of linear difference operators (see Definition 3.3.78) is a ring of generalized polynomials in the indeterminates 0'1, ... , O'm. As a ranking we can take any ordering, satisfying the conditions (4.1.1)-(4.1.2).

4.1. 9. EXAMPLE. The ring of differential-difference operators over a field. We can generalize Examples 4.1.6 and 4.1.8 and consider the ring R = K[d l , ... , dm ,

0'1, ... ,O'ql, where the indeterminates d l , ... , dm correspond to derivation operators, and the remaining indeterminates correspond to automorphisms (see Definition 3.5.41)

Let D be a ring of generalized polynomials in the indeterminates X = {Xl, ... , Xm} over a field K and F the free D-module with the basis B = {b l , ... , bn }. The K-vector space F has as a basis the direct (Cartesian) product T x B of the sets T = T(X) and B. This product we shall call the set of terms of the module F,

GROBNER BASES FOR D-MODULES 193

We cannot multiply terms, but we can define the product of a term by a monomial from the corresponding ring of polynomials. Below we shall often identify a term

i. i . h (. . .) m+l Xl ... x,;-bj WIt the vector J, 11, ... , 1m E f:J .

4.1.10. DEFINITION. A ranking on the set TF of terms is a relation < of a well-order on TF, satisfying the following conditions:

u :S Ou for any term u E TF and for any monomial BET;

if u:S v, where u, v E TF, then Bu:S Bv for all BET.

(4.1.3)

( 4.1.4)

4.1.11. DEFINITION. A ranking will be called orderly if the condition ordTl < ord T2 (Tl, T2 E T) implies Tl b; < T2bj for all 1 :S i, j :S n.

4.1.12. EXAMPLE. Let a ranking on the set T of monomials be given. We shall order the terms of the form (0, i) by their last coordinates i, and only if the last coordinates of two terms coincide we shall compare their monomials. Such ranking on TF is not orderly.

4.1.13. EXAMPLE. Let tl, t2 E TF. We set tl = (01, it) < t2 = (B2 , i 2) if and only if

either ord Bl < ord O2 ,

or ordBl = ord02 and i l < i 2 ,

or ord 01 = ord O2 , i l = i2 and 01 < B2 with respect to lexicographic order with a fixed order of variables.

This ranking is orderly. We shall call it standard.

Note, that by the definition of a ranking, the set TF is well ordered with respect to any ranking.

4.1.14. DEFINITION. Let a ranking on the set TF be given, and let u,v E TF. We shall say that the rank of u is lower (respectively, higher) than the rank of v, if u < v (u> v, respectively).

Besides the order relation < on TF we define a partial order relation « in the following manner:

(4.1.5)

In this case we shall say that 01 divides B2 . The relation (4.1.3) implies that the order < is compatible with «. i.e .•

( 4.l.6)

4.1.15. DEFINITION. Any f E F \ {O} has a unique representation as a finite sum:

r

f = L cU. t;)t;. 0 -# cU, til E K, t; E TF • (4.1.7) ;:=1


We define the leader of f as uJ = t1 and the leading coefficient as Hcoeff(f) = c(f, t1). We set Uo = 0, Hcoeff(O) = O. Analogically, UB = {uJI fEB} for any finite subset B ~ F, UM for any submodule M ~ F denotes the module generated by {uJ I f EM}.

Let F be a free D-module and f, 9 E F. We shall say that f has lower rank than 9 and write rk f < rk g, if U J rkg, ifuJ > u g . IfuJ = u g , then we shall say that f and 9 has the same rank. It is obvious, that different elements may have the same rank.

4.1.16. DEFINITION. Let B C F \ {O} be a finite set of generators of a Dmodule M ~ F (w.l.o.g. we can suppose that Hcoeff(g) = 1 for any 9 E B). Define the reduction relation in the following way: f -+ f', if f, f' E F, and there exist

B t E Tp, (E T(X) and 9 E B such that c = c(f,t)::j:. 0, t = (ug , f' = f - c(g.

Note, that f' in Definition 4.1.16 is not uniquely defined by f.

4.1.17. LEMMA. Let f,J' E F and f -+ f'. Then rkf > rkf'· B -

PROOF. The assertion follows from the fact, that> is a relation of a linear order on Tp, and the property (4.1.4). 0

Below, we shall often omit index B in the reduction relation. The symbols 2t B

and ~ denote transitive and reflexive-transitive closures of -+, respectively. An B B

element f is called reducible if there exists f' ::j:. f such that f -+ f', in the opposite B

case f is called irreducible.

4.1.18. EXAMPLE.

1. Let m = 0, i.e., F is a vector space over a field J{ and let a standard ranking on Tp be given. If v is a vector, which has the first nonzero entry in ith position, then the reduction of a vector w with respect to v consists of the subtraction from w the multiple of v, so that w - kv has 0 in ith position.

2. Let J{ be a field, n = 1 and m = 1, i.e., F is a ring of univariate polynomials over J{ and let a standard ranking on Tp be given. Let B = {x 3 - 1; x2 - I}. Then x4 -+ x and x4 -+ x 2 -+ 1.

B B B

4.1.19. DEFINITION. Let a reduction relation -+ on a free D-module F be B

given and suppose that we have a computable function Sel : F -+ F such that f -+ Sel(f) for any reducible f E F. Consider the computable function S defined

B recursively by the formula

S(f) .- { f, .- S(Sel(f)),

if f is irreducible

if f is reducible.

We shall call an S of this kind a normal reduction process (normal-form algorithm)

for ~. For example, we can choose the reduced terms in the descending order B

(with respect to a well ordering of terms), and for a fixed term we try apply the reducing elements in the order, they are listed in B.


4.1.20. DEFINITION. A partial reduction process we define as a normal reduction process, when reduction steps are fulfilled only while the leader may be reduced.

4.1.21. LEMMA. If I ~ I', then I and I' are in the same congruence class of B

F/M, where M is the D-submodule generated by the set B.

PROOF. 9 E B, hence, c(g E M. 0

4.1.22. LEMMA. Let F be a free D-module and B be a finite subset of F. Then the reduction relation --+ is Noetherian (--+ has the finite termination property),

B B i.e., there is no infinite chain of the form f --+ II --+ ... --+ fk .... Hence, for any

B B B

f E F there exists (may be not unique) irreducible I' such that f ~ I'. B

PROOF. Suppose we have infinite chains of reductions. By definition, every ranking is a well ordering of the set TF. Therefore, we can choose an infinite chain of reductions such that the leader t of the first element 9 in this chain is minimal with respect to our ranking. We have two possibilities: either there exists in our chain a step, where t is reduced; then the remaining part of the chain is an infinite chain of reduction; or t is not reduced in all this chain; subtracting t from every element in this chain we receive a new infinite chain of reductions. In the both cases the leader of the first element in the new chain has lower rank than t, and this contradicts the minimality of t. 0

4.1.23. PROPOSITION. The set of irreducible with respect to --+ elements forms B

a vector K -space.

PROOF. We have to check that if f and 9 are irreducible and c E K, then f + 9 and cf are also irreducible. This follows immediately from the fact that 1+ 9 and cf contain with nonzero coefficients only those terms, which are present in f or g. 0

4.1.24. DEFINITION. Let S be the function on F x F such that S(f, 1') = 0, if f = 0, or I' = 0, or LCM{uJ, UJ') is not defined, in other cases S(f'/') = Hcoeff(f')<pf - Hcoeff(f)(I', where <p, ( E T{X) and <puJ = LCM{uJ, UJI) = (u!'.

4.1.25. DEFINITION. Let D be a ring of generalized polynomials in indeterminates X = {Xl' ... ' X m }, F a free D-module. Suppose that M ~ F is a submodule of F, GeM is a finite set and < is a ranking on the set of terms TF. The set G is called a Grabner basis (G-basis) of M, if there exists for any nonzero f E M a Grabner representation (G-representation):

r

1= L Ci<Pigi, 0 =I Ci E K, <Pi E T(X), gi E G, ( 4.1.8) i=l

that, in particular, implies


One and the same element may have several different G-representations. For example, if gl = t2 - 1, g2 = t 3 - 1, then (t 2 - l)(t3 - 1) = t 3 . gl - gl = t 2 . g2 - g2 are two different G-representations of the same polynomial. On the other hand, it is not easy to check that an element has no G-representation. We can put the restriction that any monomial may be used in G-representation as the leader of a summand CPigi not more than for one element gi E G. In particular, we may assume, that G is an ordered set and while choosing linear independent elements CPigi we use the rules given in Definition 4.1.19. Any representation of this form we shall call a normal G-representation.

To formulate the main result of this section we need some definitions and three lemmas.

4.1.26. DEFINITION. For I, J' E F we shall write l'il I', if there exists an I" E F such that I ~ I" and I' ~ 1".

4.1.27. LEMMA. Let 1,f',1" E F and I ~ I'· Then 1+ 1"'ilJ' + 1". PROOF. Let 1= J' + e·TJ· g, where u'I.g = cP and e = e(cp, I) i= 0, e(cp, I') = o.

If c" = e(cp, 1"), then I" = e" ·TJ . 9 + h, where e(cp, h) = 0, hence,

I + I" = !' + e ·TJ . 9 + e" ·TJ . 9 + h = !' + h + (e + c") . TJ . 9 ~ !' + h

and !' + I" = !' + h + e" . TJ . 9 ~ !' + h. 0

4.1.28. LEMMA. If a set G generates a submodule M C F and I - I' E M, then there exist an integer s ~ 0 and elements I = 10, It, ... ,I. = J' such that for all i from 1 to seither li-l -+ Ii or Ii -+ Ii-I.

PROOF. Since G generates M, the element I - I' may be represented as a sum

r

L:: ei . TJi . gi , i=O

where Ci are coefficients, TJ E T(X), gi E G (they may coincide for different i). We shall prove the lemma by induction on the minimal length r of such representation. If r = 0, then I = I' and the assertion of the lemma is valid. For arbitrary r we can assume that cp = u" •. g. ~ U"i.gi for all i. Set It = 1- e(cp, I) . TJr . gr, h = It -(cr -c(cp, I)) ·TJr ·gr· Then I -+ It f- hand 12- J' = 1- J' -Cr ·TJr ·gr = L:;':~ Ci • TJi • gi, so we can apply inductive hypothesis. 0

4.1.29. DEFINITION. We shall say that a reduction relation -+ is confluent, if for any I the conditions I ~ I' and I ~ I" imply J''il 1".

4.1.30. DEFINITION. We shall say that a reduction relation -+ is locally confluent, if for any I the conditions I -+ J' and I -+ I" imply I''il 1".

4.1.31. DEFINITION. We shall say that a reduction relation -+ is pseudo-locally confluent, if for all I, J' ,I" E F such that I -+ I' and I -+ I" there exist an integer s 2:: 0 and elements I' = 10, It, ... ,I. = I" such that I ~ Ii and Ii -1 'illi for all i = 1, ... , s.

GReBNER BASES FOR D-MODULES 197

4.1.32. LEMMA. If Noetherian relation ~ is pseudo-locally confluent, then ~ is confluent.

PROOF. We shall use transfinite or "Noetherian" induction, i.e. we shall show that if the lemma is valid for all 9 such that I ~ 9, then it is valid also for I. This is enough for the proof, because in the contrary case an element I, for which the statement of the lemma is false, can be chosen as the first element of an infinite sequence I ---t 11 ~ ... ---t In ---t ... such that the lemma does not hold for all elements of this sequence.

So, we fix I and suppose, that for all 1# such that I ~ 1# the statement of the lemma is valid. We shall show that it is valid also for I. W .l.o.g we can suppose that the given elements f' and f" are distinct from I, i.e., the reductions I ---t 91 ~ f' and I ---t 92 ~ f" hold. Then 91 and 92 satisfy pseudo-local confluence condition for some integer s.

/~ f'/g~,/,~f" ~/

g,~

95

Fig. 1

We shall use induction on s. If s = 1, then 91 and 92 are locally confluent. This means that there exists 93 such that 92 ~ 93 and 91 ~ 93. By inductive hypothesis for I' and 93 there exist 94 such that I' ~ 94 and 93 ~ 94, and 95 such that I" ~ 95 and 94 ~ 95· So, in this case the lemma is valid (see Fig. 1).

The transition from s to s + 1 is illustrated by the diagram in Fig. 2. Let 91 and 92 satisfy pseudo-local confluence condition with a sequence of s + 2 elements: 91 = 10,11,···,/,,/,+1 = 92· By inductive hypothesis 11 and 92 are confluent (element 94). The existence of 95, 96 and 97 in this diagram follows from assumption that elements, which can be obtained as a reduction of elements following I, in particular, 91, 11,92, satisfy the confluence condition. 0

The following theorem contains a number of conditions, which are equivalent to the definition of Grobner basis. It should be emphasized, that it contains conditions 6' and 7', which allow in a finite number of steps to verify if the given system of generators of a module M is its Grobner basis.

4.1.33. THEOREM. Let F be a free D-module, M ~ F be a D-submodule, GeM be a finite set, < be a ranking of the set of terms TF. Suppose, that


Hcoeff(9i) = 1 for all 9i E G, and some normal-form algorithm ===> for the reduction relation -+ is fixed. Then the following conditions are equivalent:

G

1 G is a G-basis of M; I' every element of M admits a normal G-representation; 2 UG generates UM;

3 for any f E M the relation f ~ 0 holds; G

3' for any f E M the relation f ::b 0 holds; G

4 if f - f' E M and f, I' are irreducible, then f = 1'; 5 if f E M and f is irreducible, then f = O.

The following conditions are necessary for preceding ones and if G generates M, they are also sufficient:

6 if f, f' E G and SU, f') f- 0, then SU, f') admits G-representation; 6' iff, f' E G and SU, 1') f- 0, then SU, 1') admits normal G-representation; 7 if f, f' E G and LCM(uf, ufl) is defined, then there exist in G elements

f = fa, ... , fi, ... , f. = f' such that

LCM{Uf; : i = 0, ... , s} = LCM(uf, ud (4.1.9)

and every S-element SUi-I, 1;), i = I, ... , s, admits G-representation; 7' if f, I' E G and LCM(uf, ud is defined, then there exist in G elements

f = fa, ... , f;, ... , fs = I' satisfying (4.1.9) and such that every S-element SUi-I, fi), i = 1, ... , s, admits normal G-representation;

8 if f ~ 1', f ~ f" and 1', f" are irreducible, then I' = f"; G G

9 if f ~ f', f ~ f",then there exists h E F such that I' ~ h, f" ~ h, G G G G

i.e., -+ is confluent; G

10 SU, f') ~ 0 for any f, I' E'G; G

10' SU, 1') ::b 0 for any f,J' E G; G

11 if f, I' E G and LCM(uf, ud is defined, then there exist in G elements

f = fa, ... , f;, ... , fr = 1', satisfying (4.1.9) and such that SUi-l,J;) ~ 0 G

for all i = 1, ... , r; 11' if f, f' E G and LCM(uf, ud is defined, then there exist in G elements

f = fa, ... , f;, ... , fr = 1', satisfying (4.1.9) and such that SUi-I, f;) ::b 0 G

for all i = 1, ... , r.

PROOF. We shall prove the following implications: 3-T1O-T11, 3' -T 10' -T 11' -T 11, 3 -T 4 -T 5 -T 3' -T 3, 3 -T 2 -T I' -T 1 -T 6 -T 7 -T 11 -T 9 -T 3, I' -T 6' -T 7' -T 7, 4 -T 8 -T 9. 3 -T 10. It is trivial, because SU, 1') E M.

GROBNER BASES FOR D-MODULES

3' -+ 10'. Analogically. 10 -+ 11. It is sufficient to set r = 1. 10' -+ 11'. Analogically. 11' -+ 11. Trivial.

199

3 -+ 4. By Proposition 4.1.23 the set of irreducible elements is a vector space, hence, if I and f' are irreducible, then their difference is also irreducible. Since I - I' E M, it follows from 2 and the preceding remark that I - I' = 0, i.e., 4.

4 -+ 5. Set I' = O. 5 -+ 3'. It is sufficient to apply Lemmas 4.1.21 and 4.1.22. 3' -+ 3. Obviously. 3 -+ 2. Let u E M and U u 1: UG. Then u cannot be reduced to 0, that contradicts

3. 2 -+ 1'. Suppose there exists 0 f. u EM, which have no normal G-representation.

Let choose among all such elements an element with minimal U u . By condition 2 we can apply a reduction step which reduces U u . This contradiction with minimality of U u proves I'.

I' -+ 1. Obviously. 1 -+ 6. It is sufficient to note that if lEG, f' E G, then SU, 1') EM. I' -+ 6'. Analogically. 6 -+ 7. Obviously. 6' -+ 7'. Obviously too. 7' -+ 7. Obviously. 7 -+ 11. Let 1 ~ i ~ r, U = SUi-1,/;) and u = L:j=l CjCPjgj, 0 f. Cj E ]{,

CPj E T(X), gj E G, CPiUgi > CPi+1Ugi+1 be a G-representation of u. Set Uk = L:j=k CjCPjgj· Then

U = U1 --+ U2 --+ ... --+ U r --+ O. G G G G

11 -+ 9. By Lemma 4.1.32 it is sufficient to prove that the reduction relation is pseudo-locally confluent. Let I -7 I', I -7 f". It means that there exist g', g" E G, r}', TJ" E T(X), cP' = TJ'U~, cP" = TJ"U~ such that I' = 1- C'TJ' g', I" = 1- C"TJ" g", where c' = cU, cp') f. 0, c" = cU, cp") f. 0, but cU', cp') = cU", tp") = o. We may suppose that cp" ~ cp'. We denote R(€) = € - Hcoeff(€)u{ for any € E F.

Take in I the summand c' cP', i.e., I = It + c' cP' + h, where It consists of terms which are higher than c' cP', and h consists of terms lower than c' cp'. We have to consider two cases: tp" < cP' and cp" = tp'. In the former one, assuming 12' = h - C" TJ" g" and 10 = It - c'TJ'R(g') + 12', by Lemma 4.1.27 we have f' = (It - c'TJ'R(g')) + h'l(1t - c'r}' R(g')) + 12' = 10, hence I''l f".

In the case when cP' = tp", we have u~ « cP' and u~ « cp". Therefore, LCM(u~, u~) is defined. By Condition 11 of the theorem there exists in G a se-

quence g' = go, . .. ,gj, ... ,gt = g", satisfying (4.1.9) and such that S(g, gi) ~ 0 for any i. So, u gi « tp' for any i, hence, there exist TJi E T such that TJiUgi = cP', c'cP' -7 c'TJiR(g;) and I -7 It - C'TJiR(9i) + h = hi, i = 1, ... , t. We shall show that hj -1 'l hi. It follows from the fact that hi - hi -1 = C' TJi -1 R(gi -1) - C'TJi R(gi) = c'(}S(/i ,1i-1) ~ 0, where () E T(X) and () . LCM(ugi , U gi _ 1 ) = cp'. Therefore, the relation -7 is pseudo-locally confluent.


9 -7 3. If G generates M then by Lemma 4.1.27 there exist I = 10, ... , J;, ... , I. = 0 such that for any i either li-l -7 J;, or /; --t Ii-I' Let k denote the greatest index for which the condition /; ~ 0 does not hold. Then h+l ~ 0 and IHI --t Ik. By Condition 9 we have 0\1 h, and we got a contradiction with the choice of k, since 0 cannot be reduced.

4 -7 8. f' - f" = (I' -!) + (I" -!) EM, hence, f' = f" 8 -7 9. Let I ~ I' and I ~ f". Choose irreducible elements If and If' such

that I' ~ If. f" ~ If'· Condition 8 implies If = ff', i.e., the relation --t is confluent. 0

Since the question about G-representability of an element can be solved algorithmically, Item 4.1.33.6' allows us to formulate an algorithm for checking if a given system of generators of a module is its Grabner basis. Condition 7' of this theorem allows to optimize this algorithm, checking G-representability not all S-elements, but a subset of them. Detailed algorithms will be given in the following section.

4.1.34. Exercises

1. Show that the polynomials

h = x3yz - xz2,

h = xy2z - xyz,

Is = x2y2 - z2

do not form a Grabner basis of the ideal, which they generate (we use the total degree, then lexicographic, ordering of monomials, x > y > z).

2. Show, that the polynomials

h = x3yz - xz2,

h = xy2z - xyz,

Is = x2y2 - z2,

14 = x2yz - z3,

Is = xz3 - xz2,

16 = yz3 - z3,

h = xyz2 - xz2,

Is = z4 - x2 z2 ,

19 = x3z2 - xz2

form Grabner basis of the ideal, considered in the preceding exercise.

3. Show, that using Theorem 4.1.33.11, it is sufficient in the preceding exercise to consider S-elements only for the pairs (2,3), (2,4)' (5,6), (4,7), (2,7)' (5,7), (5,8), (6,8), (4,9), (5,9).

4. Let D be a ring of generalized polynomials in a finite number of indeterminates over a field, F a free D-module, and let an orderly ranking on Tp be given. Let


H be a submodule of F and G a Grabner basis of H. Show, that for every f E H there exists a representation f = Ej=l >'j9j, where >'j E D, 9j E G, such that deg f 2: deg >'j9j for all j = 1, ... , r.

If a given system of elements does not form a Grabner basis of the submodule, generated by this system, then a Grabner basis can be obtained by adjunction of elements, obtained by reducing the S-elements, to the system.

A Grabner basis of any submodule M is not uniquely defined. In particular, adjuncting to a Grabner basis of M an arbitrary element hEM we have again a Grabner basis of M. Naturally, the question about minimal Grabner bases arises.

The following terminology came from differential algebra.

4.1.35. DEFINITION. A subset G = {9i : i E I} of a free module F is called an autoreduced set, if every 9i EGis irreducible with respect to G \ {9;}.

This definition immediately implies that the leaders of all elements in an autoreduced set are distinct.

4.1.36. PROPOSITION. Let D be a ring of generalized polynomials in the indeterminates X = {Xl, ... , Xm} over a field K and F the free D-module with the basis B = {bl , ... , bn }. Any autoreduced set in F contains only a finite number of elements, hence, its elements can be ordered by increasing of the leaders.

PROOF. The proof follows immediately from Lemma 2.2.1. 0

Let a ranking < of TF be fixed. Then we can introduce a partial order on the set of autoreduced sets in the following way.

Let A = {h, ... , fp} and B = {91, ... ,9q} be autoreduced sets, whose elements are ordered by increasing of their leaders. We shall write A < B, if

(1) either there exists i, 1 ::; i ::; min(p, q) such that uJ, q and Ufj = U gj for 1 ::; j ::; q.

4.1.37. LEMMA. Any set A = {Ai, i E I,} of autoreduced sets contains a minimal element with respect to the partial order introduced. A minimal element in the set of all autoreduced subsets of a submodule M of a free D-module is a Grabner basis of M.

PROOF. By Proposition 4.1.36 we can assume that the elements in these autoreduced sets are ordered by increasing of the leaders. At first we find the minimalleader among the first elements in autoreduced sets considered (it is defined uniquely, since the set of terms is well ordered). Denote this leader by t 1 . In the system A = {Ai, i E I,} of autoreduced sets we take the subsystem Pi = {Ai, i E 1',} of sets Ai = {Ai, ... , Aki} such that uA' = tl. In Pi we find the minimal leader

• 1

of the second elements of the autoreduced sets; denote it by t2. Continuing in this way, we have at every step an autoreduced set of terms ordered by increasing of the rank of their leaders. By Proposition 4.1.36 this process terminates after a finite number of steps. The choice of leaders had been fulfilled in such a way that at every step there exists an autoreduced set whose elements have as leaders t1 , ... , ti. By construction, the autoreduced set A corresponding to the complete system tl, ... , tn is minimal.


In order to prove that A is a Grobner basis of M, we use Theorem 4.1.33.2. Suppose, that there exists gEM, whose leader is reduced with respect to A. We can assume that 9 is reduced with respect to A. Consider the set

A' = {Ai E A I Ai < g} U {g}.

This set is autoreduced and its rank is lower than the rank of A, that contradicts the assumption on minimality of A. 0

4.1.38. COROLLARY. Let D be a ring of generalized polynomials over a field, F be a free finitely generated D-module. Then in every submodule M of F there exists a Grabner basis.

4.1.39. COROLLARY. Every ring of generalized polynomials over a field is (left) Noetherian.

PROOF. By Corollary 4.1.38 every left ideal I in such a ring has a Grobner basis. By Definition 4.1.25, Grobner basis of I is finite and generates I. 0

4.1.40. DEFINITION. Grobner basis G of a module M ~ F will be called auto reduced , if the set G is autoreduced.

4.1.41. PROPOSITION. Autoreduced Grabner basis of a module M is unique up to multiplication of its elements by constants from the field K.

PROOF. Among all autoreduced subsets of M choose a minimal one. Denote it by A and suppose that its elements are normed (multiplied by constants) so that all their leading coefficients are equal to 1 (multiplication by constants does not change autoreducibility and minimality). We shall show that these conditions define A uniquely.

Let A = {Ai, ... , Ar} and 8 = {Bl' ... , B.} be two sets, satisfying these conditions. The minimality implies that r = sand UAi = UBi for every i. Suppose that there exists i such that Ai "I Bi. Then Ai - Bi E M is irreducible with respect to Aj for j "I i, since irreducibility depends only on the leaders of the sets A and 8 and these leaders coincide. By Lemma 4.1.37, A is a Grobner basis of M and we got the desired contradiction. 0

4.1.42. DEFINITION. Let 8 = {It, ... ,!.} be a G-basis of a module M ~ F with respect to some ordering of terms in TF. The basis B will be called reducible, if for some i, 1 :S i :S s, there exists G-representation fi = Ej;I!i bjfj, in the contrary case the basis B will be called irreducible.

4.1.43. DEFINITION. A G-basis 8 of M containing s elements is said to be minimal, if there exists no G-basis 8 ' of M, containing less than s elements.

4.1.44. PROPOSITION. Every minimal G-basis is irreducible and vice versa. Every autoreduced G-basis is minimal and every minimal G-basis is quasi-autoreduced, i.e., the set of its leaders is autoreduced (this set is defined by the module M uniquely).

PROOF. It is clear, that a reducible G-basis is not minimal. So, it is sufficient to show, that the leaders of the elements in an irreducible basis are defined uniquely.

4.2. BASIC ALGORITHMS OF COMPUTATION OF GROBNER BASES 203

The proof is similar to the proof of Proposition 4.1.41 and we leave it to the reader as an exercise. 0

4.1.45. EXAMPLES.

1. Let [{ be a field and m = O. If the matrix of a system of linear polynomials has the row echelon form, then this system forms a Grabner basis of the ideal generated by this system. Show that the converse is, in general, false, i.e., there exist Grabner bases of linear polynomial systems, whose matrix is not of row echelon form.

2. Let [{ be a field, n = m = 1. A set 8 is a Grabner basis of the ideal (8) if and only if it contains gcd (greatest common divisor) of all its elements. A minimal Grabner basis in this case consists of a single element.

3. The Grabner basis in Example 4.1.34.2 is not minimal. After omitting its first element, it becomes minimal and even autoreduced.

4.1.46. EXERCISES.

1. Show that a system of algebraic equations has no solutions in the algebraic closure of its field of coefficients if and only if a Grabner basis of the ideal generated by this system, contains a constant (polynomial of degree 0).

2. Show that a system of algebraic equations in [{[Xl, ... , xm] has only a finite set of solutions in the algebraic closure of [{ if and only if a Grabner basis of the ideal generated by this system contains a polynomial whose leader is a power of Xi

for any i = 1, ... , m.

4.1.47. PROBLEM. Construct a theory of Grabner bases for ideals in rings of polynomials with the coefficients in the ring Z.

4.2. Basic Algorithms of Computation of Grabner Bases

Let D be a ring of generalized polynomials over a field [{ and F a free D-module. Suppose that the standard ranking on the set of the terms TF is given.

4.2.1. Algorithm RED(J, s, G) Input: /, s, G= {gl, ... ,g.} Output: /

Begin q :=.false. do while q =.false.

q :=.true. do for i = 1 to s while q =.true.

End

if Hterm(g;) « Hterm(J) cp := Hterm(J)/Hterm(g;) / := / - Hcoeff(J) . cp . gi q :=.false.


Let G = {gl,'" ,g,} be a finite set of elements of F such that Hcoeff(gj) = 1 for all gi E G. Let M be the submodule of F generated by G and let f be any element of F. In this section we shall denote the leader of f by Hterm(J). First of all, we give an algorithm for the partial reducing of f with respect to G.

Algorithm 4.2.2 verifies whether a given sequence of elements is a G-basis for a submodule of a free D-module. This algorithm is based on Theorem 4.1.33.6'. The subalgorithm 8(g, h) for computing the 8-element follows. We use H(i,j) for the LCM(gj, gj) in the algorithms.

4.2.2. Algorithm GR-CHECK-l(s, G, answer) Input: s, G Output: answer

Begin B:= ((i,j): 1 ~ i < j ~ s & H(i,j) # O} q :=.true. while q =.true. & B # (21

let (i,j) E B B := B \ {(i,j)} 8 := 8(g;,gj) RED(8,s, G) if 8 # 0

answer:= q End

q :=.false.

4.2.3. Algorithm 8(g, h) Input: g, hj Hcoeff(g) = Hcoeff(h) = 1 Output: 8 Begin H := LCM(Hterm(g), Hterm(h)) jJ := H/Hterm(g) v:= H/Hterm(h) 8 := jJ . 9 - v . h End

Algorithm 4.2.4 is a version of the algorithm GR-CHECK, based on Theorem 4.1.33.11'. It uses the possibility to skip some pairs without computing the corresponding 8-element.

Algorithms 4.2.5 and 4.2.6 allow us to complete a given set G of elements to a G-basis ofthe submodule generated by G (a simple version and an optimized one).

Before we attempt to formulate an algorithm for constructing a minimal G-basis for a D-module, we have to introduce a data structure system which will play a key role in this algorithm. By definition, a system consists of a finite set of indices I, of a set of elements G = {go< : a E I} of the D-module sorted in ascending order of their leaders, of a set of "critical pairs" B = ((i,j) : i,j E I, i < j, H(i, j) # OJ, sorted in ascending order of H(i, j), and of a stack 8T for elements of


4.2.4. Algorithm GR-CHECK-2(s, G, answer) Input: s, G = {91, ... ,g.} output: answer

Begin B:= ((i,j): 15: i < j 5: s & H(i,j) =F O} to sort B in ascending order of H(i, j) q :=.true. while q =.true. & B =F 0

let (i,j) E B : rk(H(i,j)) is minimal B := B \ {(i,j)} if em : 1 5: I 5: s & i =F I =F j & Hterm(gi) « H(i, j)

& (i, I) ft B & (i,j) ft B) f:= 5(g;,9j) RED{t,s,G) if f =F 0

q :=.false. answer:= q End

4.2.5. Algorithm G-BASIS-1(s, G) Input: s, G Output: s, G

Begin B := ((i,j): 15: i < j 5: s & H(i,j) =F O} while B =F 0

End

let (i,j) E B B := B \ {(i,j)} f:= 5(g;,gj) RED(f,s,G) if f =F 0

s:= s+ 1 f := f / Hcoeff (f) g. := f B := B U {(i, s) : 1 5: i < s & H(i, s) =F O}

our module. We suppose that at any moment, the set G, is quasi-autoreduced, i.e. the application of the algorithm RED to an element go E G and the set G \ {go} does not change the go. Besides, the system contains a variable t of the type integer, which contains the maximal value of Q E I. Let SYS be a variable of the type system.

The algorithm for the construction of a minimal G-basis is essentially recursive and its main sub algorithm is "to join (element) to (system)".

Now we suppose that the field K is the quotient field of an integral domain I and the module's generators are normed so that their coefficients belong to I. In this case, we cannot use the algorithm RED of the partial reduction formulated above.


4.2.6. Algorithm G-BASIS-2(s, G) Input: s, G Output: s, G

Begin B:= ((i,j): 1 ~ i < j ~ s & H(i,j) ¥= O} to sort B in ascending order of H (i, j) while B ¥= 0

End

let (i,j) E B: rk(H(i,j)) is minimal B := B \ {(i,j)} if (,Bl: 1 ~ l ~ s & i ¥= I ¥= j & Hterm(9d« H(i,j)

& (i,/) (j. B & (/,j) (j. B) 1:= S(9i,9j) RED(J,s,G) if I ¥= 0

s:= s + 1 I := 1/ Hcoeff(J) 9. := I B := B U {(i, s) : 1 ~ i < s & H(i, s) ¥= O} to sort B in ascending order of H ( i, j)

4.2.7. Algorithm MIN-BASIS-2(s, G) Input: s, G Output: s, G

Begin to form the empty system SYS for i = 1 to s

to join 9i to SYS while B ¥= 0

End

let (i,j) E B: rk(H(i,j)) is minimal B := B \ {(i,j)} if (,Bl: 1 ~ 1 ~ s & i ¥= I ¥= j & Hterm(9i) «H(i,j)

& (i,l) (j. B & (/,j) (j. B) 1:= S(9i,9j) to join I to SYS

We will change this algorithm so that it gives us a pseudo-reduction, which can be used for the computation of G-elements, the coefficients of which belong to I and cannot be used for the process of reduction in the general case, because it gives us only a normal (not canonical) simplifier (see [BL82]). The result f' of a pseudoreduction I -t I' may be not congruent to I with respect to the module generated

G by G, but if I and G are contained in a module, then the result of pseudoreduction lies in this module too.

For polynomial and differential modules the algorithm RED takes the following form.


4.2.8. Algorithm to join (J) to (SYS) Input: j, SYS Output: SYS

Begin if j =1= 0

End

for ga. E G, such that Hterm(ga.) ~ Hterm(f) if Hterm(ga.) « Hterm(f)

t := t + 1 to join t to I to join j to G

RED(f, 1, ga.) to join j to SYS return

to join {(O!, t), O! E I, O! < t, H(O!, t) =1= O} to B for ga. E G, such that Hterm(ga.) > Hterm(f)

if Hterm(g",) » Hterm(f) RED(g"" 1, j)

for any JEST

if g", =1= 0 to push go. into ST

to delete O! from I to delete g", from G to delete all pairs (O!, i) and (j, O!) from B

to join j to SYS

4.2.9. Algorithm RED(f, s, G) Input: j, s, G = {gl," ., gs} Output: j

Begin q :=.false. while q =.false.

q :=.true. for i = 1 to s while q =.true.

if Hterm(g;) « Hterm(f)

End

<p:= Hterm(f)/Hterm(g;) j := Hcoeff(gi) . j - Hcoeff(J) . <p . gi q :=.false.

The algorithm for the construction of the S-element has also to be changed in this case.

The algorithms GR-CHECK-l and GR-CHECK-2 are not changed. In the algorithms G-BASIS-l and G-BASIS-2, the line

j := f/ Hcoeff(J)

must be deleted.


4.2.10. Algorithm S(g, h) Input: g, h Output: S

Begin H := LCM(Hterm(g),Hterm(h)) J.l:= H/Hterm(g) v:= H/Hterm(h) S := Hcoeff(h) . J.l . 9 - Hcoeff(g) . v . h End

If we want to compute a G-basis in a difference module, we have to remember that the multiplication in the ring of difference operators is noncommutative and it remains noncommutative in the associated graded ring. So we have to apply the corresponding operators to the Hcoeff (g;).

4.2.11. Algorithm RED(J, s, G) Input: f, s, G= {gl, ... ,g.} Output: f

Begin q :=.false. while q =.false.

q :=.true. for i = 1 to s while q =.true.

End

if Hterm(g;) « Hterm(J) t.p:= Hterm(J)/Hterm(g;) f := t.p(Hcoeff(g;)) . f - Hcoeff(J) . t.p. g; q :=.false.

The algorithm for the construction of the S-element must also be changed.

4.2.12. Algorithm S(g, h) Input: g, h Output: S

Begin H := LCM(Hterm(g), Hterm(h)) J.l := H / Hterm(g) v := H / Hterm(h) S := v(Hcoeff(h)) . J.l. 9 - J.l(Hcoeff(g) . v· h) End

APPLICATION OF GROBNER BASES

4.3. Application of Grabner Bases to the Computation of Characteristic Polynomials

209

It will be shown in this section how, using the Grabner bases, we can compute the characteristic polynomial of a graded module or an excellently filtered module over a ring of generalized polynomials. Firstly, we consider the case of N-filtration (if in this case the ring considered is a ring of commutative polynomials, then we have the classic Hilbert polynomial), after this we shall introduce a notion of the characteristic polynomial in several variables.

In this section D is a ring of generalized polynomials in indeterminates X = {Xl, ... , Xm} over a field K (see Definition 4.1.4), and T = T(X) is the semi group of monomials in Xl, •.. , Xm . In particular, we can take as D the ring of commutative polynomials (Example 4.1.5), or a ring of differential (Example 4.1.6), or difference (Example 4.1.8), or differential-difference (Example 4.1.9) operators.

Recall that the set T(X) is ordered with respect to the relation < (see Section 4.1), we shall suppose that this ranking is orderly (see Definition 4.1.11), i.e. the condition ord T1 < ord T2 implies T1 < T2.

By N-filtration on T = T(X) we shall mean an ascending chain of subsets (T(S)).eN of T such that T = U.eNT(s). If the contrary is not said explicitly, then we shall suppose that N-filtration (T(S)).eN on T is given such that

T(s) = {X~l ... x!; I i1 + ... + im S s},

for any sEN. This filtration induces a N-filtration of D:

Do C Dl C ... C D. C ... (4.3.1)

where D. = {2=~=1 kiT; In EN, ki E K, Ti E T(s) for i = 1, ... , n}.

4.3.1. EXERCISE. Prove that (4.3.1) is a filtration of the ring D, i.e., D = U.eND. and D.Dq C D.+q for any s, q EN.

Let F be a free D-module with a bases B = {b1 , ... , bn }. The filtration (4.3.1) of D generates a N-filtration of F:

where

Fo C F1 C ... C F. C ... ,

n

F. = LD.bj

j=l

It is clear, that F = U.eNF. and D.Fq ~ F.+q for any s,q EN.

(4.3.2.)

Consider an orderly ranking on the set of terms TB = {6bj}geT,bieB. Then the filtration (4.3.2) is compatible with the ranking, i.e., I E F. is equivalent to ord uJ S s (as before, uJ denotes the leader of I, and ord 6 is the order of 6). Indeed, if I E F., then UJ has the form >"bj, where>.. E T{s), hence, orduJ S s. Conversely, if ord uJ S s, then since ord T1 < ord T2 (T1, T2 E T) implies T1bi < T2bj


for all 1 S; i,j S; n (we assume the ranking to be orderly), we can write f in the form

f = k· uJ +

hence, f E F •. In particular, if the ordering of monomials is standard, then we can consider the

standard ranking on TB, although the lexicographic ordering of the set T is not orderly.

4.3.2. EXERCISE.

(a) Compute dimK D •. (b) Compute dimK F •.

4.3.3. DEFINITION. The Hilbert function of a filtered D-module F with a filtration (Fs)sEf~ is the function i.pF : N -+ N such that i.pF(S) = dimK Fs for any sEN.

Suppose that nH is a submodule of F; then H has the induced filtration: H. = F. n H (s EN). It is easy to check that we have a N-filtration, and if we set (F/H). = F./H. (s EN), then we obtain a N-filtration on F/H, with which the Hilbert function i.pF/H : N -+ N is associated such that i.pF/H(S) = dimK(F/H). for any sEN.

4.3.4. EXERCISE. Show that the Hilbert function of the module D/ D>., where >. E D, ord>. = a, is equal to

The theory of Grabner basis allows us to reduce the problem of Hilbert function computation to the combinatoric problem considered in Theorem 2.2.5.

4.3.5. THEOREM. Let D be a ring of generalized polynomials in indeterminates Xl,"" Xm over a field K. Let F be a free D-module with the basis {b l , ... , bn }

and H a D-submodule of F. Consider an orderly ranking on T, and let F. = {f E Fiord uJ S; s} (s EN) be aN-filtration of F, H. = FsnH, and A be a Grabner basis of H (see Definition 4.1.25). Denote by Ej the set ofm-tuples (el, ... ,em ) E Nm

such that X~l ... x;;,.~ bj is the leader of an element in A. Then for alllarge enough integers s the Hilbert function i.pF/H of F/H is equal to "L;=l WEj(S), where WEj

is the Kolchin dimension polynomial of the set Ej .

PROOF. Define the mapping,: H -+ Nm x Nn in the following way: if hE H and the leader of h has the form Uh = X~l ... x~ bj , then ,(h) = (i l , ... , im , j). By Theorem 4.1.33.3 any h E H is reducible with respect to A, hence we have 1m, C Uj'=l (VEj' j). (Recall, that for E C Nm , we have defined VE = {x E Nm I 3e E E, X ~ e}, see Section 2.2.) Since the ranking is orderly, we have ,(Hs) = U;=l (VEj (s), j). It is easy to see that if {h;}iEl is a system of elements of F and ,(hi) :f: "f(hj ) for all i :f: j, i,j E I, then the system {hd is linearly

APPLICATION OF GROBNER BASES 211

independent over K. Therefore, dimK H. ~ "2:}=1 Card VEj(S). In order to prove the inverse inequality, we denote by A. a set maximal with respect to the inclusion, consisting of elements of the form X~l ..... x~ f, where f E A, (i1 , .. . , i m ) E Nm, X~l ' ... ' x~ f E H. and the leaders of different elements of A. are distinct. Since the filtration U. F. is compatible with the ranking, CardA. = "2:']=1 Card VEj(S). By Theorem 4.1.33.3 for any f E H. there exists a decomposition f = "2:kjUj, where kj E K, Uj E A •. Hence, dimH. ::; CardA •. Therefore, 'PF/H(S) = "2:']=1 WEj(S) for all large enough sEN. 0

4.3.6. COROLLARY. Under the hypotheses of Theorem 4.3.5, the Hilbert function of the module M = F / H for all large enough sEN is a polynomial W M of degree no higher than m.

PROOF. See Theorems 4.3.5 and 2.2.7. 0

4.3.7. DEFINITION. Let H be a submodule of a free D-module F, then the polynomial WM described in Corollary 4.3.6 is called the characteristic Hilbert polynomial of the module M = F / H.

4.3.8. EXERCISE. Let D be a ring of polynomial s over a field K, a E D, M . . { a'Pl = 0, . be aD-module glVen by the generators 'PI, 'P2 and equatIOns Fmd the

a'P2 = O. characteristic polynomial W M.

We shall now consider NP-graded modules. Always below in this section we fix some partition of the set Nm = ~=1 Uj into nonempty disjoint subsets u, ... , up.

For any p-tuple (S1, ... , sp) E NP we consider in T = T(X) the subset

Obviously, U(.l .....• p)ENP T'(s1, ... , sp) = T and T'(SI, ... , sp) n T'(r1, ... , rp) = 0

for (S1, ... , sp) ::j:. (rl, ... , rp).

4.3.9. DEFINITION. Let D be a ring of generalized polynomials over a field K in the indeterminates X = {Xl, ... , xm }, Nm = ~=1 Uj a partition of Nm. We suppose the ranking of T = T(X) to be orderly, and

D(.l .....• p) = { L: arT I a r E K and almost all coefficients }. rET'(.l .....• p) ar are equal to 0 (4.3.3)

The ring D will be called NP -graded if

D= and


4.3.10. EXERCISE. Let D be a ring of commutative polynomials in the indeterminates X = {Xl"", Xm} over a field K, Nm = U~=1 (Tj be a partition of Nm . We shall call a polynomial f(xl, ... , xm) E D homogeneous of degree (SI, ... , sp) E NP, if in the ring D[t1 , .•. , tp] of commutative polynomials we have f(tj(I)Xl, ... , tj(m)Xm) = t~' ..... t~P f(xl, ... , xm), where j: Nm -t Np is the mapping which maps any i E Nm onto the index of subsets, where i lies. Show that the linear K-subspace D(Sl, ... ,8 p)' given by (4.3.3), is the set of homogeneous polynomials in D of degree (SI, ... , sp), and D is a NP-graded ring.

4.3.11. EXERCISE. Let D be a ring of difference operators over a field K. Prove, that we can consider D as a NP-graded ring, using the method of Exercise 4.3.10.

Note, that not for any ring of generalized polynomial D, a partition of the set Nm defines by the formula (4.3.3) a NP-grading of D.

4.3.12. EXAMPLE. Let D be a ring of differential operators over a field K = C(x), A = {d}, d = a/ax and Dj = K . di . Then the condition DIDo C Dl does not hold, because dx = xd + 1 f/:. Dl , therefore, D = E9 Dj is not a grading of D.

4.3.13. EXERCISE. Let K be a differential field, A = {dl , ... , dm} be the basic set of derivation operators on K, Nm = ~=1 (Tj be a partition of Nm , tl, ... , tp be commutative variables over K. Suppose that the multiplication in the ring D of generalized polynomials in indeterminates dl , ... , dm, t l , ... , tp is defined by the rule: ifi E (Tj, then k·di = dj·k-d;(k).tj (k E K) (see Example 4.1.7). By D(81, ... ,8p) we denote the linear K-subspace of D with the basis {d~' ..... d:';'t~' ..... t~P} such that EiE<7j ei = Sj for every j = 1, ... ,p. Prove that E9(81, ... ,8p)EJlIP D(81,""Sp) is a NP-grading of D.

4.3.14. EXERCISE. Let D be a NP-graded ring of generalized polynomials in m indeterminates over a field K. Compute dimK D(Sl,""Sp)'

4.3.15. DEFINITION. Let D be a NP-graded ring of generalized polynomials over a field K. AD-module M will be called NP-graded, if for any (SI, ... , sp) E NP a K-subspace M(Sl,""Sp) of M is defined such that M = E9(""""p)ENP M(81, .. "'p) and

for all (SlJ ... , sp) E NP. The elements of M(Sl, ... ,8 p) will be called the homogeneous elements of M.

4.3.16. EXERCISE. Prove, that if vM is a finitely generated NP-graded Dmodule, then dimK M(", ... ,Sp) < 00 for any (SI, ... , sp) E NP.

4.3.17. DEFINITION. Let vM be a finitely generated module over a ring of generalized polynomials and M = E9(", ... ,8p)ENP M(", ... ,.p) be a NP-grading of M. The function r,o~, whose value at any p-tuple (81, ... , sp) is equal to dimK M(", ... ,8p) will be called the characteristic function of the NP -graded module M.

We shall show now, that the theory of Grobner bases allows us to compute the characteristic function of a NP-graded module.


4.3.18. EXERCISE. Let DM be a NP-graded D-module, and {ml, ... ,mk}, where mi E M(",;, ... ,,,,~) (i = 1, ... , k), be a set of its homogeneous D-generators. Prove that

for all (SI' ... 'Sp) E NP (we assume, that D('l, ... ,'p) = 0, if Sj < 0 for some 1 ~ j ~ pl.

4.3.19. EXERCISE. Suppose that under conditions of Exercise 4.3.18 the set {ml' ... ' mk} is linearly independent over D. Compute dimK M(Sl, ... ,Sp).

4.3.20. THEOREM. Let DM be a NP-graded module, {ml, ... ,mk} be a finite set of its generators such that mj E M(",~, .. ,a~) (j = 1, ... , k). Then there exist sets E; C Nm (i = 1, ... , k) such that for all large enough (SI, ... , Sp) E NP the characteristic function of M is equal to

P k

<p~ (SI, ... , Sp) = II 81 (j) EWE, (SI - ai, ... , Sp - a~), j=1 ;=1

where WE (SI, ... , sp) is the dimension polynomial of the matrix E, associated with the partition ofNm (see (2.3.7) for the definition of 81(j)).

PROOF. Consider the D-module homomorphism

1/;: F --+ M --+ 0,

where F is a free D-module with the basis {/i};=I, .. ,k and 1/;(1;) = mi (i = 1, ... , k). We shall view F as a NP-graded module, setting

k

F(Sl, ... ,sp) = L D('l-",;, ... ,sp-a~jli. i=1

( 4.3.4)

So,1/; is a homomorphism of NP-graded modules, therefore, Ker 1/; = H is a homogeneous submodule of F (i.e. H has a set of homogeneous generators). Hence, H is a NP-graded module and

<p~ (SI , ... , sp) + <p~ (SI' ... , sp) = <plj/ (SI , ... , sp)

= 81(1) ..... 8 p(1) t (SI + kl - a{) ..... (sp + kkP - at) j=1 kl + P +

= tIT (Sj +;. ~ 7~ -1) ;=1 j=1 J +

(see Exercise 4.3.19); here Nm = Uj=1 (J"j is a partition which defines a NP-grading of D, (kj = Carduj).

214 N. GROBNER BASES

Consider an orderly ranking on the set of terms x~' ... x~ Ii, and let A be a Grobner basis of H with respect to this ranking. We may assume that every element of A is homogeneous with respect to NP-grading (4.3.3). Indeed, in order to find a Grobner basis we can apply Algorithm 4.2.5 or 4.2.6. We can suppose the input generators {hi} to be homogeneous; if hi E H(/Jl, ... ,/Jp)' then Uhi E F(fJI, ... ,{Jp)'

Therefore, while reducing and computing S-elements, we always receive homogeneous elements of H, and as a result we have a homogeneous Grobner basis A of H. Then the reasoning are similar to the ones in Theorem 4.3.5. Denote by Ej the set of m-tuples (e1, ... , em) such that x? ... x~m Ii is the leader of an element from A, and let,: F -t Nm x Nk be the mapping, which maps f E F onto a vector (e1"'" em,j) such that x? .. . x~m Ii is the leader of f. Since A is a Grabner basis of H, we have "Y(H) = U~=l(VE;,j). Since the basis A is homogeneous,

where by VE (Sl, ... , sp)g r we denote the set of m-tuples (i1 , ... , i m ) E VE such that EkE"'; ik = Sj (j = 1, ... , p). Since the leaders of K-linearly independent elements of K are distinct, this implies

k

dim H(", ... ,.p) ~ L Card VE; (Sl - a{, ... , Sp - a~)gr. j=l

Denote by A('l, ... ,'p) a maximal by inclusion set of elements {X~' ... x~mh} such that h E A, X~' ... x~mh E H(", ... ,.p) and the leaders of different elements from A(Sl,""Sp) are distinct. Since A is a homogeneous Grobner basis, we have

dimH(", ... ,8 p ) ::; CardA(", ... ,.p) = E~=l Card VE;(Sl - a{, ... , sp - at)gr. Note, that CardVE;(sl, ... ,sp)gr + CardVE(sl, ... ,Sp)gr = CardNm(sl, ... ,Sp)gr = n~=l (';t,k..!;l) (since VE = Nm\E), therefore, for all sufficiently large (Sl, ... , sp) E NP we have

k

dimH(", ... ,.p) = L Card VE; (S1 - a{, ... , Sp - a~)gr j=1

k p i k _ '" II (Sj + kj - 1 - aj) '" C d TF (j j)gr -L.J k.- -L.J ar vE S1- a1,···,sp-ap i=1 j=1 J 1 j=1

~ IIP (Sj + kj - 1 - a i.) ~ . . = {;tj=1 kj -l J -f;;tWE(S1-ai, ... ,sp-af,)

k

+ LWE;(S1 -1- a{,s2 - at .. . ,sp - a~) j=1

k

+ ... + LWE;(S1 - a{, S2 - at .. ·, sp - 1- a~) j=1

Hence,

APPLICATION OF GROBNER BASES

k

- L W Ej (S1 - 1 - a{, ... , sp - 1 - a~) j=1

k p ( .) = L II Sj + ~. = ~ -aj

i=l j=l J

k

-~1(1) ... ~p(I)LWEj(Sl-a{, ... ,sp -a~) j=1

_ ~IIP (Sj +kj -I-a}) gr( ) -~. k'-1 -<PM Sl,""Sp .

• =1 J=l J

p k

<pX; (Sl, .. . , Sp) = II ~1(j) LWEj (Sl - aL .. ·, Sp - a~) j=1 i=l

for all sufficiently large (Sl, ... , Sp) E NP, and the theorem is proved. 0

215

We see that the Grabner bases theory allows to compute characteristic polynomials of N-filtered and NP-graded modules. Now we are going to consider NPfiltered modules. As before, we suppose that we have an orderly ranking of the set of monomials T = T(X) of the ring D, and we consider a NP-filtration on T associated with a partition Nm = Uj aj,

T= u (Sl, ... ,sp)EJ'liP

T(S1, ... ,sp) = {X~' .. . x:'"m I Lei::; Sj for all j = 1, .. . ,p}. iEo,]

4.3.21. DEFINITION. Let D be a ring of generalized polynomials over a field f{ in indeterminates X = {Xl, . .. , xm} and Nm = U;=1 Uj a partition of the set Nm .

The ring D is called NP-filtered, if D = U(sl, ... ,Sp)EJ'liP D(S1, ... , sp), where

and

(e" ... ,em)ENm

k(el, .. ,e m ) E K, Lei::; Sj for all j = 1, ... , p} iE(J j

D(Sl, ... ,sp)D(q" ... ,qp) ~ D(Sl+q" ... ,SP+qp)

(4.3.5)

for all (S1, ... ,sp) and (ql, .. . ,qp) E NP. We suppose that D(Sl,""Sp) = 0, if Sj < 0 for 1 ::; j ::; p.

4.3.22. EXERCISE. Let D be a NP-filtered ring and (Sl, ... ,Sp)::; (q1, ... ,qp) in the sense of the product order on NP. Then D(Sl,""Sp) ~ D(q" ... ,qp)'


4.3.23. EXAMPLE. Let D be a ring of commutative polynomials in the variables Xl,·'" Xm , over a field K, Nm = U~=l (1'j be a partition of Nm . Let D(s" ... ,sp) = {f E D I degO'j f :s Sj for all j = 1, ... ,p}, where degO'j f denotes the degree of f as a polynomial in the variables {Xi}iEO" Then D = U(o )E"'p D(s, ...• ) is a

J "'1, ... ,8p.l~' ) p

NP-filtration of D.

4.3.24. EXAMPLE. Let D be a ring of differential-difference operators over a field K with a basic set X = {Xl"'" xm}, where each Xi is either a derivation operator or an automorphism of K. Let Nm = U~=l (1; be a partition of Nm . Then (4.3.5) gives a NP-filtration of D.

Note, that similarly to the graded case, not every partition of Nm gives a NPfiltration on an arbitrary ring of generalized polynomials.

4.3.25. EXAMPLE. Let D be the ring given in Exercise 4.3.13. If m =1, p = 1, then D is a ring of generalized polynomials in X = {Xl, X2} over a field K. In D X2 commutes with Xl and elements of K, and multiplication by Xl is defined by the rule k· Xl = Xl' k - d(k)X2' where d is the derivation operator on K.

Consider the partition N2 = {I} U {2}, and let D(S"S2) = K . T(Sl, S2). If the derivation operator is not trivial, then the condition D(1,o) . D(o,o) ~ D(1,o) does not hold, since Xl E D(1,o) , k E D(o,o), but Xl . k = k . Xl + d(k)X2 ~ D(l,O) for d(k) =1= O.

4.3.26. EXERCISE. Let D be a f:JP-filtered ring over a field K. Compute dimK D(s" ... ,sp)'

4.3.27. DEFINITION. Let D be a NP-filtered ring of generalized polynomials over a field K. A left D-module M will be called f:JP-filtered if a system of linear K-subspaces M(s" ... ,sp) (Sl,"" sp E f:J) of M is given, satisfying the following conditions:

(1) D(s" ... ,sp)M(q" ... ,qp) ~ M(.,+q" ... ,sp+qp) for all (Sl,"" sp), (q1,"" qp) E NP;

(2) M(s" ... ,sp) ~ M(q" ... ,qp) for all (Sl,"" sp) E NP, (ql,"" qp) E NP such that Sj :s qj for j = 1, ... ,p;

(3) M = U(s" ... ,Sp)ENP M(s" ... ,sp)'

The system M(s" ... ,sp) (Sl," .,sp E f:J) is called a NP-filtration of M.

4.3.28. EXAMPLE. Let DM be a finitely generated module and {m1," .,md be a system of its generators. For any (Sl," .,sp) E f:JP we set M(s" ... ,sp) = L~==l D(s" ... ,sp)mj. It is easy to check, that it is a f:JP-filtration of M; we shall call it the f:JP -filtration associated with the choice of generators {m1' ... ,mk}.

4.3.29. EXAMPLE. Let DM be a NP-filtered module and DN be a submodule of DM. Set N(s" ... ,sp) = M(s" .. ,sp) n N for any (Sl,"" sp) E NP. It is easy to check that the linear K-spaces N(s" ... ,sp) form a NP-filtration of M. We shall call it the f:JP-filtration of N induced by the NP-filtration of M.

Let a NP-filtration on a ring D of generalized polynomials over a field K be given. Since the D-module D ®K K[Yl, ... , Yp] is isomorphic as a D-module to the ring D[Yl, ... , Yp], we can consider D ®K K[Y1' ... , Yp] as a ring. Define a subring


jj as the set of elements D ®K K[Y1, ... , Yp] of the form L(., ..... sp)El\IP T(., •...•• p) ®

y~' ... y;P, where T(., ...... p) E D(., ..... sp) for all (Sl' ... , sp) E NP and the sum contains only a finite number of nonzero terms.

4.3.30. EXERCISE. Show that jj is a ring.

4.3.31. THEOREM. Let D be a NP-filtered ring of generalized polynomials over a field K. Then the ring jj is left Noetherian.

PROOF. We shall show that jj is a ring of generalized polynomials over K in the indeterminates X = {Xl. ... , Xm , Zl. ... , zp}, where Zi = 1 ® Yi (i = 1, ... ,p) and if i E 0), then Xi = Xi ® Yj. Indeed, the elements of X pairwise commute and every element of jj can be expressed in the form

Consider the following order on the set T = T(X) of monomials of jj: every element BET we can write in the form B = Tzf' ... z;P, where T E T = T(X), and we order such elements lexicographically with respect to

(ord B, T, a1, ... , a p ) (4.3.6)

(recall, that we suppose the ranking on T(X) to be orderly). We have to prove that for any A, B E jj the equation UAB = UA . UB holds. First of all, prove that if A E T is a monomial and B = k E K, then

A . B = k' . A + L: kjTj,

jEJ

(4.3.7)

where k',kj E K, Tj E T' and Tj < A for all j E J. Indeed, if A = Zj, then Zj . k = 1 ® kYi = k· (1 ® Yi) = k . Zi. Let A = Xi, i E 0'/. Since the ring D is NP-filtered, we have Xi . k E D(o .... • 0.1.0 .... • 0)D(0 .... • 0) C D(o .... • 0.1.0 ... · .0) (where 1 in (0, ... ,0,1,0, ... ,0) is in lth position), hence, Xi . k = ko + LjEJ k j . Xj, where ko, k j E K, kj =f:. 0, J C 0'/. Since D is a ring of generalized polynomials we have U(xi. k ) = U Xi • Uk = Xi, therefore, the sum LjEJ kjxj contains a term kiXi,

and if j E J, j =f:. i, then Xj < Xi (in the sense of the ranking of T). Therefore, AB = Xi·k = Xi ·k®y/ = (ki 'Xi+kO+LjEJkjXj)®Y/, where J ~ 0'/, i rt. J and AB = kiXi + koz/ + LjEJ kjxj = kiA+ L kjTj, where Tj E T' and Tj < A. So, (4.3.7) is valid if ord A = 1, and B is an element of K. Now, (4.3.7) can be proved for an arbitrary monomial A E T' by induction on ord A. If A, B E jj, then we can write

A = ko . UA + L: kjTj,

JEJ B = TOUB + L: TjBj ,

jEJ'

where kj,Tj E K, Tj,Bj E T', Tj < UA for all j E J' and Bj < UB for all j E J'. Then

AB = k o ' UA . TO' UB + L: kj Tj TOUB + L: kOUATjBj + L: kjTjTjBi.

jEJ jEJ' jEJ. iEJ'


. Using (4.3.7), we have AB = couAuB+I>jrJuB+LbijAi8j, where cj,bij E I<, rJ < UA, Ai < uA, OJ < UB· By the definition of a ranking, in this case rJuB < UAUB and AiOj < UAUB, therefore, AB = COUAUB + Lcj1'j, 1'j < UAUB, hence, UAB = UAUB·

SO, we have proved that D is a ring of generalized polynomials over I< in the indeterminates X = {Xl, ... , Xm , Zl, .. " zp}, if T(X) is ordered by the formula (4.3.6). By Theorem 4.1.39, the ring D is Noetherian. 0

Let D be a f;!P-filtered ring of generalized polynomials. In Theorem 4.3.31 we proved that D is also a ring of generalized polynomials, and the ranking of monomials of the ring D is also orderly. With the notation of Theorem 4.3.31, consider a partition of the set f;!m+p = Uf=l iTi such that iTi = Ui U {i + m}. By (4.3.3), we have

for all j = 1, ... ,p} =D( ) @ y'l y'P SI, ... ,Sp l' .. P .

Let (Sl' ... , sp) and (rl' ... ,rp) be distinct elements of f;!p. Then, obviously,

D(sl""'Sp) n D(rl, ... ,rp) = 0, D D (D @ 'I Sp) . (D @ rl rp ) (SI,""Sp) (rl' . .,rp) = (aI, .. ,sp) Yl'" Yp (rl"'.,rp) Yl'" Yp

CD @ 'l+rl, .. ·,Sp+rp _ (al+rl, ... ,ap+rp) Yl

= D(sl+rl, ... ,ap+rp)

and D = E9 D(sl""'sp)' so, we have the following result.

4.3.32. THEOREM. Let D be a NP-tiltered ring of generalized polynomials,

D = U(sl, ... ,sp)EJlIP D(sl, ... ,a p )' Then the ring D is f;!P-graded with respect to the

linear I< -subspaces D(sl , ... ,sp).

Let now M be a NP-filtered module. Consider the D-submodule

(4.3.8)

of M @ I< [Yl, ... , Yp].

4.3.33. EXERCISE. Show that the sum in (4.3.8) is direct, and jjM IS a f;!P-graded module.

4.3.34. DEFINITION. A f;!P-filtration of a module DM is called excellent if the module jj M is finitely generated.


4.3.35. THEOREM. Let D be a NP-filtered ring of generalized polynomials. Then every finitely generated D-module can be supplied with an excellent NP-filtration.

PROOF. Consider the filtration of Example 4.3.28, associated with the generators m1,"" mk. The module jjM is generated by m1 ® 1, ... , mk ® 1. Indeed, any m ® y~' .. . y;p, where m E M(", ... ,.p), can be expressed in the form LjP'j ® y~' .. . y;p) . (mj ® 1), where Aj E D(.l, ... ,.p)' hence, the filtration M =

U M(", ... ,.p) is excellent and the theorem is proved. 0 ('" ... ,'p)ENp

4.3.36. EXERCISE. Show that if under the hypotheses of Theorem 4.3.35, we assign to every generator a "weight" , then the resulting filtration

k

M(", ... ,.p) = L: D(.,-a{ •...•• p-a~) mj, j=1

where (at ... , at) E NP (j = 1, ... ,p) is also excellent.

4.3.37. PROPOSITION. Let M be an excellently NP-filtered module. Then the module D M is finitely generated.

PROOF. Since the module jjM is NP-graded, we can choose its homogeneous

generators {mi};el, where mi = mi ® y~~ ... y;~, mi E M('i, ... ,'~) and Card! < 00. The set {mi};el generates the D-module M, because, if m E M, then m E M(", ... ,.p)' m ® y~' ... y;p E M(., .....• p), therefore,

hence, m = :Liel Aimj, Ai ED and Aimj E M(., .... ,.p). 0

4.3.38. EXERCISE. Show that in the case p = 1, Definition 4.3.34 is equivalent to the following one: a filtration (Mr )rEN of aD-module M is called excellent if dimK Mr < 00 for all r E N and if there exists ro EN such that Mr = Dr-roMro for all r > ro.

4.3.39. THEOREM. Let DM be an excellently NP-filtered module, and let tp(t1, ... , t p) be a function of the real variables t 1, ... , tp such that

tp(S1, ... , sp) = dimK M(., •...•• p)

for all (S1""'Sp) E NP. Then tp(S1, ... ,Sp) < 00 for all (SI""'Sp) E NP and tp(tl, ... , t p) for all sufficiently large (tl, ... , t p) can be represented by a polynomial, whose degree in tj does not exceed CardtTj for all j = 1, ... ,p (i.e. there exist apolynomialw(tl, ... ,tp), degtjw ~ CardtTj andp-tuple (rl, ... ,rp) such that tp(t1, ... , t p) = W(tl, ... , t p) for all tj ~ rj (j = 1, ... , p)). This polynomial will be called the characteristic polynomial of the excellently NP -filtered module D M and will be denoted by WM(tI, ... , tp).

PROOF. First of all, note that dimK M(." ....• p) = dimK M(.l .....• p) for all (SI, ... , sp) E NP. Indeed, let {/;};El be a K-basis of M(.l •...• &,,). Then


{Ii 0 yfl ... y;P hEf is a K-linearly independent system of elements in the module £1(."""sp)' Conversely, if we choose a K-linearly independent system {mihEJ of

elements in £1(" ''''''p), then mj = mi 0y~' ... y;P , where mi E M(" , ... ,sp)' therefore, elements {m;}iEJ are linearly independent.

The statement of the theorem follows now from Theorems 4.3.20 and 4.3.32, since by assumption n£1 is a finitely generated NP-graded module. 0

4.3.40. THEOREM. Let DM be an excellently NP-filtered D-module, DN be a NP-filtered D-module and t/J: N -+ M be an injective homomorphism ofNPfiltered modules (i.e. t/J(N(", ... "P)) ~ M(",,,,8p) for all (SI,"" sp) E NP). Then NP-filtration of N is also excellent.

PROOF. It is easy to see that the injective homomorphism t/J induces an injection nil '-+ n£1. By Theorem 4.3.31 the ring D is Noetherian and by hypotheses of the theorem the module £1 is finitely generated over D. Therefore, the module nil is finitely generated, hence, NP-filtration DN is excellent (see Definition 4.3.34). 0

4.3.41. COROLLARY. Let D be a NP-filtered ring of generalized polynomials over a field K and F be a free D-module. Let F('l, ... ,'P) (SI,"" sp E N) be the filtration on F, associated with free generators and let H be a D-submodule of F. Then the induced NP-filtration on H: H( ...... ,sp) = H n F(",,,,,,P) is excellent. Furthermore, in H there exists a finite system of elements {hihEf such that every hE H(S1,".,Sp) may be expressed in the form

h = L Ajhi , Aj ED, iEI

where Ajh i E H(S1,"',sp) for all i E I.

(4.3.9)

PROOF. The fact that the linear K-spaces H(", ... ,Sp) form an excellent NP

filtration follows from Theorem 4.3.40 ('IjJ == 1). Hence, the module n if is finitely generated. The existence of the system {hi} follows from the proof of Proposition 4.3.37. 0

4.3.42. EXAMPLE. Let :F be a differential field, ~ be its basic set of derivation operators, 9 = :F ("11, ... ,"In) be a finitely generated ~-extension of :F. Suppose that a partition Nm = U7=1 (J'i (m = Card~) of Nm is given. Consider the module of differentials D fJg /:F, where D is the NP -filtered ring of differential operators over 9; this module is finitely generated by 15"11, .. ' ,tS"In' By Theorem 4.3.39 (see also Example 4.3.24 and Theorem 4.3.35), there exists a polynomial Wg / :F(tl' ... ,tp) such that Wg/ :F(Sl, ... , sp) = dimg 2::;=1 D(" , ... ,sp)tS"Ij for all sufficiently large (Sl' ... , sp) E NP. In Chapter 5 we shall see that

for all sufficiently large (SI' ... , sp) E NP.

4.3.43. REMARK. If p = 1, then Definition 4.3.21 of NP-filtration on a ring of generalized polynomials coincides with the definition considered in the beginning of the section (see (4.3.1)). Therefore, the existence of a system with property


(4.3.9) is guaranteed by the existence of a Grobner basis in a submodule of a free module (see Exercise 4.1.34.4). But, ifp > 1 and {hd is a Grobner basis of H, then property (4.3.9) may be not valid. Note also, that in case p = 1, Theorem 4.3.5 reduces the computation of the characteristic polynomial of a submodule of a free D-module to a combinatoric problem, but the analogous statement for p > 1 does not hold, since an orderly ranking on the set of monomials may be incompatible with the NP-filtration.

4.3.44. EXAMPLE. Let D be a ring of commutative polynomials [([Xl, X2] over a field [( and H be the ideal of D generated by 9 = X I + X2. Consider the partition N2 = {I} U {2} of N2 and let D = U D('l,") be a N2-filtration of D. Obviously, 9 forms a Grobner basis of H, and H('l"2) = D('l,") n H = {f E H I degxl 1 ~ Sl, degx, 1 ~ S2} = D('l-1,B2-1) . 9 for all sufficiently large (Sl' S2) E N2. Therefore, dimK H('l,") = dimK D('l-1"2-1) = Sl . S2 and dim(D(",.,)/ H('l,.'») = (Sl + 1)(S2+1)-Sl'S2 = Sl +s2+1, but Wu .(Sl' S2) = W(l,O)(SI, 82) = S2+1. The difference between Wu .(Sl,S2) and dimH(",.,) is due to the fact that 9 = (Xl + X2) E H(1,l) and u g = Xl E F(l,O)' By Theorem 4.3.39, in order to compute dimH('l,") it is sufficient to compute the characteristic polynomial of the corresponding N2-graded ideal H of D. From the proof of Theorem 4.3.31 we can see that D is a ring of polynomials D = [([Xl, X2, X3, X4] in commuting indeterminates with a N2-grading such that 0\ = {1,3}, (T2 = {2,4}. Since H('l"2) is the filtration associated with the generator (Xl + X2) of the weight (1,1) (see Exercise 4.3.36), the ideal H of D is generated by (Xl + X2) ~ Y1Y2 = (Xl ~ yt) . (1 ~ Y2) + (X2 ~ Y2) . (1 ~ Y2) = X1X4 + X2X3 E H(l,l) (see the proof of Theorem 4.3.31). Obviously, this element forms a Grobner basis of i, therefore (see Theorem 4.3.19) dim(D/H)('l,.2) = A1(1)A1(2)w(1,O,1,O) = Sl +82 + 1, or dimH( .... ,) = (Sl + 1)(s2 + 1) - (Sl + S2 + 1) = Sl S2 for all sufficiently large (81, S2) E N2.

This example shows that the transition from an ideal H to H is a "homogenization on several variables" .

4.3.45. EXERCISE. Let D = [([Xl, ... , xm] be a ring of commutative polynomials, Nm = Uf=l C1'i be a partition of Nm , D = U('l"""p)ENP D(",oo.,'P) be a NP-filtration of D (see Example 4.3.23) and J be an ideal of D. Consider NPfiltration of J induced by the NP-filtration of D: J = U('l,oo"'p)ENP J(",oo.,.P)' Then the ideal i of D = [([Xl, ... , Xm , Zl, ... , Zp] may be described in the following way: 1 E i if and only if there exists 9 E J(",oo"Sp) such that I(X1, ... , Xm , Zl, ... , zp) =

S - -Z~l ... zpP g(...£L, ... , ~) (here the mapping j : Nm -+ Np maps a number to the

%J(I) %J(m)

index of the partition containing this number). If p = 1, then j is the homogenization of J.

CHAPTER V

DIFFERENTIAL DIMENSION POLYNOMIALS

5.1. Characteristic Polynomials of Excellently Filtered Differential Modules

Let R be a differential ring with a basic set ~ = {db .. " dm }, D be the ring of linear differential operators over R (see Definition 3.2.38). As before, by T we denote the set of monomials of D (see Example 4.1.6 and Definition 4.1.4), and T(r) denotes the set of monomials whose order does not exceed r. Consider on D an ascending filtration (Dr )rEZ, where Dr = {f E D lord f ~ r} = R . T(r) for r ~ 0, and Dr = 0 for r < 0 (see Exercise 4.3.1). Below, if the contrary is not said explicitly, by a filtration on D we shall mean this filtration. By a filtered differential R-module we shall mean aD-module M with exhaustive and separable filtration (Mr )rEZ, It means that M = UrEZ Mr and there exists ro E :&; such that Mr = 0 for all r < ro, M; ~ Mi+l and D;Mr ~ Mr+; for all r, i E:&;.

5.1.1. DEFINITION. Let M be a filtered ~-R-module with a filtration (Mr)rEZ and suppose that Mr are finitely generated over R for any r E :&;. Then we say that the filtration (Mr)rEZ is finite and we call M a finitely filtered ~-R-module.

5.1.2. DEFINITION. Let M be a filtered ~-R-module with the filtration (Mr )rEZ, If there exists an integer ro E:&; such that M. = D.-roMro for all s > ro, then the filtration (Mr)rEZ is called good, and M is called a good filtered ~-R-module.

5.1.3. DEFINITION. A finite and good filtration of a ~-R-module M is called excellent. In this case M is called an excellently filtered 6.-R-module (compare with Definition 4.3.34).

5.1.4. EXAMPLE. Let R be a differential ring, D be the ring oflinear differential operators over Rand P be a R-module with a nondecreasing chain of R-submodules (Pr )rEZ such that UrEZ Pr = P and Pr = 0 for all small enough r E:&;. Then the tensor product D®R P may be naturally treated as a left D-module. For any r E :&; consider the R-submodule (D ®R P)r of D ®R P generated over R by all elements of the form u ® x, where u ED;, x E Pj and i + j = r. It is easy to see that ((D ®R P)r)rEZ is a filtration of the ~-R-module D ®R P.

5.1.5. DEFINITION. Let M,N be filtered differential R-modules. AD-homomorphism r.p : M -+ N will be called a homomorphism of filtered ~-R-modules, or a homomorphism of filtered D-modules, if r.p(Mr) ~ Nr for all r E Z.

5.1.6. DEFINITION. A sequence M -+ N -+ P of homomorphisms of filtered differential modules is called exact if for any r E Z the sequence of R-module homomorphisms Mr -+ Nr -+ Pr is exact.

223


224 v. DIFFERENTIAL DIMENSION POLYNOMIALS

5.l.7. EXAMPLE. Let M be a finitely generated D-module, and {mjhEl be a finite system of its generators. The filtration associated with these generators (see Example 4.3.28) is excellent.

5.l.8. EXAMPLE. Let M be a filtered .6.-R-module and N be a submodule of M. Consider the induced filtration on N (see Example 4.3.29) Nr = N n Mr. Then the homomorphism of injection z : N --+ M is a homomorphism of filtered D-modules. Introducing on the D-module F = M/N the filtration Fr = Mr/Nr, we obtain the sequence N ~ M -4 F (J(m) = m (mod N)) which is an exact sequence of homomorphisms of filtered .6.-R-modules.

Recall we can associate a graded ring gr D with a filtered ring D.

5.l.9. THEOREM. The ring gr D is isomorphic to the ring R[Xl, . .. , xml of commutative polynomials in m indeterminates (m = Card.6.) over R.

PROOF. The definition of the filtration (Dr )rEZ implies dj E Dl for all i = 1, ... , m. Denote Xj = dj (mod Do) (i = 1, ... , m). By the definition of D we have kd; = djk - dj(k) for all k E R, hence, kXj = xjk. Any f E gr D, as it is easy to see, may be expressed in the form

(jl, ... ,j~)EI'~~, k·l···i~ER

that completes the proof. 0

. x:';,

5.l.10. PROPOSITION. Let A be a filtered ring, M an A-filtered module, and the filtration (Mr )rEZ be exhaustive and separable. Suppose that gr M is a finitely generated module over gr A. Then M is finitely generated over A.

PROOF. Let {mj hEl be a finite set of generators of the gr A-module gr M. Every mj is equal to mj (mod Mj') for some mj E Mj .+1. Then {m;} is a system of Agenerators of M. Indeed, since the filtration (Mr) is exhaustive, if m EM, then mE Mr for some r E IZ, therefore, m (mod Mr-d = l:iAjm;, where Aj E grA. Hence, m = l:iEl ajmj + h, where aj E A, h E Mr - l . Now we expand h and so on. By assumption the filtration is separable, therefore, M; = 0 for all i < ro, and the proposition is proved. 0

5.l.1l. THEOREM. Let R be an Artinian .6.-ring with a basic set .6. = {dl , ... , dm } of derivation operators, let (Mr )rEZ be an excellent filtration of a .6.-R-module M, and IR(W) denote the length of a R-module W. Then there exists a polynomial XM such that XM{S) = IR(M.) for all sufficiently large sEN. Besides, degXM ::; m.

PROOF. Obviously, IR{Mr) = l:.<rIR(gr. M) (the sum in the right-hand side is finite, since we consider only separable filtrations). We shall show that there exists an univariate numerical polynomial <p(t) such that deg <p(t) ::; m - 1 and <p{s) = IR(gr. M) for all sufficiently large s E IZ.

Since the filtration (Mr )rEZ is excellent, there exists ro E IZ such that M. = Ds-roMro for all s E IZ, s > roo By Example 5.1.4, the D-module D 0R Mro may

5.1. CHARACTERISTIC POLYNOMIALS OF DIFFERENTIAL MODULES 225

be viewed as a filtered .1.-R-module with the filtration (D ®R Mro)qEZ described the filtration ((Mro)q)qEZ of Mro is such that (Mro)q = Mro n Mq for any q E ~). In this case the mapping 71" : D ®R Mro --t M such that 7I"(u ® x) = ux (u E D, x E Mro) is well defined and is a surjective homomorphism of filtered D-modules, since 7I"((D ®R Mro)q) = Mq for all q E ~, q ~ roo As Proposition 1.3.41 shows, the exactness of the sequence D ®R Mro ~ M --t 0 implies the exactness of the sequence gr(D ®R Mro) gr") gr M --t 0 of graded gr D-modules. Taking into

account the natural epimorphism gr D ®R gr Mro ..f!..". gr(D ®R Mro) --t 0, we see that the sequence of graded gr D-modules

is exact. Since the gr D-module gr D ®R gr Mro is finitely generated over gr D (by assumption, Mr is finitely generated over R), we see that gr M is a finitely generated gr D-module. By Theorem 5.1.9, gr D ~ R[x}, ... , xm], hence, gr M may be viewed as a finitely generated graded R[Xl, ... , xm]-module. By Theorem 1.3.21, there exists an univariate numerical polynomial rp(t) such that deg rp(t) ~ m-l and rp(s) = lR(gr. M) for all s E ~, s > So, where So is some integer. By Proposition 2.1.5, there exists a numerical polynomial X(t) such that deg x(t) = deg rp(t)+1 ~ m and x(r) = lR(Mr ) = [R(M'o-tl + L:=.o lR(gr. M) for all integers r > So. The theorem is proved. 0

5.1.12. DEFINITION. Let (Mr)rEZ be an excellent filtration of a .1.-R-module M over an Artinian .1.-ring R. The numerical polynomial X(t), whose existence is proved in Theorem 5.1.11, is called dimensional or characteristic polynomial of M, associated with the excellent filtration (Mr )rEZ and will be denoted by XM (t).

Let R be a differential ring with a basic set .1. = {d1 , ... , dm } of derivation operators, D the ring of .1.-operators over R, M a .1.-R-module with a filtration (Mr )rEZ, and R[x] the ring of univariate polynomials with coefficients in R. As in Section 4.3, we denote by D the subring LrEZ Dr ®R Rxr of D ®R R[x], and by

Nt we denote the left D-module LrEZ Mr ®R Rxr.

5.1.13. LEMMA. Let (Mr)rez be a finite filtration of a .1.-R-module M. The filtration (Mr )rez is good if and only if the module Nt is finitely generated over D.

PROOF. Let the filtration (Mr)rez be good. Then there exists an integer ro EN such that M. = D.-roMro for all s > roo By assumption, the R-module Mro is finitely generated. Let {m? heI be a finite system of its R-generators. For any o ~ h ~ ro, we denote by {mfheI the system of R-generators of Mh. Then the D-module Nt is generated by the elements

{mh ® xh}. i .el, h=O, ... ,ro· (5.1.1)

Indeed, it suffices to prove that m®x' for any m E M. belongs to the D-submodule Nt generated by system (5.1.1). If s ~ ro, then by the choice of {mfheI there exists a representation m = LiE! Aim!, Ai E R, therefore, m ® x' = LieI(Ai ® 1) . (mt ® x')' where Ai ® 1 E D. Suppose s > roo Since the filtration (Mr )rez is good, we


have M. = D.-roMro, i.e. there exists a representation m = 2:iEI Aimi, where

mi E Mro , Ai E D.-ro . Let A; = 2:;;;;~o a~O~, where a~ E R, O} E T(j). Then

m®x' = LA;m; ®x· ;EI

iEI,j=O, ... ,.-ro

:L (a~O; ® x·- ro ) . (mi ® xrO) j=O, ... ,.-ro,;EI

= L Xim; ® xro , ;EI

where X; E D. SO, system (5.1.1) D-generates the module M. Conversely, let {m;};EI be a finite system of generators of iJM. We can choose

this system so that m; = m;®xj ;, m; E Mj;. Let r = maXiE! j;. We shall show that M. = D.-roMro for all s 2:: roo Indeed, let m E M. and s 2:: roo Since m®x' EM, here exists a representation m®x' = 2:;EI X;m; ®xj;, where X; ED, Xi = A; ®xk;,

or m ® x' = 2:;EI A;m; ® xi;+k; = 2:~o h, ® x', where h, = 2:;:k;+j;=1 Aim;. The R-module R[x] is free over R, therefore, [Kash77 Prop. 10.2.5] m = h. = 2:;:k;+i;=. A;m;, i.e., m E 2:;~o D._pMp. It is easy to check that D. ~ D.-roDro for any s 2:: ro, hence, m E 2:;~o D._pMp = 2:;~o D.-roDro-pMp ~ D.-roMro· So, the filtration (Mr )rEZ is good. 0

5.1.14. LEMMA. Let R be a Noetherian differential ring with a basic set d = {d1, ... , dm } of derivation operators. Then the ring of d-operators D and the ring D constructed above are left Noetherian.

PROOF. Let L be a left ideal of D, and L; = LnD; be the induced filtration on L. Then gr L is an ideal of gr D. The injection is fulfilled in the following way: if I E Li, then cp(1 (mod L;-d) = I; (mod D;-d. If I; E D;-l then I; E L n D;-l = Li-l, therefore, cp(1 (mod L;-d) = O. Hence, cp is an injection of gr L into gr D. By Theorem 5.1.9, gr D ~ R[Xl, ... , xm ], therefore, the ideal gr L is finitely generated. Proposition 5.1.10 implies that the ideal L is also finitely generated over D.

In order to prove the second statement we note that D is the ring of linear d'-operators over the polynomial ring R[x'], where d' = {di, ... , d~}, and the derivation operators act on R[x'] by the rules:

dHk) = d;(k)x', d~(x') = 0

see Theorem 4.3.31. Since the ring R[x'] is Noetherian, it suffices to apply the first statement of the lemma. 0

5.1.15. PROPOSITION. Let R be a Noetherian differential ring with a basic set d = {d1 , ... , d,.}. 1ft: N ~ M is an injective homomorphism of filtered differential R-modules and the filtration of M is excellent, then the filtration of N is excellent.

PROOF. Let (Mr )rEZ and (Nr )rEZ be the given filtrations of M and N respectively. The filtration (MrEZ) of M is excellent, hence, finite. Therefore, the Rmodule Mr is finitely generated, hence, Noetherian for any r E Z (since R is a

5.1. CHARACTERISTIC POLYNOMIALS OF DIFFERENTIAL MODULES 227

Noetherian ring). For any r E 7l the module Nr is isomorphic to a R-submodule of Mr (the homomorphism ~ is compatible with the filtrations), so, Nr is finitely generated over R, hence, the filtration (Nr )rEZ is finite.

In order to prove that the filtration (Nr )rEZ is good, we consider the modules

M = LrEZ Mr ®R Rxr and if = LrEZ Nr ®R Rxr over D = LrEZ Dr ®R Rxr (as before, D is the ring of .6.-operators over R). Since R-module Nr for any r Ellis isomorphic to a submodule of Mr , we see that if is isomorphic to a D-submodule of M. The filtration (Mr)rEZ is excellent, hence, good, and Lemma 5.1.3 implies that M is a finitely generated D-module. By Lemma 5.1.14 the ring D is left Noetherian, therefore, M is a Noetherian D-module. So, the D-module if is finitely generated. Once more applying Lemma 5.1.13, we see that the filtration (Nr )rEZ is good. The theorem is proved. 0

Theorems 5.1.11 and Lemma 5.1.14 imply that the ring of differential operators over a field is an Ore ring.

5.1.16. COROLLARY. Let R be a differential field with a basic set .6. = {d1, ... , dm } of derivation operators and let D be the ring of .6.-operators over R. Then D is a left Ore's ring, i.e., for any u, v E D there exist nonzero I, g E D such that lu = gv.

PROOF. If u = 0, then the assertion is trivial. Suppose that u and v are distinct from zero and let p = ord u, q = ord v. For any sEll, let D(s) denote the filtered .6.-R-module (D(S)r)rEZ, where D(s)r = Ds+r for any r E 7l. Then the mapping <p : D(-p)$D(-q) --t D such that (x, y) --t xu-yv (x, y E D), is a homomorphism of filtered D-modules. Let M denote the image and N denote the kernel of <po The filtration of the .6.-R-module D( -p)$D( -q) is excellent (its rth component (r E 7l) has the form D(r - p) $ D(r - q)), therefore, the induced filtrations on M and N are also excellent (for M it is obviously, and for N it follows from Proposition 5.1.15). Let XM(t), XN(t) and X(t) denote the characteristic polynomials of M, N and D( -p) $ D( -q), respectively, associated with the given excellent filtrations. The exactness of the sequence of excellently filtered .6.-R-modules

o --t N --t D( -p) $ D( -q) --t M --t 0

implies that

( r+m-p) (r+m-q) XN(r) + XM(r) ~ x(r) = m + m

for all sufficiently large r E 7l (see Example 5.1.8), therefore

XN(t) ~ e+:- p) + e+:- q) -XM(t)

~ e+:- p) + e+:- q) - e:m)

(since M ~ D, and XD(t) = (I"!:') according to Exercise 4.3.4). The last inequality shows that degXN(t) ~ m, hence, N i- O. If (f,g) EN, then <p(f,g) = 0, therefore lu =gv. 0

228 V. DIFFERENTIAL DIMENSION POLYNOMIALS

5.2. Differential Dimension

In this section we shall prove the theorem on the existence of the Kolchin's differential dimension polynomial, using Theorem 5.1.11 on the existence of the characteristic polynomial of an excellently filtered differential module.

5.2.1. DEFINITION. Let Q be a differential field with a basic set ~ of derivation operators, F be a ~-subfiled of Q. A filtmtion of Q over F is an asceI'.ding sequence ... C Qr C Qr+1 C ... (r E IZ) of (nondifferential) subfields of Q such that

(1) Qr = F for r < 0; (2) if d E~, 77 E Qr, then d(77) E Qr+1; (3) UrEZ Qr = Q.

5.2.2. DEFINITION. Let (Qr )rEZ be a filtration of a ~-extension Q over :F. It will be called finite, if for any r E IZ the field Qr is finitely generated over:F. It will be called good, if there exists ro E IZ such that Q. = Qro(T(s - ro)Qro) for any s > ro. A filtration is called excellent, if it is finite and good.

5.2.3. EXAMPLE. Let Q = F(771,"" 77n} be a finitely generated differential extension of:F. Let Qr = F(T(r)771' ... , T( r)77n) denote the subfield of Q, generated by the elements {77j }j=l, .. ,n and all their derivatives up to rth order. Then (Qr )rEZ is an excellent F-filtration of Q.

Let a filtration (Qr )rEZ of a ~-extension Q of F be given. If Or denotes the Q-subspace of the module of differentials Og/:F, generated by 677, where 77 E Qr, then we obtain a filtration of the Q[~]-module Og/:F.

5.2.4. EXERCISE. Let (Qr)rEZ be an excellent filtration of a ~-extension Q of :F. Prove, that the filtration of the module of differentials Og/:F is also excellent.

5.2.5. THEOREM [J069b]. Let (Qr)rEZ be an excellent filtration ofa ~-extension Q over F. Then there exists a polynomial X(s) such that X(s) = trdeg:F Q. for all sufficiently large integers s and the polynomial X(s) coincides with the characteristic polynomial of the module of differentials Og/:F (see Definition 5.1.12).

PROOF. By Theorem 5.1.11, there exists a polynomial X(s) such that X(s) = dimg 0. for all sufficiently large s. Therefore, it suffices to prove that dimg Os = trdeg:F Q •.

If (77a)aEA is a transcendence basis of Q. over F then the elements 677a (Q: E A) generate the Q-space 0.. We shall prove that the system 677a (Q: E A) is linearly independent over Q. For any Q: E A we find a derivation operator D a of Q. over F such that Da(77{3) = 0 for (3 E A, (3 =1= Q: and Da(77a) = 1. By Proposition 1.5.6, Da can be extended in a unique way to a derivation operator Da on Q. Now, if LaEA Aa677a = 0, where Aa E Q, then LaEA A",677a(D{3) = 0 for all (3 E A, hence, Aa = 0 for all Q: E A and the system {6 77",} is Q-linearly independent (we consider here Og/:F as a subspace of (Der:F Q)*, where (Der:F Q) is the linear Q-space of F-linear derivation operators). 0

5.2.6. DEFINITION. The polynomial X, whose existence is proved in Theorem 5.2.5, we shall call Kolchin's differential dimension polynomial of the ~-field Q over :F.

5.2. DIFFERENTIAL DIMENSION 229

5.2.7. EXAMPLE. Let :F be ad-field, cp be a d-independent over :F element and 9 = :F(cp). If (g')'EZ is the filtration described in Example 5.2.3, i.e., g. = :F(()CP)8ET(s) , then Og/F is a free g[d]-module with the generator dcp and the differential dimension polynomial for this filtration is equal to X(t) = e~m), where m = Cardd.

If we fix a finitely generated d-extension 9 of ad-field :F and consider different excellent filtrations of 9 over :F, then we shall obtain different differential dimension polynomials. Nevertheless, they have some invariants.

5.2.8. LEMMA. Let R be a differential field, d the basic set of its derivation operators, D = R[d] the ring of d-linear operators over R. If(Mr )rEZ and (M:)rEZ are two excellent filtrations of a a d-R-module DM, then there exists kEN such that Mr ~ M:+k for all sufficiently large r.

PROOF. Since both filtrations are excellent there exists p such that Ds Mp = Mp+. and D.M; = M;+. for all sEN. Since they are exhaustive, there exists kEN such that Mp ~ M;+k. Then Mp+. = D.Mp ~ D.M;+k = D.+kM; = M;+.+k for all sEN. It means that if r > p then Mr ~ M:+k , and the lemma is proved. 0

5.2.9. THEOREM ON d-INVARIANTS OF THE CHARACTERISTIC POLYNOMIAL OF A DIFFERENTIAL MODULE. Let (Mr )rEZ be an excellent filtration of a d-R-module M and X(t) be the characteristic polynomial of M, associated with this filtration. Then the integers (dl)mX(t), r = degx(t) and (dlVX(t) (m = Cardd) do no depend on the choice of excellent filtration of M (see (2.3.7)).

PROOF. Let (Mr)rEZ and (M:)rEZ be two excellent filtrations of M, and let X, Xl(t) be the characteristic polynomials of M, associated with the filtrations (Mr )rEZ and (M:)rEZ, respectively. By Lemma 5.2.8, there exists pEN such that Mr C M:+p for all sufficiently large r, hence, X(t) S Xl(t + p), and conversely, xdt) S X(t + q) for some q EN. Therefore the degrees and the leading coefficients of the polynomials X(t) and Xl (t) coincide. Theorem 5.1.11 implies that deg X::; m, hence, (dI)mX(t) = const = (dl)mX1(t). 0

This proposition jstifies the following definition.

5.2.10. DEFINITION. Let M be a d-R-module, where R is a differential field and d = {d1 , ... , dm } is its basic set of derivation operators. If X(t) is a characteristic polynomial, associated with an excellent filtration of M, then (dt)m X(t), r = degx(t) and (dltX(t) are called differential dimension, differential type and typical differential dimension of the d-R-module M, respectively. The differential dimension of M will be denoted by d(M).

5.2.11. PROPOSITION. Let R be a differential field with a basic set of derivation operators d = {db ... , dm}. If

is an exact sequence of finitely generated d-R-modules, then d(N) + d(P) = d(M).

PROOF. Remark, that the filtration of M, associated with the choice of the finite set of generators, is excellent (see Example 5.1.7), so we may speak about d(M).


Let (Mr )rEZ be an excellent filtration of M. If we set Nr = z-l (z( N) n Mr) and Pr = J(Mr) for any r E Z then, obviously, the filtration (Pr )rEZ of P is excellent, and by Proposition 5.1.15 the filtration (Nr)rEZ of N is also excellent. Let XN(t), XM(t) and Xp(t) be the characteristic polynomials of N, M and P, respectively, associated with these filtrations. Since for all sufficiently large r E Z the sequence of linear R-spaces

0--+ Nr --+ Mr --+ J(Mr) --+ 0

is exact, therefore dimR Nr +dimRJ(Mr) = dimR Mr, and XN(t) + Xp(t) = XM(t). Hence,

d(M) = (~dmdM(t) = (~dm(XN(t) + Xp(t))

= (~l)mXN(t) + (~dmxp(t) = d(N) + d(P).

The proposition is proved. 0

5.2.12. PROPOSITION. Let R be a differential field with a basic set ~ = {d1 , ... ,dm } of derivation operators, D be the ring of ~-operators over R. If M is a finitely generated ~-R-module, then its differential dimension d(M) is equal to the maximal number of elements of M which are linearly independent over D (i.e. d(M) = rk DM).

PROOF. First of all, we shall show that d(M) = 0 if and only if every element of M is linearly dependent over D. Suppose that d(M) = 0, and there exists x E M which is linearly independent over D. In this case the mapping <p : D --+ M, which maps any u E D into ux E M, is a monomorphism of D-modules. Applying Proposition 5.2.11 to the exact sequence of finitely generated ~-R-modules 0 --+ D -4 M --+ M/<p(D) --+ 0, we have d(D) = d(M) - d(M/<p(D)) ::; d(M) = 0, and Example 5.2.7 shows that d(D) = (~dmC"!;.m) = 1. This contradiction proves that any element of M is linearly independent over D.

Conversely, suppose that any element of M is linearly dependent over D. Let 6,.·.,~k be generators of the D-module M. Let Ni = D~i (1::; i::; k), and Li denote the kernel of the mapping tPi : D --+ Ni (1 ::; i ::; k) such that tPi (u) = U~i for any u E D. By assumption, ~i is linearly dependent over D, therefore, Li = Ker tPi =I- 0 (1::; i ::; k). We have d(L;} =I- 0, since in the contrary case the ring D has zero divisors. Applying Lemma 5.2.11 to the exact sequence of finitely generated

D-modules 0 --+ Li --+ D ~ Ni --+ 0, we see that d(Ni) = d(D) - d(Li) ::; 0, i.e. d(Nj ) = 0, for all i = 1, ... , k. Since there exists an epimorphism of D-modules

ffiNi --+ M --+ 0, by Lemma 5.2.11 we have d(M) ::; d(ffi~=lN;) = E~=l d(Ni) = o. Thus, the condition d(M) = 0 is equivalent to the condition that every element of M is linearly dependent over the ring of ~-operators D.

Now we can complete the proof of the proposition. Let p be the greatest integer such that there exist p elements Xl, ... ,Xp E M linearly independent over D. Introduce an excellent filtration (Ef=l DrXi)rEZ on the free ~-R-module F = Ef=l Xi· Using Theorem 2.2.7, we can show that the characteristic polynomial associated with this filtration has the form X(t) = pc"!;.m), therefore, d(F) = (~dmx(t) = p. The maximality of the system Xl, ... , xp of linearly independent over D elements implies that every element of the finitely generated ~-R-module M / F is linearly

5.3. AUTOREDUCED AND CHARACTERISTIC SETS 231

dependent over D. It has been shown that this implies d(M/ F) = O. Applying Proposition 5.2.11 to the exact sequence of D-modules

O-tF-tM-tM/F-tO,

we obtain d(M) = d(F) + d(M/F) = d(F) = p. The proposition is proved. 0

5.3. Autoreduced Sets of Differential Polynomials. Characteristic Sets

Let :F be a differential field of zero characteristic with a basic set ~ = {db ... , dm } of derivation operators, T = T(~) be the free commutative semigroup with generators d1 , ... , dm. Consider a finite family (Yl, ... , Ym) of differential indeterminates over :F.

5.3.1. DEFINITION. By a ranking of {Yl, ... , Yn} we shall mean a total ordering of the set of all derivatives (}Yj (() E T, 1 ~ j ~ n) that satisfy the two conditions:

(1) u ~ (}u for any () E T and any derivative u; (2) if u ~ v then (}u ~ (}v, for all derivatives u, v, and any () E T.

In this case the ordering of the set of terms (}Yj (() E T, 1 ~ j ~ n) will be also called ranking (in accordance with Definition 4.1.10 of the ranking of a set of terms).

In the case u> v (u < v) we shall say that u has higher (lower) rank than v. Section 4.1 contains some examples of ranking satisfying this definition, it con

tains also definitions of orderly ranking (such that ord (}l < ord (}2 implies (}l Yi < (}2Yj) and standard ranking.

Let a ranking of {Yl, ... , Yn} be fixed.

5.3.2. DEFINITION. Let:F be a ~-field, A E :F{Yl, ... ,Yn}, A fI.:F, and let a ranking of {Yl, ... , Yn} be given. The derivative (}Yj of highest rank present in A is called the leader of A (we shall denote it by UA). If d = deguA A then

A = r:f=o liu~, where 10 , ... , Id are uniquely defined polynomials free of UA.

The ~-polynomial lA = fd is called the initial of A, and the ~-polynomial SA = r:f=l iliU~-l is called the separant of A. If A E :F\O then we set UA = fA = SA = 1, and if A = 0 then we set UA = fA = SA = O.

5.3.3. DEFINITION. Let A, BE :F{Yl"'" Yn}. We shall say that A has higher rank than B (and write rk A > rk B) if either A fI. :F, B E :F, or UA has higher rank than UB, or UA = UB = U and degu A > degu B.

Evidently, distinct ~-polynomials may have the same rank, and if A fI. :F then SA and lA have lower rank than A.

5.3.4. DEFINITION. Let A,F E :F{Yl, ... ,Yn}, A fI.:F. The ~-polynomial F is called partially reduced with respect to A if F is free of every proper derivative (}UA of the leader of A. If F is partially reduced with respect to A and deguA F < deguA A, then F is said to be reduced with respect to A. If A E :F then an element F E :F {Yl, ... , Yn} is called reduced and partially reduced with respect to A iff F = O. If A C :F{Yb ... , Yn} and F is (partially) reduced with respect to every element of A, then F is called (partially) reduced with respect to A.


5.3.5. DEFINITION. A nonempty subset A c F {YI, ... , Yn} is called autoreduced if any element of A is reduced with respect to any other element of A.

In particular, any set containing a single element of F {Y1, ... , Yn} is autoreduced. If an autoreduced set A contains an element F "# 0 from the coefficient field :F, then A contains no other elements, i.e. A = {F}. Lemma 2.2.1 implies that every autoreduced set is finite.

5.3.6. LEMMA. Let A be an autoreduced set in F{Y1, ... ,Yn}, and F E F {YI, ... , Yn}. Then there exist a ~-polynomial F E F {YI, ... , Yn} (par-tial remainder of F with respect to A) and integers tA EN, A E A such that F is partially reduced with respect to A, the rank of F is no higher than that of F and TIAEA S~A F == F (mod [A]).

More exactly, TIAEA S~A F - F can be expressed as a linear combination of derivatives OA (A E A) with coefficients in F{YI, ... , Yn} such that OUA has no higher rank than UFo

PROOF. If F is partially reduced with respect to A, then we set F = F, tA = 0 (A E A). In the contrary case F contains a derivative OUA of the leader of some (A E A). Let VF be such derivative of the maximal rank. We shall prove the lemma by induction on VF. If VF can be expressed as derivatives of several leaders UAj' then let Ue be the maximal among such UAj, VF = Oue. If e = Ei Iiu~ and d E ~ then de = Sedue + Ei(d1i) . u~. Therefore, T = oe - SeOue has lower

rank than Oue = VF. Denoting I = degllF F, we have F = E~=a JiV}" where J; is lower than VF, hence

1 1

SI F "SI-i J (S )i "SI-i J Ti e = L.J e i eVF = L.J e i

i=O i=O

(mod Oe)

If G = E~=o S:;i JiTi is not partially reduced with respect to A then UG < VF, therefore, the lemma holds for G, i.e., TIAEA S~AG = 0 (mod [A]). Now it suffices

to set F = 0, te = ke + I, tA = kA (A E A, A"# e). 0

5.3.7. THEOREM. Let A be an autoreduced subset of F{YI, ... , Yn}. If F E F {YI , ... , Yn} then there exist a ~-polynomial Fo (remainder of F) and r A, t A E N, A E A such that Fa is reduced with respect to A, the rank of Fa is no higher than the rank of F, and TIAEA I'l S~A F == Fo mod [A]. More exactly, TIAEA I~A S~A FFo may be expressed in the form of a linear combination of derivatives OA with coefficients in F{YI, ... , Yn} such that A E A and OA has rank no higher than UFo

PROOF. Let A = {A1, ... ,Ar}. Let us denote Uk = UAk' 1k = IAk , and Sk = SAk (1 :::; k :::; r). Suppose that Uk, < Uk. if k1 < k2 • We can write Ak = hU~k + h,1U~k-1 + ... + h,dk • Denote lr = degur F, where F is the partial remainder of F with respect to A, and let ir = lr - dr + 1, if Ir > dr and ir = 0, if lr < dr. Then l:r F == FI mod (Ar), where F1 is partially reduc-;d with respect to A, is reduced with respect to Ar, and has rank no higher than F. In the case when r = 1 this implies the statement of the theorem. If r > 1, then we can find a ~polynomial F2 == 1;:"-1' FI mod (Ar-d, which is reduced with respect to Ar- l , Ar,


partially reduced with respect to A and is no higher than F. By induction on r we obtain the proof of the theorem. 0

Let us summarize the results in the form of an algorithm. Let A be an autoreduced subset of :F {Yl, ... , Yn}, s be its cardinality, and F E :F {Yb ... , Yn} be a ~-polynomial. Let Hterm denote the function of taking the leader, Sep denote the function of taking the separant, Ini denote the function of taking the initial coefficient, and if we treat F as a polynomial in v, F = E~=o Ijvi , where v = ()Yi,

and Ik i- 0, then by definition Ini,,(F) = h.

S.3.S. Algorithm RED(A, s, F) Input: F, s, A= {A1, ... ,A.} Output: F (the remainder of F)

Begin if A = {A} where A E :F then

F:=O else

End

v:= 1 j:= ° while v = 1 and j < s

j := j + 1 UA := Hterm(Aj) if 3() E T such that F contains ()UA then

find () of highest rank such that F contains ()UA

v := ()UA

C :=Aj d:= deguA C S:= Sep(C) 1:= Ini(C)

if vi-I then 1:= deg" F II :=Ini,,(F) if () i- 1 then F := S . F - II . vl - 1 . ()C RED(A,s, F) else if I> d then

else

F := I . F - II . u~-d . C RED(A,s, F)

v:= 1

According to this algorithm, the remainder F of a ~-polynomial F reduced with respect to an autoreduced set A is an uniquely defined ~-polynomial (sometimes we shall say that F is a result of reduction of F with respect to A and write F ~ F),

{A}


although not every polynomial Fo, whose existence is stated in Definition 4.3.7, coincides with F.

We shall extend now the notion of comparative rank onto the set M of all autoreduced subsets of .1' {Y1, ... , Yn}. If A = {A 1, ... , Ar} EM is an autoreduced subset, then any two leaders UAi' UAj for 1 ~ i # j ~ n are distinct, and we shall suppose in the following definition, that elements of any autoreduced set are arranged in order of increasing rank of their leaders UA, < UA, < ... < UA r •

5.3.9. DEFINITION. Let A, BE M, A = {A1"'" Ar }, B = {B1, ... , B.}. We shall say that A has lower rank than B and write rk A < rk B, if either there exists a kEN such that rkA; = rkB; (1 ~ i < k) and rkAk < rkBk, or r > sand rkA; = rkB; (1 ~ i ~ s). If r = sand rkA; = rkBi (1 ~ i ~ s), then A is said to have the same rank as B (rk A = rk B).

5.3.10. PROPOSITION. Any nonempty set of autoreduced subsets of the ring .1' {Y1, ... , Yn} contains an autoreduced subset of lowest rank.

PROOF. Let Ii,. be a set of autoreduced subsets of .1'{Y1, ... ,Yn}. Define a sequence /10.;, i E N of subsets of Ii,. in the following way: An = Ii,., and for i > 0 let /10.; be the set of elements A E /10.; -1 such that Card A ~ i and if the elements of A = {A 1 , ... , Ai, ... ,} are arranged in order of increasing rank then Ai has the lowest possible rank. It is obvious that if A = {A 1 , ... , Ai, ... } E /10.;, then the leader UAi is defined uniquely. If all /10.; # 0, i E N, then the set of the leaders UA. forms an infinite sequence of derivatives BYI such that no UA. is a derivative BUAj of another leader UAj, i # j, that contradicts Lemma 2.2.1.

Therefore there exists a smallest i such that /10.; = 0. Since An = Ii,. # 0, we have i > O. Any element of /10.;-1 is clearly an autoreduced subset in Ii,. of lowest rank. 0

5.3.11. DEFINITIO N. Let .1' be a Ll-field of characteristic zero and f ~ .1' {Y1, ... , Yn} be a Ll-ideal. An autoreduced subset of f of lowest rank is called a characteristic set of f.

5.3.12. LEMMA. Let A be a characteristic set of all-ideal f ;; .1'{Y1, ... , Yn}. Then fA f/. f, SA r/:. f for any A E A and f contains no nonzero elements reduced with respect to A.

PROOF. Let P E f, P f/. .1' and suppose that P is reduced with respect to A. Then P and elements A E A such that rk UA < rk Up form an autoreduced subset, which has lower rank than A. Since SA and fA for all A E A are reduced with respect to A, we have fA r/:. f, SA f/. f. 0

As it was noted in Section 3.2, the theorem on Ll-algebraicity of a tower of Llalgebraic fields holds for differential fields. We shall prove below a special case of this theorem, the remaining part is left to the reader as an exercise.

5.3.13. PROPOSITION. Let.1' be a differential field of characteristic 0, a and (J be elements of a Ll-extension of .F. If a is Ll-algebraic over .1' and (J is Ll-algebraic over .1'(a), then (J is Ll-algebraic over .F.

PROOF. Suppose an orderly ranking of Y to be fixed. Let P be the defining ideal of a. By Lemma 5.3.12, in .1'{y} there exists a Ll-polynomial A such that


A(a) = 0 and SA(a) =1= O. Since dA(y) = SAduA(y) + R(y), where rkR < rkduA for any dEll, substituting a instead of y we obtain that there exists (it E T such that 91a E :F((9a)9Y<91Y)· Obviously, also 9'91a E :F((9a)By<BIB 1Y) for any 9' E T. Setting r1 = ord 91, we have for all r ~ rl

Similarly, there exists 92 E T such that

hence, 92 f3 E :F((9a)BET(q), (9f3)By rl, then

The number of derivatives 9f3 in the left-hand side of this inclusion is equal to Card T(s) = ('~m) = sm 1m! + o(sm-l), and the general number of generators 9a and 9f3 in the right-hand side is equal to ('+:;..+m) _ ('+q-;:;l +m) + ('~m) _ ('-r;.+m), and, as a polynomial in s, has degree m - 1. So, for all sufficiently large s, the number of generators in the left-hand side is greater than in the right-hand side. Hence the family (9f3)9ET(.) is algebraic over:F. It means that f3 is Ll-algebraic over:F. 0

As it was noted in Section 3.2, for some differential fields an analogue of the theorem on a primitive element holds. We shall prove below a very important for us special case of this theorem, when :F is a Ll-extension of C(Xl, ... , xm) (see Theorem 5.3.15). The general result has been formulated in Theorem 3.2.35.

5.3.14. LEMMA. Let Ll-field :F be a Ll-extension of C(X1,"" xm) and A E :F{y} \:F be a nonzero Ll-polynomial in the indeterminate y. Then there exists an element c E:F such that A(c) =1= O.

PROOF. Let UA = di' .. . d~y and d = deguA A. We shall seek c in the form

c= C · . xit xim J,···Jm 1 ... m'

(il, ... ,jm)S(i, , ... ,i m )

where cit ... im E C and :s denotes the total degree then lexicographic ordering

of vectors (it, ... , im). Substituting c = L(j" ... ,im)S(i" ... ,im) Ci, ... imX{' ... x~ in A(y) = 0 instead of y and setting Xl = 0, ... , Xm = 0, we obtain an equation for the elements Cj, ... jm' The left-hand side of this equation has degree d with respect to Ci , .. .im , hence, it is not identically equal to O. Therefore the inequality A(y) =1= 0 has a solution in C. 0


5.3.15 (Ll-THEOREM ON A PRIMITIVE ELEMENT). Let Ll-tield :F be a Ll-extensian of q Xl, ... ,xm ) and let Ll-tield g be a Ll-extension of:F. Then every tinitely generated Ll-algebraic extension of:F in g is generated by a single element (see [Ko173, Chapter II, § 8, Proposition 9].

PROOF. It suffices to show that if a and f3 are Ll-algebraic over :F then there exists e E :F such that :F(a, f3) = :F(a + ef3). Let t, y, z denote differentially independent over :F(a, f3) variables and fix some orderly ranking of the family y. We shall show that a + tf3 is Ll-algebraic over :F(t). Indeed, it belongs to the Llfield :F(t, a, f3) = :F(t)(a)(f3). Taking into account that f3 and a are Ll-algebraic over :F(t) and twice applying Proposition 5.3.13, we see that a + tf3 is Ll-algebraic over :F(t). By Lemma 5.3.12 there exists a Ll-polynomial A E :F(t){y} such that A(a + tf3) = 0 and SA (a + tf3) ::j:. O. Let UA = (JoY denote the leader of A. Find a polynomial B E :F {y, z} such that B(a +tf3, t) = 0 and (8Bj8«(Joy))(a +tf3, t) ::j:. 0, and which is free of derivatives (Jy such that (Jy > (JoY. We have

(Jo(a + tf3) = (Jot· f3 + v, (5.3.1)

where v is free of (Jot and for every (Jy, (J ::j:. (Jo present in B the polynomial (J(a+tf3) is free of (Jot. Differentiating the both sides of (5.3.1) with respect to (Jot, we obtain

8B 8B 8«(Joy) (a + tf3, t) . f3 + 8«(Joz) (a + tf3, t) = O.

Since the set t is independent over :F, there exists e E :F such that

8B 8«(Joy) (a + ef3, e) ::j:. O.

(5.3.2)

Substituting e for t in (5.3.2), we see that f3 E :F(a + ef3), therefore, :F(a,f3) = :F(a + ef3). 0

In conclusion of this section we shall prove the Ritt-Raudenbush basis theorem. We shall say that a perfect differential ideal I is of finite type if there exists a finite system al, ... , an of elements of I such that I = {al,"" an}.

5.3.16. LEMMA. Let S be a subset of a Ritt algebra R. If there exists a E R such that {a, S} is an ideal of tinite type, then in S there exist elements bl , ... , br

such that {a, S} = {a, bl, ... , br }.

PROOF. Let {a, S} = {CI,"" cq }. By Theorem 3.2.12 the ideal {S, a} coincides with the radical of [S, a], therefore, some power of any Ci may be expressed as a linear combination (with coefficients in R) of a, elements of S and their derivatives. It is clear, that the elements of S, present in this representation may be taken as bl , ... , br . 0

5.3.17. THEOREM. Let :F be a differential tield of characteristic 0, R = :F {YI, ... , Yn} be the ring of differential polynomials over :F in indeterminates YI, ... , Yn' Then R satisties the ascending chain condition for perfect differential ideals.

PROOF. We shall show that every perfect differential ideal in R is of finite type. Suppose the contrary. Then by Zorn's lemma we may pick up a perfect differential ideal I which is maximal among those not of finite type. We shall show

5.4. DIMENSION POLYNOMIAL OF A FIELD EXTENSION 237

that I is prime .. Suppose that ab E I, but a fI. I, b rt. I. Then A.-ideals {I,a} and {I,a} properly contain I. Hence, they are of finite type. By Lemma 5.3.16 {I, a} = {a, Cl, ... , cr } and {I, b} = {b, el, ... , e.}, where all Cj and ej are in I. By Lemma 3.2.18, {I,a}{I,b} ~ {ab, ... ,cre.} ~ I. For any z in I the element z2 1ies in {I, a}{ I, b}, hence, in {ab, ... , cre.}. Therefore z lies in {ab, ... , cre.}, and I coincides with {ab, .. . , cre.}. Thus, I is of finite type. This contradiction proves that I is prime.

Choose in I a characteristic set A with respect to some ranking of the family Y1, . .. , Yn, and let A E A. By Lemma 5.3.12, the separant SA fI. I and the initial IA fI. I. Since I is prime, SA . IA rt. I therefore, {SAh, I} is a perfect A.ideal, properly containing I. Hence, {SAIA, I} is of finity type. By Lemma 5.3.16, {SAIA, I} = {SAIA , Gl , ... I Cq }, where all Cj are in I.

Let F be an element of I. By Theorem 5.3.7, there exist integers rand t and a A.-polynomial G, reduced with respect to A, such that SAI~F - G lies in the A.-ideal generated by A, and all the more so lies in I. Thus G is an element of I. By Lemma 5.3.12, I contains no elements reduced with respect to A, hence, G = O. It follows that SAIAF is in {A}. Since this is true for all F E I, we have SAIAI ~ {A}. Therefore,

Thus, we see that I ~ {A, Cl, ... , Cq }. Since {A, Cl, ... I Cq } ~ I, we see that I = {A, Cl , ... , Cq }. This is a contradiction, since I is not of finite type. Theorem 5.3.17 is proved. 0

5.4. Differential Dimension Polynomial of a Finitely Generated Differential Field Extension

In Section 5.2 we have proved the existence theorem (Theorem 5.2.5) for differential dimension polynomial of a finitely generated extension of a differential field. Here we give historically earlier proof by Kolchin.

5.4.l. THEOREM (KOLCHIN). Let 1/ = (1/1, ... , 1/n) be a finite set of elements of a A.-extension of a A.-field F. Then there exists a numerical polynomial w,,/7'"(t) with the following properties:

(1) the transcendence degree of F((01/j )9ET(.),1~j~n) over F is equal to w,,/7'"(s) for all sufficiently large sEN;

(2) degw"/7'"(t) ~ m (= Card A.), hence W,,/F(t) = 2:~oajetj), where aj E~; (3) if P is the defining prime A.-ideal of a point 1/ in F{Yl, ... , Yn}, and A is

a characteristic set of p with respect to an orderly ranking of (Y1 , ... , Yn), and if Ej denotes for any Yj the set of points (II, ... ,1m) E Nm such that d~l ... d~Yj is the leader of an element of A, then w'l/7'"(t) = 2:';=1 WE; (t).

PROOF. First of all, note that A(1/) = 0 for any A E A and by Lemma 5.3.12 IA(1/) =f:. O. Therefore, UA(1/) is algebraic over F((01/j)8ET,1~j~n,rk8y;<rkuA). Consecutive derivations show that if v = OUA is the derivative of the leader of an A E A, then v(1/) is algebraic over F((01/j)8ET,1~j~n,8y;<v), since S(1/) =f:. 0 by Lemma 5.3.12. Let V denote the set of all derivatives OYj (0 E T, 1 ~ j ~ n),


which are not derivatives BUA of the leaders of elements of A, and let V(s) denote the subset of elements BYj E V, such that ord B ~ s. It is clear that the field F((B1]j)eeT(&),1$j$n) is algebraic over F(v(1])ve v (&)). On the other hand the system v(1]) (v E V) is algebraic independent over :F, since in the contrary case there exists a ~-polynomial REP n F[V], and p contains an element reduced with respect to A, that contradicts Lemma 5.3.12. Therefore, the system v(1]) for v E V(s) forms a transcendence basis of F((B1]j)eeT(&),1$j$n) over F For any Yj, the number of derivatives d~l ... d~Yj in V(s) is equal to the number of points e = (i1' ... ,im ) E Nm with ord e ~ s such that e is less than any point in Ej with respect to the direct product of orders on N. Hence, W,,/F(S) = Ei=1WEj(t). 0

5.4.2. DEFINITION. The polynomial W,,/F is called the differential dimension polynomial of the system 1] over F. If P C F{Y1, ... , Yn} is a prime ~-ideal and 1] = (1]1, ... , 1]n) is its generic point, then W,,/F is called the differential dimension polynomial ofp and denoted by wp. J. Johnson has shown (see Theorem 5.2.5) that W,,/F coincides with the characteristic polynomial of the module of differentials f"I.(i/F, where 9 = F(1]1 , ... , 1]n). Below this polynomial will be denoted woQIF •

5.4.3. DEFINITION. The differential dimension (~-dimp) of a prime ~-ideal pC F{Y1, ... , Yn} is ~-trdegF(1])/F, where 1] = (1]1, ... , 1]n) is a generic zero of p.

As it was noted in Section 3.2, every subset E of F {Y1, ... , Yn} defines a system of algebraic differential equations: A(Y1, ... , Yn) = 0, A E E. Let U be a ~-extension of F A point 1] E un is called solution (or zero) ofthis system, if A(1]) = 0 for every A E E. Note, that ~-dimp is the "correct" definition of the number of arbitrary functions in m = Card ~ variables in a generic solution of B = 0 (B E p), as it is usually said in classical literature.

The notion of differential dimension of a prime ~-ideal or of a corresponding irreducible subset in un is analogous to the notion of dimension of an affine algebraic set. But in contrast to the algebraic dimension the ~-dimension of a subset of un does not coincide with its topological dimension. For example, ~-dimUn = n, and the topological dimension of un is equal to 00; if P and q are prime ~-ideals in :F{yl, ... ,Yn} and p ~ q, then ~-dimp ~ ~-dimq, but the strict inclusion does not imply the strict inequality. The differential dimension polynomial is a more delicate "measure" of the set of zeros of a ~-ideal. Theorem 5.4.4 states that its leading coefficient is equal to the differential dimension.

5.4.4. THEOREM. Let 1] E un, and W'I/F(t) = E7::o ai Cti). Then

PROOF. Fix an orderly ranking of (Y1, ... ,Yn). Let d = ~-trdegF(1]}/F. We may assume that 1]1, ... , 1]d form a ~-basis of F(1]}/F For any index j such that d < j ~ n, the element 1]j is ~-algebraic over F(1]1. ... , 1]d}. Therefore, there exists a polynomial Aj E K{y}, where K = F(1]1 , ... , 1]d}, such that Aj(1]j) = 0, SAj(1]j) f. 0 and if Aj = Eliu~j' then


where t5 E ~. Hence, there exists (Jj E T such that

Fixing a sufficiently large hEN, we have

Consecutive derivations show that if (JI Yj is a derivative of (JjYj and ord(J' = r' 2: ord (Jj = rj then

where (Jj M means that (JYj is not a derivative of (JjYj. Therefore, if sEN and s 2: max(rd+l, .. . ,rn), then

Hence, for all sufficiently large sEN we have

WI)/F(s)::;d·CardT(s+h)+ L Card(T(s)-T(s-rj)) d<j'!:.n

( s+m) = d m + o(sm-l),

and am ::; d. On the other hand, the system ((J1/;)eET(s),l'!:.i'!:.d is ~-algebraic independent over

F, therefore,

( s+m) WT//F(S) 2: d· CardT(s) = d m

for all sufficiently large s and am 2: d. 0

Differential dimension polynomial WT//F is not a ~-birational invariant of an extension of a ~-field, i.e., the condition F(1/l, ... , 1/d) = F(f.t, ... ,f,h) does not imply WI)/F = W{/F·

5.4.5. EXAMPLE. Let m = 2, and p = [dlYl] be a prime ~-ideal in F{yt}, generated by dlYl. Consider the standard ranking on F{yt}. It is easy to see that p contains no elements reduced with respect to d1Y1, so the characteristic set A of p consists of the single element, A = {d1yd, and by Theorem 5.4.1(3), wp(s) = W(Ol) = S + 1. If 1/1 is a generic point of p, then F(1/1) = F(6,6,6), where ((,1,6,6) is a generic point of the ~-ideal q ~ F {Yl, yz, Y3}, q = [d1Y1, Y2 -dlYl, Y3 - d2Yl]. It can be easily computed that wq(s) = wp(s + 1) = s + 2.


5.4.6. LEMMA. Let 1'/= (1'/l, ... ,1'/n) and~= (6, ... ,~r) be subsets of aLlextension of a Ll-field :F.

(1) If h E :N and 1'/; E :F((O~/c)9ET(h),lS/cSr) (1 ~ j ~ n), then w,.,/:F(t) ~ we/:F(t + h).

(2) If :F(1'/) = :F(~), then there exists hEN such that wF./:F(t - h) ~ w,.,/:F(t) ~ we/:F(t + h).

PROOF.

1) Since :F((01'/;)9ET(.),lS;Sn) ~ :F((O~/c)9ET(.+h),lS/cSr), for all sufficiently large S we have w'l/:F(s) ~ we/:F(s + h).

2) The condition implies the existence of h E :N such that

and

This and 1) imply 2). 0

5.4.7. COROLLARY. The polynomial w'l/:F is a birational invariant of extensions of Ll-fields (i.e., :F(1'/) = :F(~) implies w'l = we). A Ll-birational invariant is the coefficient am (differential dimension of g over :F), the degree (differential type) and the leading coefficient (typical Ll-dimension) of the differential dimension polynomial in its decomposition with respect to the standard basis Cti ), i EN. They shall be denoted by diffdim:F g, T, and ar , respectively.

Let p, q ~ :F {Y1, ... , Yn} and p C q. As it has been noted, the proper inclusion does not imply Ll-dimp > Ll-dim q. Nevertheless, it will be proved below that in such case the equality wp(t) = wq(t) cannot hold.

5.4.8. EXAMPLE. Let n = 1, m = 1, P = [d2Yd, q = [dyd. Then Ll-dimp = Lldimq = 0, but wp =2, Wq = 1.

Consider the ordering of numerical polynomials introduced in Section 2.4.

5.4.9. PROPOSITION. Let p, q be prime Ll-ideals of :F{Y1, ... , Yn}, P S; q. Then wp(t) > Wq(t).

PROOF. Consider the filtration R. = :F((OYi)9ET(.),lSiSn) on :F{Y1, ... , Yn}. It induces a filtration on p: p = Up., P. = P n R •. Note, that P. is a prime ideal of the polynomial ring R. and by Theorem 5.4.1, wp(s) = dimp. for all sufficiently large s. Similarly, wq(s) = dimq., P. C q. ~ R •. Since P. =I q. for all sufficiently large s, we have wp(s) > Wq(s). 0

All results given below in this section are due to W. Sit (see [Si78]). Let 1'/ = (1'/1, ... , 1'/n) be a finite system of Ll-generators of all-field g over a

Ll-field :F, let d be the differential dimension of g over:F. Let f3'1/:F(t) denote the polynomial w'l/:F(t) - de'!;,m) (m = Card Ll).


5.4.10. PROPOSITION. Suppose that 171, ... ,"1d form a fl.-transcendence basis of 9 over T. Then

PROOF. If d = 0, then w'I/:F(t) = f3'1/:F(t) and the proposition holds. Suppose that d 2: 1. As usual, let T (respectively T(s)) be the set of all deriva

tives 8 = d~' .. . d:'r,m, where (e1, ... ,em) E Nm (e1 + ... + em :S s, respectively). Denote the sets (8"1j)8ET(s),19~d, (8"1j)8ET(s),1~j~n and (8"1j)8ET(s),d+l~j~n by q, r~ and r~, respectively. For all sufficiently large sEN we have

W(1)d+l, ... ,'1n)/:F('11, ... ,1)d)(S) = trdeg:F('11, ... ,'1d) F("11, ... , "1d)(rj)

:S trdeg:F(rj) T(rf, r 3) = trdeg:F(rn F(n) = trdeg:F F(r2) - trdeg:F F(rD

= w'I/:F(s) - d(s: m) = f3'1/:F(s). 0

5.4.11. COROLLARY. Let T = deg f31)/:F, where "11, ... , "1d is a fl.-transcendence basis of 9 over:F. Then the differential type of 9 over F("11, ... , "1d) does not exceed T and if it is equal to T, then the typical differential dimension of 9 over F ("11, ... , "1d) is less than or equal to ar ·

5.4 .12. THEOREM. Let 9 = F ("11, ... , "1n) be a finitely generated fl.-extension of a fl.-field F, fl. = {d1, ... , dm }. Then there exists a permutation of "11, ... , "1n such that "11, ... , "1d form a fl.-transcendence basis of 9 over F and

PROOF. Let an orderly ranking of (Y1, ... , Yn) be fixed, and let p be a prime defining ideal ofthe point "1 = ("11, ... , "1n) in F{Y1' ... ' Yn}. Let A be a characteristic set of p with respect to this ranking. Let Ej for each j (1 :S j :S n) denote the set of points (e1, ... , em) E Nm such that d~' ... d:'r,m Yj is the leader of an element from A. Theorem 5.4.1 implies that wp(t) = 'L'j=1 WEj(t), where degwEj < m, if

Ej =1= 0 and WEj (t) = c;;..m), if E j = 0. Since wp(t) = w1)/:F(t) = de;;..m) +o(tm - 1 ),

this implies that Ej = 0 exactly for d indices. Without loss of generality we can assume that these indices are j = 1, ... , d (may be after some permutation). The leader of any element of A is a derivative of some Yk (d < k :S n). Therefore, "11, ... , "1d form a fl.-transcendence basis of 9 over T. By Proposition 5.4.10, it suffices now to prove that

for all sufficiently large s (the sets r: are defined in the same way as in the proof of Proposition 5.4.10).

Suppose the claim to be false. Then for any So E N there exist s > So and a set B of derivatives of "1d+1, ... , "1n of order less than or equal to s such that


the set B is algebraically independent over .1'(fn and algebraically dependent over .1'(171, ... ,TJd). Let B' denote the set of corresponding derivatives of Yd+1, ... , Yn' Let q be the smallest integer such that B is algebraically dependent over .1'(fi). Then q > s and there exists f E .1'(Q)[(V)vEBI], f #- 0 which vanishes on the set B. Eliminating the denominator of f, we obtain a nonzero differential polynomial Q E .1'[(BYj)8ET(q),1<j<d, B'l such that Q(TJ) = O. Let us take such Q of lowest possible rank. Since-the ranking is orderly and q > s, the leader uQ of Q has the form B1YI, where ord B1 = q and 1 :s 1 :s d. Lemma 5.3.12 implies that the separant SQ(TJ) #- O.

Let Q = L~l 9jMj, where the sum is taken over a finite set of monomials {M;}l~i~N in variables (V)VEBI and 9j #- 0 for all i and 9j E .1'[(BYj)8ET(q),1::;j~d]. Since the leader of any element of A is a derivative of an Yl (d < I :s n), we see that 9j are reduced with respect to A. By Theorem 5.3.7, for any Mj there exists a differential polynomial MI, reduced with respect to A whose rank is not higher than the rank of Mj (hence, lower than uQ) and there exists H tI. p (H is a product of initials and separants of elements A E A) such that for any Mj we have H Mj == Mf mod [A]. Therefore, the differential polynomial Q' = L~l 9jMf is reduced with respect to A. Furthermore, Q E P and HQ == Q' mod [A], hence, Q' E p. Since p contains no nonzero polynomials reduced with respect to A (see Lemma 5.3.12), we have Q' = o.

We have 8M'/8uQ = 0 and 8M/8uQ = 0 (since the leader uQ = B1YI, where 1 :s d). Furthermore,

N N '" 8gj _ '" 8gj I [ 1 HSQ=H.~-.Mj=~-.Mj mod A ;=1 8uQ j=1 8uQ

_ 8Q' = 8uQ mod [A] == 0 mod [A].

Thus, H SQ E P and SQ E P. This contradicts the condition SQ (17) #- 0 and proves the theorem. 0

5.4.13. THEOREM. Let 171, ... , TJn be elements of a A-extension of.1', V a set of some (maybe not all) derivatives of 171, ... , TJn, and V' the corresponding set of derivatives of Y1,"" Yn' For any sEN, let V(s) (respectively, V'(s)) be the elements of V (respectively, of V') of order :s s. Suppose that for some q EN, we have

(1) the field .1'((BTJj)8ET(qp::;j::;n) is algebraic over .1'(VVEV(q)); (2) there is a d E A such that the set {VVEV(q)UdV (q)} is algebraically indepen

dent over .1'.

Then the field .1'(171"'" TJn) is algebraic over .1'( VVEV(O))' In particular, if Card V(O) = k, then Card V(q) :s k(q;;..m) and w'11'''','1n/:F(t) :s ke;;..m).

PROOF. First we prove that .1'(171, ... , TJn) is algebraic over .1'(VVEV(O)). Let h be the smallest number s such that .1'((BTJj)8ET(.),1~j~n) is algebraic over .1'(VVEV(s))' Then h :s q. If h = 0, then the theorem holds.


Suppose that h 2: 1. By the choice of h, for some w E Uj=l T(h-1)l1i \ V(h-1)

the element w is not algebraic over F(vvEV(h-1»). There exists I = E~=og;Xi E F[V' (h), X) such that

(1) g; E F[V'(h)), gl =f:. 0 and gcd(go, ... ,gl) = 1 (i.e. the polynomials go, ... ,gl are relatively prime);

(2) (gl(VvEV(h»))-l I(VvEV(h), X) is the minimal polynomial for w over F(VvEV(h»);

(3) ai/au =f:. 0 for some u E V'(h) \ V'(h -1). (In order to find such j, it is sufficient to take the minimal polynomial for an element w algebraic over F(vvEV(h»).) We shall denote ag/au for any 9 E F{Y1, ... , Yn}[X) by g'. We shall prove that f'(VVEV(h), w) =f:. O.

Suppose the contrary were true. Since the system {VvEV(h)} is supposed to be algebraically independent over F (see hypothesis (2) of the theorem), condition (3) implies that I' (VvEV(h), X) is a nonzero polynomial in F(VvEV(h»)[X), whose degree does not exceed l, which by assumption vanishes at w. By virtue of minimality of I, for some q E F(VvEV(h»). q =f:. 0 we have

ql(VvEV(h),X) = f'(VvEV(h),X).

Comparing the coefficients by Xi, we have for i = 0, ... , I

q9i(VvEV(h») = gHvvEV(h»).

In particular, gi =f:. 0 implies g: =f:. O. Furthermore,

gi(V"EV(h»)gHvvEV(h») = gHvvEV(h»)gl(V"EV(h») hence, gigr = g:9I. Since deg 91 > deg gr, there would be an irreducible factor of gl that divides gi for any i. This contradicts condition (1) on j. Thus, it is proved that f' (v"EV(h), w) =f:. O.

Appying the derivation operator d to the relation I(VvEV(h), w) = 0, we obtain

'" al _ d al L.J av (VvEV(h), w) . dv + I (vvEV(h), w) + aX (VvEV(h), w) . dw = 0, tiEV(h)

where Id is the polynomial obtained by applying d to each coefficient of the polynomial I considered as an element of F[X U V'(h)). From the last relation and condition (3) we have

dU(7J) E F( VvEV(h), (dv(I1))VEV'(h)" w, dw). v;tu

Since w, dw E F(((Jl1i )SET(h), 15i:Sn ), by virtue of the choice of h both these elements are algebraic over F(VvEV(h»). This implies that du is algebraic over F(VvEV(h), (dv(I1))VEV'(h))· This contradicts condition (2) of the theorem. Thus,

v;tu h = 0, and the first part of the theorem is proved.

Permuting, if necessary, the elements 7J1, ... , I1n, we assume that V(O) consists of 111, .. ·,l1k· For any sEW, (J' E T(s),(J' =f:. 1 and 1 ~ j ~ n, we have (J'l1i E F(((J1U)SET(s),l:Si:Sk), and hence,

trdegF F( ((J7Ji )SET(s),l:Si:Sn) = trdegF F(((JI1; )SET(.),l:Si:Sk) ~ k (s : m) . This implies the required inequality. 0


5.4.14. COROLLARY. If'T}l,"" 'T}n are elements of a fl.-extension of a fl.-field F then the condition w l1 /:F(t) = de~m) holds if and only if

(5.4.1)

PROOF. Let condition (5.4.1) hold. Without loss of generality we may assume that 'T}1, ... , 'T}d form a fl.-transcendence basis of F('T}l, .. . , 'T}n) over F. Then F (7]1, ... , 'T}n) is algebraic over F ('T}l , ... , 'T}d), and in the proof of Theorem 5.4.13 we showed that wl1 /:F(t) = de~m).

Conversely, suppose wl1 /:F(t) = dc~m). Then by Theorem 5.4.4 the differential dimension of F ('T}1, ... , 'T}n) over F is equal to d. Let 'T}1, ... , 7]d be a fl.transcendence basis. Applying Theorem 5.4.13 to the set V, consisting of the derivatives «(JYj )9ET,1:5:j:5:d, we see that F('T}l, ... , 'T}n) is algebraic over F('T}l, ... , 'T}d). Thus, trdeg:F F('T}l,"" 'T}n) = d. 0

5.4.15. COROLLARY. If P is a prime fl.-ideal of the ring F {Y1, ... , Yn} and wp(t) = dc~m), then P = {Po}, where Po = P n F[Y1, ... , Ynl is a prime ideal of dimension d.

PROOF. It is clear, that Po is a prime ideal. Let 'T} = ('T}1, ... , 'T}n) be a generic zero of p. Then 'T} is a generic zero of Po, and by Corollary 5.4.14 dimpo = d. We have proved above that {Po} is a prime differential ideal such that W{Po}/:F(t) = dc~m). Since {Po} c p and wp(t) = de~m) we have {Po} = p. 0

5.4.16. REMARK. The converse statement holds also: if Po is a prime ideal of F[Y1, ... ,Ynl then wp(t) = de~m), where p = {Po} ~ F{Y1, ... ,Yn} (see [Kol73, Chapter IV, §17. Proposition 10]).

5.5. Coherent Autoreduced Sets. Ritt-Kolchin's Algorithm

Consider now necessary and sufficient conditions for an autoreduced set to be a characteristic set of a prime ideal in the ring of differential polynomials R = F {Y1 , ... , Yn}. Let a ranking of the family (Y1, ... , Yn) be fixed.

If A is an autoreduced set then we set HA = TIAEA SA/A' If / is an ideal of R, fER, then / : foo denotes the set of elements 9 E R such that 9 . r E / for some n E N.

5.5.1. DEFINITION. An autoreduced set A C F{Y1, ... ,Yn} is called coherent iff for any fl.-polynomials A, A' E A and any common derivative of their leaders v = (JUA = (J'UA' the condition SA,(JA - SA(J' A' E (Av) : H'A holds, where Av is the set of fl.-polynomials (J" A" (A" E A, (J" E T) for which (JIIUA" s: v.

5.5.2. REMARK. It is sufficient to check the given in Definition 5.5.1 condition only for v equal to the least common derivative of the leaders UA, UA' (in this case the fl.-polynomial SA (A, A') = SA,(JA - SA (J' A' is called S-polynomial for (A, A')). Indeed, let w = dv = d«(JUA) = d«(J'UA') (d E fl.). Differentiating the relation (SA,(JA - SA(J'A')H1 E (Av) for some hEN and multiplying it by HA, we have d(SA,(JA - SA(J' A') E (Adv) : H'A = (Aw) : H'A hence

SA,d«(JA)-SAd«(J'A') = d(SA,(JA-SA(J'A')-d(SA,)(JA+d(SA)(J'A' E (Aw): H'A.

Now we have to use induction on ord v.

5.5. COHERENT AUTOREDUCED SETS. RITT-KOLCHIN'S ALGORITHM 245

5.5.3. LEMMA. Let A be a coherent autoreduced subset of F{Y1' ... ' Yn} and G E [A] : He: be an element partially reduced with respect to A. Then G E (A) : He:.

PROOF. Let A = {A!, ... , A,}. The condition G E [A] : He: implies the existence of a representation

, r(j) ,

H~G= LLCijB;jAj + LDjAj , (5.5.1) j=1i=1 j=1

where hEN, Cij, Dj E F {Y1, ... , y,}, Gij =I- 0, ord Oij > O. If r(j) = 0 in (5.5.1) for all j = 1, ... , s, then H~ G E (A) and the statement is proved.

Let r(j) > 0 for some j and let v be a maximal element with respect to the fixed ranking of the set U9ijAj (j = 1, ... , s, i = 1, ... , r(j)). Choose among all possible representations of the form (5.5.1) a representation with minimal v. Let l(j) = {I $ i $ r(j) I v = U9ijAj} for j = 1, ... , s. Since 1(/) =I- 0 for some 1 $ 1 $ s, we can take k E 1(/). After multiplying both sides of (5.5.1) by SAl> we can rewrite (5.5.1) in the form

, , + L L Gij· (SAIOijAj - SAjOkIAz) + L L CijSAjOkIAI.

j=1 iE/(j) j=1 iE/(j) (5.5.2)

Since the set A is coherent and v = U8ijAj = U8klAI for j = 1, ... , s, i E l(j), we have SAIOijAj - SAjOklAI E (Av) : He:. Multiplying (5.5.2) by H, where HA = SAl· H, we obtain

, r(j) ,

H!AG = LLOiAjAj + LDjAj + EOkIAI, (5.5.3) j=1i=1 j=1

where E, Ojj, Dj E F{Y1, ... , Yn}, Oij =I- 0, ord8ij > 0, ordOkl > 0 and U9ijAj < v.

Let OklAI = SAl V + T, where rkT < v and SAl =I- o. We can replace v in (5.5.3) by - T / SAl. Since H!A and G do not contain v, after multiplying both the parts of obtained relation by some powers of SAl and H we obtain a representation of the form (5.5.1), in which r(j) = 0 for alII $ j $ s by the choice of v. 0

5.5.4. LEMMA. Let A be an autoreduced subset of F{Y1, ... , Yn}. Then [A] : H'f is a d-ideal. If the set A is coherent and the proper ideal (A) : H'f is prime then [A] : H'f is also prime.

PROOF. Let fH~ E [A]. Then dfH~ + hfH~-1dHA E [A] for any d E d. Multiplying by HA, we see that dfH~+1 E [AJ, hence a = [AJ : H'f is d-ideal and a : H'f = a. Let us denote ao = (A) : H'f .


Let FG E a and F, G E F {YI, ... , Yn}. Denoting the remainder of F (G, respectively) with respect to A by Fo (Go, respectively), we see that FoGo E a, and since FoGo is partially reduced with respect to A, Lemma 5.5.3 implies FoGo E ao. By assumption either Fo or Go is in ao, hence H:"F E a, or H:"G E a for some tEN, and therefore, a is a prime ideal (1 does not belong to a, because in the opposite case the partially reducibility of H:" with respect to A and Lemma 5.5.3 imply H:" E (A) and 1 E ao). 0

5.5.5. LEMMA. Let A be an autoreduced set ofF{YI, ... , Yn} and a = (A) : H'f be an ideal of the ring F{YI, ... , Yn}. Suppose that U is a set of derivatives (JiYj such that A C F[U], and b = (A) : H'f is an ideal of F[U]. Then:

(1) a is generated in F{YI, ... , Yn} by the set b, a = (b); (2) b = an F[U); (3) a is a prime ideal iff b is a prime ideal; (4) a contains no nonzero elements reduced with respect to A iffb contains no

such elements.

PROOF. It suffices to consider the case when F[U] = F[xl, ... , x/c], and a is an ideal of polynomial ring F[xl, ... , X!c+l).

(1) Obviously, (b) ~ a. Let! E a, then

!(XI, ... , x/c+l) . H:"(XI' ... , x/c) = L hj(XI, ... , x!c+dAj (Xl, ... , x/c). j

Expand! and hj with respect to X/C+I:

then obviously all !i E b, hence, a ~ (b). (2) The inclusion b ~ an F[XI' ... ,x/c) is evident. If !(XI, ... ,x/c) E a, then in

the previous notation

hence !(XI, ... , x/c+l)H:" = L hojAj and therefore! E b. (3) By (2) it suffices to prove that if b is a prime ideal of F[Xb ... ,x/c], then a =

(b) is a prime ideal of F[Xb ... , x/c+d. This is true, since the ring F[Xb ... , x/C+l]/a is isomorphic to (F[XI' ... , x/c]/b)[X/C+I].

(4) By (2) it suffices to prove that if b contains no nonzero elements reduced with respect to A, then a also contains no such elements. Suppose the contrary: G E a, and G = L GiX~+1 is reduced with respect to A. By (1) the elements Gi E bare reduced with respect to A, therefore, G = o.


5.5.6. THEOREM. If A is a characteristic set of a prime differential ideal p of a Ll.-polynomial ring F{Yl, ... , Yn}, then p = [A] : H'f, A is coherent, and the ideal (A) : H'f ofF {Yl, ... , Yn} is prime and contains no nonzero elements, reduced with respect to A. Conversely, if A is a coherent autoreduced set such that the ideal (A) : H'f of F {Yl, ... , Yn} is prime and contains no nonzero elements, reduced with respect to A, then A is a characteristic set of a prime Ll.-ideal of the ring F{Yl, ... , Yn}.

PROOF. Let A be a characteristic set of p C F {Yl, ... , Yn}. Then by Lemma 5.3.12, p contains no nonzero elements reduced with respect to A, hence HA rt. p, therefore [A] : H'f ~ p. The remainder of any element of p with respect to A (see Theorem 5.3.7) is reduced and belongs to p, hence it is equal to 0, therefore p ~ [A] : H'f. If v = (}UA = (}'UA', where A, A' E A, then the rank of the Ll.polynomial SA()' A' - SA,(}A E P is lower then the rank of v (note that in this case ord() > 0, ord(}' > 0), hence SA(}'A' - SA,(}A E (All) : H'f and A is a coherent autoreduced set. Denote by V the set of derivatives (}iYj, which are not proper derivatives of the leaders UA (A E A); then A ~ F[V] and by Lemma 5.5.3 the ideal (A) : H'A of F[V] coincides with pnF[V]. Lemma 5.5.5 implies that (A) : H'A is a prime ideal of F {Yl, ... , Yn} and contains no nonzero elements reduced with respect to A.

Conversely, let A be a coherent autoreduced set such that (A) : H'A is a prime ideal containing no nonzero elements reduced with respect to A. By Lemma 5.5.4, p = [A] : H'f is a Ll.-ideal of F {Yl, ... , Yn}. Every element of p reduced with respect to A by Lemma 5.5.3 belongs to (A) : H'f and by assumption equals O. Let B = {Bl , ... , B.} be a characteristic set of lJ, and suppose that UBi < UB. < ... < UB.

and A = {Al. ... , Ar }, where UAi < UA. < ... < UA r • Since Bl is reducible with respect to A element, the Ll.-polynomials Bl and Al are of the same rank. Since B2 is reducible with respect to some Aj , and since rk B is minimal, the polynomials B2 and A2 are also of the same rank and so on. Hence A is a characteristic set of the Ll.-ideal p. 0

5.5.7. REMARK. Let F be a Ll.-field, where all derivation operators dEll. act trivially, d(f) = 0, f E F Then linear Ll.-ideals of the ring F{Yl, ... , Yn} correspond to submodules of a free module with n generators over the polynomial ring F[Xl, ... , xm] (m = Card Ll.), and characteristic sets are minimal Grobner bases (see Definition 4.1.43). So, we can consider characteristic sets as a generalization of Grobner bases. However, reduction algorithm for the characteristic set (even of a prime ideal) in the general case does not satisfy the characteristic property of a Grobner basis (see Theorem 4.1.33(8)).

5.5.8. EXAMPLE. Let A = (y")2 + 2y, HA = y", p = [A] : H'A. If the set V consists of the two elements {y, y"}, then ((A) : H'A) n F[V] is isomorphic to the ideallJl = (xi + 2X2) : xl'" of F[Xl, X2]. It is easy to see that lJl = (xi + 2X2) is a prime ideal containing no nonzero elements reduced with respect to xi + 2X2. By Lemmas 5.5.4 and 5.5.5 the Ll.-ideal p is prime.

Consider G1 = y lII + 1, G2 = Y" - y'. Denote by 0 1 and 02 the remainders of Gl and G2 with respect to {A} (see the definition ofremainder in Theorem 5.3.7). Then 0 1 = y" - Y' = O2 and G1 1. G2 (mod lJ), since in the contrary case Gl - G2


is in p, therefore, may be reduced to 0, but

ylII + 1 - y" + y' ~ -y' + y" - (y")2 + y'y" ~ y'y" + y" + 2y - y' f::. o. {A} {A}

5.5.9. PROPOSITION. Let I' be a prime ~-ideal of F{Yl, ... , Yn} and A be its characteristic set. A ~-polynomial F belongs to I' iff its remainder with respect to A is equal to O.

PROOF. If FE 1', then H~F == F (mod 1'), therefore, F E I' and F is reduced with respect to A, hence, F = 0 (see Lemma 5.3.12). Conversely, if F = 0, then H~F E I' and FE 1'. 0

5.5.10. PROPOSITION. Let a be a ~-ideal and A C a be an autoreduced set. Then A is a characteristic set of a iff any ~-polynomial / E a may be reduced to o with respect to A, / --t O.

A

PROOF. If A is a characteristic set then j = 0 for any / E a, since j E a and j is reduced with respect to A (see Lemma 5.3.12).

Let now A C a, A = {AI, ... , Ar }, rkA l < ... < rk Ar , be an autoreduced set and / --t 0 for all / E a. Suppose that B = {Bl, ... ,B,}, rkBl < ... < rkB,

A is a characteristic set of a. We shall prove that r ~ sand rk Ai = rk Bi for all 1 < i < s. We shall use induction on i. Since Bl f::. 0 and Bl --t 0, then Bl is

- - A reducible with respect to an Aj, j ~ 1, therefore, rk Bl ~ rk Aj ~ rk AI. Since B is a characteristic set of a, we have the inverse inequality: rk Bl ~ rk AI, therefore, rkA I = rkB l . Let now rkAi = rk Bi for i = 1, ... , k -1, and B/c be reducible with respect to some Aj . If j < k, then the condition rk Bj = rk Aj implies that B/c is reducible with respect to Bj, a contradiction to the definition of characteristic set. If j ~ k, then rkB/c ~ rkAj ~ rkA/c. Thus, rkB/c ~ rkA/c. IfrkB/c < rkA/c, then the autoreduced set {AI' ... ' A/c} is of lower rank than B. 0

Now we are going to describe the Ritt-Kolchin algorithm. It allows us for a given finite set <J> C F{Yl, ... , Yn} to find a set A = {AI, ... ,A/c} of autoreduced subsets such that Ai is a characteristic set of a prime ideallJi for all i = 1, ... , k and {<J>} = 1'1 n ... n 1J/c. More exactly, it reduces the problem to the following problem of commutative algebra: Let an ideal I in a polynomial ring be given. Find out whether I is prime, and if not, then to find polynomials /,g, ft I such that /g E I. This problem is not discussed in this book. It should be noted that Ritt-Kolchin algorithm does not allow us to find all prime components of a ~-ideal {<J>}, since the following problem is not solved.

5.5.11. PROBLEM. Let characteristic sets A and B of prime ~-ideals I' and q be given. Find out whether the inclusion I' C q holds.

Ritt-Kolchin algorithm will be divided into two subalgorithms. First, we describe an algorithm which for any finitely generated ~-ideal a = [It, ... , /q] finds a coherent autoreduced set A such that [A] ~ a ~ [A] : H'f. If <J> is a set {gl, ... ,g/c}, then A(<J» denotes its autoreduced subset oflowest possible rank (it is defined uniquely).

5.5. COHERENT AUTO REDUCED SETS. RITT-KOLCHIN'S ALGORITHM 249

5.5.12. Algorithm RKI (<I), A) Input: <I) = {fl, ... , ir} is a finite set of .1.-polynomials Output: A is a coherent autoreduced set of .1.-polynomials

[A] ~ [<I)] ~ [A] : H'f

Begin A := A(<I)) if A = {f}, i E :F then

else

End

return

G:= <I) \A W:=12I for any 9 E G

r := remainder of 9 with respect to A if r 1= 0 then

W:= WU{r} for any pair ii, f; E A

r := remainder of SA (Ii, f;) with respect to A if r 1= 0 then

W:= WU{r} if W 1= 121 then

<I) := <I) U W Algorithm RK1 (<I),A)

We describe the Ritt-Kolchin algorithm in the recursive form. First of all we prove that Algorithm 5.5.12 terminates. Indeed, if W 1= 121 on

some step of the algorithm, then we replace the set <I) by ~ = <I) U <I)', where any element of <I)' is reduced with respect to A( <I)). Then rk A( ~) < rk A( <I)), since the autoreduced set B = 9 U {h E A( <I)) I h is reduced with respect to g} C ~, where 9 is any element of <I)', is oflower rank than A( <I)). Since the set of autoreduced subsets of :F{Yl, ... , Yn} is well ordered, we deduce that Algorithm 5.5.12 will terminate.

Now, since the set <I) on every step of the algorithm may be only extended and A(<I)) ~ <I) ~ I, we have I = [<I)] and A C I. Obviously, when the set W becomes empty, the set A(<I)) becomes coherent. Then any h E <I) may be reduced to 0, h -----+ 0, therefore, for any h E <I) the condition h E [A] : H'f holds. By Lemma

A(if?)

5.5.4, [A] : H'f is a .1.-ideal, hence, I = [<I)] ~ [A] : H'f.

5.5.13. REMARK. If a set of linear .1.-polynomials is given to the input of Algorithm 5.5.12, then we obtain on its output a coherent linear set A such that I = [A], i.e., minimal Grobner basis (see Definition 4.1.43). Hence, A is a characteristic set of I. In general case, however, we do not obtain a characteristic set of I, i.e. the conditions that A is a coherent autoreduced set of a .1.-ideal I = [fl,···, ir] and [A] C I c [A] : H'f do not imply that A is a characteristic set of I.

5.5.14. EXAMPLE. Let m = 1, n = 2, Fl = dyldY2 + 1, F2 = (dY2)2, and the standard ranking be considered.

The set A = {Fl , F2} is coherent autoreduced, and [A] = [Flo F2] C [A] : H'f.


At the same time dY2 . Fl = dY2 + dYl . F2, therefore, dY2 E [AJ, hence, 1 E [AJ, i.e. {I} is a characteristic set of [A]. As a result of Algorithm 5.5.12 we obtain A = {Fl , F2 }.

Thus, for a set iP we can construct a coherent autoreduced set A c iP such that liP] c [A] : H'A. The second part of the Ritt-Kolchin method allows to find prime components of a perfect differential ideal under the condition that we can solve some problems in commutative polynomial rings.

5.5.15. Algorithm RK2 (iP, A) Input: iP = {It, ... , fr} is a finite set of Ll-polynomials Output: A is a set of coherent autoreduced subsets of :F {Yl, ... , Yn} such

that {iP} = nAEA[A] : H'A

Begin Algorithm RKI (iP,A) if A = {f}, f E :F then

A= {A} return

elseif 3P E (A) : H'A reduced with respect to A then Algorithm RK2 (iP U P, Al ) Algorithm RK3(iP, A, A2 )

A:= Al UA2

elseif 3P, Q fI. (A) : H'A such that PQ E (A) : H'A and P, Q are free of BYj, which are not present in A then P := remainder of P with respect to A Q := remainder of Q with respect to A Algorithm RK2 (iP U P, Ad Algorithm RK2 (iP U Q, A2 )

Algorithm RK3 (iP, A, Aa) A := Al U A2 U A3

else Algorithm RK3 (iP, A, A) A:= AU{A}

End

We use here auxiliary Algorithm 5.5.16. Algorithm 5.5.15 is recursive and it terminates because on every its step the rank

of the auto reduced set A( iP) is lower than the rank of this set on the preceding step. In order to prove that this algorithm gives us the result required, we formulate the

main principle, on which the algorithm is based. Let nonzero elements G l , ... , Gk

reduced with respect to A(iP) and such that G l ..... Gk E {iP} be known. If for all 1 :s i :s k we find a decomposition {iP, Gd = nBEl!,[B] : HB', then IE = Ui IB; gives us a solution to our problem: {iP} = nBEl![B] : HB' (see Lemma 5.5.17 below). Using Algorithm RKl, we can construct for iP an autoreduced set A such that A C [iP] ~ [A] : Hex. If in the ideal (A) : Hex we find an element P, reduced with respect to A (by Lemma 5.5.5, in this case P may be taken free of the derivatives BYj which are not present in A), then PH';, E (A) ~ {ell} and


5.5.16. Algorithm RK3 (, A, A) Input: = {II, ... , fr} is a finite set of D.-polynomials

A C [] is an autoreduced set Output: A is a set of coherent autoreduced set

Begin A:=0 for any A E A

Algorithm RK2 (USA,Ad Algorithm RK2 ( U fA, A2 )

A :=AUAI UA2

End

PTIAEA SA fA E {}. Since SA and fA for all A E A are reduced with respect to A, according to the main principle, we have solved our problem. Similarly, if PQ E (A) : H'A, P, Q rt. (A) : H'A (by Lemma 5.5.5, we can assume that P and Q are free of 8Yj which are not present in A), then we can replace P and Q by their remainders P and 0 with respect to A. Since P rt. (A) : H'A (otherwise P = H!A.P - EkAkAk E (A) : H'A, where Ak E F{Yl, .. ·,Yn}, because Pis free of the derivatives of the leaders of Ai E A, hence H!A.P E (A) : H'A) and PO E (A) : H'A (since PO == H!A.PQ mod (A)), we have PQH~ E (A) ~ {} and it suffices to apply the main principle. Let now (A) : H'A be a prime ideal containing no elements reduced with respect to A. By Theorem 5.5.6, [A] : H'A is a prime D.-ideal with the characteristic set A, and {} ~ [A] : H'A. In this case we have used the formula

{} = ([A]: H'A) n n {, fA} n n {, SA} AEA AEA

(see Corollary 5.5.18 below).

5.5.17. LEMMA. If Gl . G 2 E {}, then {<fI} = {, Gd n {, G2 }.

PROOF. The inclusion ~ is obvious. We prove that the hypotheses of the lemma imply that 81 Gl . 82G2 E {} for all 81 ,82 E T. Indeed, if d E D., then dG l . G2 + Gl . dG2 E {}. After multiplying by G2 , we obtain dG l . G~ E {}. Therefore, dG l · G2 E {} and Gl · dG2 E {}.

The case of arbitrary 81 , 82 follows from this by induction. Let now C E {<fI,G l } n {,G2}. Then C2 E {<fI,Gd . {,G2 } and C2 =

(91 + EAj 8jGl ) . (92 + E')'j8jG2), where 91,92 E {}, 81,8 E T and Aj,'Yj E f{ {Yl, ... , Yn}. Hence,

C2 = 91·92 + L 'Yj8jG29l + LAjB;G192 + LAj'Yk(8i G18I G2) E {<fI}

and C E {}. 0

5.5.18. COROLLARY. Let A be a characteristic set of a prime D.-ideal p, A ~ [] and {} ~ p. Then

{} = P n n {, fA} n n {, SA}. AEA AEA


PROOF. The inclusion ~ is obvious. If I E p, then I H!A E [A] C {ell} and ITIAEA SAIA E {ell}. By Lemma 5.5.17, we have

{f, ell} n n {elI,IA} n n {ell, SA} ~ {ell}, AEA AEA

and if I E {elI,I A} n { ell, SA} for all A E A, then I E {ell}. 0

5.5.19. REMARK. The construction of the set A(elI) for the set ell is reduced to the choice of autoreduced subsets of ell (i.e. to the computation of up for F E ell, the degree of up in F and the checking of the presence of Up in F' E ell, F' f:. F), and to the comparison of the ranks of these sets. In order to find the remainders G with respect to A we have Algorithm 5.3.8. Therefore, Algorithm 5.5.15 works if we can solve the following algebraic problem.

5.5.20. PROBLEM. Let a set of polynomials fr+l, ... , In E K[Xl, ... , xn] be given, where 0 S r S n, K is a field, and every /; is free of Xj+l, ... , X n , has degree dj > 0 in Xj and degree less than di in Xi (r Xn-l > ... > Xl, and ~ = 0). Define, whether the ideal I = (fr+l, ... ,In) : hOC of K[Xl, ... ,Xn] is prime and contains no nonzero polynomial of degree less than dj in Xj for all j = r + 1, ... , n, where h is the product TIr<i <n lj 8/; / 8x j, and lj is the leading coefficient of /; being considered as a polynomiaJ in Xj. In the case of negative answer, find either two polynomials gl, g2 rt. I such that glg2 E I, or a polynomial 9 E I of degree less than dj in Xj (r < j S n).

The solution to this problem which is given by E. Kolchin (see [Kol73]) assumes that we can factorize univariate polynomials over the quotient field of the ring K[Xl, ... ,xn]/p, where p is a prime ideal. Using the Kronecker algorithm (see, for example, [BW93]), it can be done over finite fields. If K = IQ, then a modification of the Kronecker algorithm allows to solve the problem, however, it seems that an implementation of this algorithm is impossible.

With the above notation, Problem 5.5.20 can be solved by induction on n - r

with the help of a process described in the following proposition.

5.5.21. PROPOSITION. [Kol73]. If r = n, then I = (0), and the answer is positive.

Consider the ideal I' = (fr+l, ... , In-I) : h'oc of K[Xl, ... xn-d, where h' = TIr<j<n Ij8/; /8xj. iffor I' the answer to Problem 5.5.20 is positive and the polynomial In(6,.·. ,en-I, xn) is irreducible in K(6,··· ,en-l)[Xn], where (6,··· ,en-I) is a generic zero of I', then I is a prime ideal which contains no elements reduced with respect to the set {/r+l, ... , In}. Conversely, if either the answer is negative, or In(6, ... ,en-I, xn) can be factorized in K(6, ... ,en-t}[xn], then we can give the negative answer to the problem for I.

PROOF. Let A' = {/r+1, .. . ,/n-l}, and A = {fr+1, ... ,/n}. Suppose that I' contains I f:. 0 reduced with respect to A'. Then I is reduced with respect to A and I . h,k E (A'), hence, I . hk E (A) and I E I.

Suppose that I' is not prime and contains no nonzero elements reduced with respect to A'. Then we can find polynomials I,g E K[Xl, ... ,Xn-l] such that


Ig El', I,g rt. I'. Let I ~ /" and 9 ~ g'. Since I' = h,k 1+ r, rEI' and A' A'

I' rt. I' (f' is reduced with respect to A'), we can assume that I,g are reduced with respect to A', hence, with respect to A. Now, if either I, or g is in I, then we have solved the problem for I. If I,g rt. I, then I is not prime, because Ig E I.

Let the ideal I' be prime and contain no elements reduced with respect to A'. Suppose that the polynomial In(6, ... ,€n-l,Xn) can be factorized over the field [{ (6, ... , €n-l), where (6, ... , €n - d is a generic zero of I'. Then there exist polynomials gl,g2 E [{[Xl, ... , Xn-l] and hI, h2 E [{[Xl, ... , xn] such that

(5.5.4)

gl fj. I', g2 El', degz hi > 0 (i = 1,2). Let hI ~ hI, h2 ~ h2. Then . ~ ~

degzn hi = degzn hi for i = 1,2 and we can assume that hb h2 are reduced with respect to A, multiplying, if necessary, (5.5.4) by an appropriate power of h'. If hI E I or h2 E I, then the problem for I is solved, and if hI, h2 rt. I, then hlh2 E I and I is not prime.

Conversely, let l' be a prime ideal containing no elements reduced with respect to A' and let In(€l, ... , €n-b xn) be a polynomial in Xn irreducible over [{(6, ... ,€n-I). Suppose that I E I, I -:j:. 0 and I is reduced with respect to.A. Then

and, since (6, ... ,€n-l) is a generic zero of I', we have

Being irreducible over K(€l, ... ,en-d, the polynomial In(el, ... ,€n-l, xn) divides either 1(6, ... ,€n-I.Xn) or h(6, ... ,€n-I,X n). But I and h are reduced with respect to A; therefore, degzn I < dn and degz " h < dn , hence, both cases are impossible.

Finally, let I' be a prime ideal which contains no elements reduced with respect to A'. Suppose that the polynomial In (€I, ... ,€n-I, xn) is irreducible over the field K(6, ... , €n-l) and the ideal I is not prime, and contains no elements reduced with respect to A. Then

(5.5.5)

for some I,g fj. I, Aj E [{[Xl, ... , xn]. Let 1-7 /', and 9 -7 g'. Then /"g' fj. I, A A

since I' and g' are reduced with respect to A and I, g fj. I. Therefore, we may assume that I and 9 in (5.5.5) are reduced with respect to A. Then

I(€I, ... , €n-b xn) . g(€I, ... ,€n-l, xn) . hk (€I, ... ,€n-l, xn)

= An(€I, ... ,€n-l, xn) . In(el, ... , en-I, Xn)

and since the polynomial In(el, ... ,en-bXn) is irreducible over [{(el, ... ,€n-d, the polynomial In(6, ... ,€n-lJXn) divides either 1(6, ... ,€n-lJXn), or


h(6, ... , ~n-l, Xn), or g(6, ... , ~n-l' xn), and as before we obtain a contradiction. The proof is completed. 0

Let us consider another approach to solving Problem 5.5.20. First of all, note that this problem "in principle" can be solved with the help of

methods based on the Grabner bases theory. In order to do this we can:

(1) compute generators of the ideal I; (2) find the elements of the ideal I which are reduced with respect to the au

toreduced set A = {/r+1, ... , In}; (3) check whether the ideal I is prime in the ring K[Xl, ... ,xn ].

Algorithmic aspects of steps (1) and (2) are simple enough (see Propositions 5.5.22, 5.5.23,5.5.24, and 5.5.25 below), but in order to fulfil step (3) we should can factorize polynomials in several variables over a field. It is possible "in principle", if we can factorize univariate polynomials over the initial field K. Without discussing the algorithms here, let us note that there exist explicit algorithms for such factorization over the field Q and iterative algorithms over the field C.

5.5.22. PROPOSITION. Let J be an ideal of the polynomial ring K[Xl, ... , xnl over a field K, hE K[Xl, ... , xn], h i- 0, It = J : h = {I E K[xl, ... , xn] I Ih E J}. Let L = (J, 1 - y. h(Xl, ... , xn)) denote the ideal of the ring K[Xl, ... , X n, y] which is generated by the ideal J and the polynomial 1 - y. h(Xl, ... , xn). Then It = L n K[Xl, ... , xn], i.e., the ideal It coincides with the set of elements of the ideal L which "do not depend on the variable y".

PROOF. The inclusion It ~ L n K[Xl' ... ,xn ] is obvious. Let

Then 1= LCXkik + ,8(1- hy),

k

where!k E J and degy I = 0, CXk, ,8 E K[Xl, .. ·, X n , y]. Expanding the polynomials CXk and ,8 in the powers of y and taking into account that degy 1= 0, we obtain

(5.5.6)

where jk E J, ,8k E K[Xl, ... , xn] for all k. Comparing the coefficients of y' in the right-hand side of (5.5.6), we obtain that j. = h,8.-1. Therefore,

j8Y' = hy,8._1y·-l = (1- 1 + hy),8._1y·-l = ,8._1y·-l - (1 - hy),8._1y·-l.

This means that for a fixed I we can replace the number s in relationship (5.5.6) by s - 1. Repeating this process, we obtain

where jo,it E J. Again we have it = h,8o and 1= jo + ,80, where ,8oh E J. Hence, ,80 E J : h and I E J : h = It. 0


5.5.23. PROPOSITION. Let J be an ideal of the ring K[X1, ... , xnl and h a nonzero element of K[X1"'" xnl. Let us introduce the notation Ik = J : hk = {I E K[X1, ... , xnll I· hk E J} and I = J : hoo = {f E K[X1, ... , xnl I 3k EN: I . hk E J}. Then

It ~ 12 ~ ...

is a non descending chain of ideals, and if J,. = Ik+1, then I = Ik (by the Zorn lemma, such k always exists).

PROOF. If i < j and I E Ii, then I· hi E J implies I· hi-i. hi = Ihi E J and f E Ij. Let J,. = Ik+1 and f E I. Then f· hr E J, hence f E Ir . If r :5 k, then the inclusion Ir ~ J,. implies IE Ik. Let r> k. Since Ihr = (f. hr- k- 1) . hk+1 E J, we have f· hr- k- 1 E Ik+l = Ik , hence I· hr- k- 1 . hk = I· hr- 1 E J, and I E Ir- 1. Repeatingly decreasing, if necessary, r, we obtain I Elk. 0

Propositions 5.5.22 and 5.5.23 allow to compute generators of the ideal 1= J : h 00, if we know generators of the ideal J. Actually, generators of the ideal It can be computed, if we know how to "eliminate an indeterminate from polynomial equations". This can be done, by computing the Grabner basis of the ideal L (see Proposition 5.5.22) with respect to the lexicographic order.

5.5.24. PROPOSITION. Let L be an ideal of the ring K[X1, ... , xnl and {91, ... , 9k} its Grabner basis with respect to the lexicographic order (we assume that Xn > Xn-1 > ... > Xl). Let 91, ... ,91 be the elements of the Grabner ba-sis of the ideal L which are free of the variables X p+1, ... , X n, 1 :5 p < n. Then {91, ... ,91} is the Grabner basis of the ideal L n K[X1, ... , xpl.

PROOF. It is clear that if f E L n K[X1, ... , xp], then reducing f with respect to the Grabner basis {91, ... , 9k}, we use only the elements which are free of the variables Xp+1, ... , Xn (because u/ < Xi for j = p+ 1, ... , n), hence I E (91, .. . ,91). 0

Thus, if we know generators of the ideal J, then, using Propositions 5.5.22 and 5.5.23, we can find generators of the ideal It = J : h. Since 12 = J : h2 = It : h, we make the same computations with h. The Grabner bases allow us to verify whether the inclusion It ~ 12 is strict (it is sufficient to check whether the generators of the ideal 12 are reducible to 0 with respect to the Grabner basis of the ideal It). When we obtain the equality, then by Proposition 5.5.23 we determine the generators of the ideal I. Let us note that the smallest k such that I = Ik can be estimated using the product dr+1 ..... dn in notation of Proposition 5.5.20 (see [Se74]). Then we are going to find out whether the ideal I contains elements irreducible with respect to A. It can be done if we know a Grabner basis of the ideal I.

5.5.25. PROPOSITION. Let a set of polynomials Ir+1, ... ,/n E K[x1, ... ,xn l be given, where 0 :5 r :5 n, K is a field, and every Ii is free of xi+1, ... , Xn has degree di > 0 in Xi and degree less than di in Xi (r Xn-1 > ... > Xl, and ~ = 0). Let I = (fr+l, ... ,/n): hoo denote the ideal of K[X1, ... , xnl, where h is the product Dr<i<n liO/j/oxi, and lj is the leading coefficient of Ii being considered as a polynomial in xi' Let G be a reduced Grabner basis of I with respect the same ordering of monomials. Suppose


that I -+ 0 for all lEG. Then the ideal I contains no elements irreducible with A

respect to A.

PROOF. Let us introduce the notation Ij = InK[X1, ... ,Xj] and Gj = Gn [{[Xl, . .. , Xj]. Suppose that I is a polynomial reduced with respect to A. Let i be an integer such that I E Ii \ Ii-I. Since G is a Grabner basis of the ideal I, we have the expansion I = 2:::7=1 aj(x1, ... , Xi)9j(X1, ... , Xi), where 9j E Gi and degxi 9j ::; degxi I < di for all 1 ::; j ::; m, because I is reduced with respect to A. Since 9j -+ 0 and 9j is reduced with respect to {/r+1, ... , In}, we have

A

l tr+l lti-l E (f F) r+l·····i-19j r+1,···,ji-l·

Denoting hI = lr+! ..... li-1 E Ii-I, we see that 9j E J = Ur+1, ... ,/;-d : hr'(Xl, ... , Xi-I) for all 1 ::; j ::; m. Since the polynomials Ir+1, ... , li-1 are free of the variables Xi, ... , xn , by virtue of Proposition 5.5.22, we can choose generators of the ideal J which are free of the variables Xi,···, xn , and then 9j = 2:::/ a/xL a/(xl, ... , xi-d E Ii-I. Since G is a Grabner basis of the ideal I, we can reduce every a/ to zero with respect to G, and while reducing we can use only the elements of Gi-l. Repeating the reduction process, we can also reduce to zero the element 9j = 2:::1 alx~. This contradicts the assumption that G is a reduced Grabner basis of the ideal I. Thus, we proved that I contains no elements irreducible with respect toA. 0

So, knowing a Grabner basis of the ideal I, we can verify whether I contains irreducible with respect to A elements, proceeding in the following way: we reduce every element lEG with respect to A and if we always obtain 0, then the answer is negative. Otherwise, some lEG is reduced to 1 #- 0 and 1 E I is an element from the ideal I which is irreducible with respect to A.

The most difficult is the problem of testing whether the ideal I is prime. An algorithm for decomposition of an ideal into an intersection of primitive ones can be found in the monograph [BW93]. Let us note that an ideal if prime if and only if it has only one primary component which is a perfect ideal.

5.5.26. THEOREM, [BW93, Theorem 8.101]. Assume that a field K is computable {i.e., there exists on [{ an effective implementation of the operations +, *, and /); and for any finite set {Ul, ... , , u r } of indeterminates, the rational function field [{ (Ul, ... , u r ) is infinite and perfect (for example, algebraically closed or Q) and allows effective factorization of univariate polynomials. Suppose F is a finite subset of [{[Xl, ... , xn] which generates a proper ideal. Then the algorithm PRIMDEC of Table 8.10 computes a finite set of primary ideals with intersection (F) as well as the corresponding associated prime ideals.

Let us note that this algorithm works "in principle" in the fields Q and C, however its effectiveness is bounded by algorithms for polynomial factorization over the corresponding fields. As for the testing of perfectness of an ideal, this problem can be solved by the algorithm RADICAL (see Table 8.9 in the same book). Let us also note that if an ideal I is not prime, then, knowing its primary components, we can find elements a, b ~ I such that ab E I.

5.6. INVARIANTS OF DIFFERENTIAL DIMENSION POLYNOMIALS 257

The algorithm given in [BW93] for primary decomposition of an ideal is based on the following ideas. First, the general case is reduced to the case when dim I = O. Let us note that by virtue of Proposition 5.5.21 this can be done, by passing from the field K to the field K (Xl, ... , X r ). If dim I = 0, then with the help of a "linear change of variables" we can pass to the case when the minimal Grobner basis of the ideal I with respect to the lexicographic order of variables has a special form, namely, {91(xI),x2 - P1(xd, ... ,Xn - Pn(X1)}. Then it is sufficient to factor the polynomial 91 into irreducible factors,

Let us note that from the computational viewpoint this approach is seems more perspective than Kolchin's one, because it reduces the problem to a less difficult problem of factorization over a purely transcendental extension of the basic field.

If there exists a satisfactory factorization algorithm for univariate polynomials over the residue fields of K, then we can test the primality of an ideal in the ring K[x1, ... , xn ], using the Ritt-Kolchin algorithm. Actually, if an ideal I is given by generators h, ... , /,., then using the approach given in Proposition 5.5.21, we can solve Problem 5.5.20 and by Algorithm 5.5.15 we can find the decomposition

where Pi are prime ideals given by their characteristic sets. The problem of inclusion Pi C Pi in the polynomial case can be effectively solved with the help of Grobner bases; therefore, we can assume that all ideals Pi in the decomposition obtained are distinct. Computing Grobner bases for the intersections of these ideals (see [BW93]), we obtain generators of the radical of I, in particular, we can verify whether the ideal I is perfect. In case of perfect I, in order to verify whether the ideal I is prime it is sufficient to remark that Algorithm 5.5.15 gives in this case only one prime ideal.

5.5.27. REMARK. The notion of autoreduced and characteristic sets and the theory presented in Section 5.5 are nontrivial in the case of polynomial rings, Le., in case A = 0. In particular, Algorithm 5.5.14 in this case is known as the Wu-Ritt process (see, for example, [Mi93]). In the book [Mi93] an irreducible characteristic set is also defined as a set satisfying the conditions formulated in Proposition 5.5.21. Some problems in the ideal theory for polynomial rings have an elegant solution by this approach without using the Grobner bases methods.

5.6. Invariants of Differential Dimension Polynomials

Let 9 be a finitely generated A-extension of a A-field F, and "11,··., TIn E g. In Section 5.2 it was noted that the differential dimension polynomial wf}l, ..• ,f}n

(see Definition 5.2.6) is not a differential birational invariant, i.e. the condition F(TJ1, ... ,"In) = F(6, ... ,e,.) does not imply wf}l, ... ,f}n (t) = we, , ... ,ek (t). The last equality follows from the more strong condition F(Tl1, ... ,TIn) = F(6, ... , e,.), in other words, the differential dimension polynomial is a birational invariant. From the point of view of the module of differentials the condition


means that different systems of L\.-generators of the module ng / F are chosen, and the condition :F('I11, ... , 'I1n) = :F(6, . .. , ek) implies the coincidence of the corresponding filtrations. Expanding the polynomial W'11 •...• '1 .. (t) by the standard basis (2.1.11)' we obtain:

(m = Card L\.)

As Proposition 5.2.9 shows, the coefficient am is a L\.-invariant of the L\.-extension :F('I11, ... , 'I1n) over :F.

5.6.1. DEFINITION. Maximal integer T such that aT # 0 is called the differential type of the L\.-extension :F ('111, ... , 'I1n) over :F (or of the corresponding prime L\.ideal) and denoted by L\.-type, and aT is called the typical L\.-dimension.

Note that the differential type and the typical differential dimension are, by Proposition 5.2.9, L\.-invariants of an extension :F('I11, ... , 'I1n) over :F.

Let 9 be a finitely generated L\.-extension of a L\.-field :F of differential type T, L\. = {d1, ... ,dm }. If L\.' = {d~, ... ,d~}, dj = L~1Cijdi (j = I, ... ,T), Cij are L\.-constants of :F, then we can consider 9 as a L\.'-extension of the L\.'-field :F. Theorem 5.6.3 below shows that the matrix C = (Cij) can be chosen such that L\.'-dimension polynomial w'11 •...• '1 .. /F(t) has the degree T.

First, we prove a lemma.

5.6.2. LEMMA. Let :F be a L\.-field, a a L\.-algebraic over :F element of a L\.extension 9 of :F. Then there exist an integer r E N and nonzero homogeneous polynomial 9 E C[X1, .. ·, Xm] of degree r over the field C of constants of 9 with the following property: if (Cij) is a nonsingular m x m-matrix over C and g(C1m, ... , cmm ) # 0, then

where d~, ... ,d~ are the derivation operators defined by the equations

m

di = Lcijdj. j=1

PROOF. Consider the standard ranking on the set of L\.-derivatives of y. Since a is L\.-algebraic over :F, there exists a L\.-polynomial A E :F{y}.o. \:F such that A(a) = 0 and SA(a) # O. Let C = (Cij) be an invertible m x m-matrix with entries in C, and let L\.' be the set of derivation operators d~, ... , d~ defined by the equations di = L~=1 Cii,d~ (i = 1, ... , m). Let d~l .. . d':nmy be a L\.-derivative of order r. Then

- el em (d')r + f - Clm ..... Cmm mY,


where ord!:S r and the ~/-polynomial! is free of (d:"'ry. Therefore the order of the ~-polynomial A is equal to its order as a ~/-polynomial, and

8A (5.6.1)

Since SA (a) =F 0, at least one of the partial deri vati ves in the right-hand side of (5.6.1) does not vanish after substituting y = o'. Consider 9 = F(O') as a linear space over C and choose a basis bdkE[' Expressing this basis in terms of the

. I d' . 8A I partIa envatIves 80 y y=o:

From (5.6.1) we have

= L9k(C1m," .,Cmm)-Yk k

where 9k are homogeneous polynomials from C[X1' ... , Xm] of degree r, and some 9k =F O. If we set 9 = 9k where 9k =t 0, then 9(C1m, ... , cmm ) =F 0 implies 8A/8(d~y)(O') =F O. Considering A as a ~/-polynomial, from d:"'A(O') = 0 we have

8A/8(d~y)(O') . (d:nr+1O' - T(O') = 0,

where T is a ~/-polynomial in F{y} of order no higher than r + 1 and T is free of d~+l y. The lemma is proved. 0

5.6.3. THEOREM. Let 9 be a finitely generated ~-extension of a ~-field F of differential type T and of typical differential dimension d.

(1) If T = -1, then 9 is an algebraic extension of F of a finite degree. (2) If T =F -1, then there exists a set ~., consisting of T linear independent

combination over C of elements of ~ such that 9 is a finitely generated ~. -extension of ~. -transcendence degree d over the ~. -field F. (Here C is the field of ~-constants of F.)

PROOF. Let 1] = (1]1, ... , 1]n) be a set of ~-generators of 9 over F. If T = -1, then WI)/F(t) = 0, and the transcendence degree of F((01]j)UET(.). l~j~n) over F is equal to 0 for all sufficiently large sEN. In particular, any Yj is algebraic over F. Therefore d1]j E F(1]j) for all d E ~, hence, 9 = F(1]) = F('1]) and (1) is proved.

Suppose that 0 < T < m. If ~-transcendence degree J. of 9 over F is greater than 0, then by Th~re~ 5.4.4, T = m and J. = d. In this case, if we pass from ~ to


t::.' with any invertible C-matrix, then the t::.'-field g has the same t::.'-transcendence degree over :F. Suppose that J = o.

Let c = (CHI) be an invertible m x m-matrix with entries in C, let d~, ... , d~ be the linear combinations of the operators db . .. , dm from t::., determined by the equations di = 2::~=1 cwdi (1 ~ i ~ m), and let t::.~ be the set consisting of d~, ... ,d~_l. We are going to show that the matrix (CH') may be chosen such that g is a finitely generated t::.~-extension of:F, of t::.~-type T and typical t::.~-dimension d. Clearly, this is sufficient for the proof of (2).

By Lemma 5.6.2, there exist integers rl, ... , rn and a nonzero homogeneous polynomial 9 E g[Xl, ... , Xm] such that if g(Clm, ... , cmm) # 0 then

for any T/j. Therefore, the family {eij = (d~T/j)}o~i~rj generates g as a t::.~l~j~n

extension of :F. Denoting the sets of derivatives d~il ... d~m such that i1 + ... + im ~ 8 by T'(8)

and d~l ... d~':..ll with i1 + ... + i m - 1 ~ 8 by T{ (8), we see that if r > rj then

Differentiating it 8 - r times, we obtain that if 8 ~ rand ()' E T'(8) , then

hence,

for all sufficiently large 8. Denoting the t::.~-dimension polynomial of the t::.~-field g = :F(eij) over :F by we/:F' we have w,,/:F ~ we/:F. On the other hand, obviously,

, ,i , :F(((}ldmT/j)6;ET{(.), O~i~rj, l~j~m) ~ :F(((} 1/j)6'ET'(.+rj), l~j~m)

~ :F(((}'1/j)6'ET'(.+ro), l~j~m),

where ro = max(rl, ... , rn), hence, we/:F(t) ~ w,,/:F(t + ro). These two inequality imply that g has t::.~-type T and typical t::.~-dimension d over:F. 0

One of the classical problems of differential algebra is: to estimate the invariants of the differential dimension polynomial Wp, on the base of estimates of the given system of differential equations E C :F {Yl, ... , Yn} (here .p is a prime component of the t::.-ideal {E}).

Ritt in [Rit32] proved the following result for ordinary differential equations: if every element of a set E is free of derivatives of Yj of order greater than ej

(j = 1, ... , n), then for any prime component .p of the ideal {E}, the condition wp = const implies wp ~ el + ... + en.

We give here the proof due to Kolchin, who generalized this result to the partial differential equations (see [Kol73]).


5.6.4. LEMMA. Let d E ~, ~o = ~ \ d and el, ... , en EN. Suppose that E C .1'{(dkYjh~j~n,o~k~eJLl.o' and P is a prime component of {E} C .1'{Yl, ... , Yn}LI.. Let 9 = .1'('r/l, ... , 'r/n) be a ~-extension of .1', where 'r/ = ('r/l, ... , 'r/n) is a generic zero of the ~-ideal p. If ~-dimension of P is equal to 0, then ~o-transcendence degree of 9 over .1' does not exceed el + ... + en.

PROOF. The classical approach consists in constructing of a system of equations of order not greater than 1, which is equivalent to the given system. Thus, the problem may be reduced to the case, when ej ::; 1. For this purpose we take (Zjkh~j~n,O~k~ej to be ~-indeterminates and consider the ~-homomorphism

(J" : .1'{(zjkh~j~n,o~k~eJLI. -t .1'{Yl, ... , Yn}LI.,

mapping Zjk into dk Yj (j = 1, ... , n, k = 0, ... , ej). Clearly, (J" maps bijectively .1'{(zjkh~j~n,O~k~eJLl.o on .1'{(dkYjh~j~n,O~k~eJ, (J" is surjective and the kernel of (J" is the ~-ideal [J{], where J{ is the set of ~-polynomials Zj,k+l - dZjk (1 ::; j ::; n, 0::; k < ej). Since E C .1'{(dkYjh~j~n, O~k~eJLl.o, there exists a unique set Ho C .1'{(zjkh~j~n,o~k~eJ such that (J"(Ho) = E. Let H = Ho U J{, then H C .1'{(zjkh~j~n,o~k~ej' (dZjkh~j~n,O~k~eJLl.o and {H} = (J"-l({E}). The ideal q = (J"-l(p) is a prime component of {H} in .1'{(zjkh~j~n,O~k~eJLI.' the point (= (dk'r/jh~j~n,O~k~ej is a generic zero of q, and .1'«)LI. = 9. Therefore, it suffices to prove the lemma for systems, where all ej are equal either to 0 or to 1. Indeed, in this case the lemma holds for H, and

~o-trdeg9/.1' < maxd-ordz " F + ... + maxd-ordz , F + ... - FEH FEH 0,

+ maxd-ordz ,F + ... + maxd-ordz F FEH n FEH nOn

=el + ... + en·

Suppose that E C Ro = .1'{Yl, ... ,Yn,dYl, ... ,dyv}Ll.o (we can achieve it by a permutation of indices; here II ::; n). We have to prove that ~o-trdeg 9/.1' ::; II.

Let Po = P n Ro, then Po is a prime ~o-ideal of Ro· Since E C Po C P we see that p is a prime component of {Po} in .1'{Yl, ... , Yn} LI. and ('r/l, ... , 'r/n, d'r/l, ... , d'r/v) is a generic zero of Po.

Let J1. = ~o-trdeg.1'('r/l' ... ''r/v)Ll.o/F. After permuting the indices 1, ... ,11, we can assume that ('r/l, ... , 'r//J) (0 ::; J1. ::; II) is a ~o-transcendence basis of .1'('r/l, ... , 'r/v)Ll. o over F. Proposition 5.3.13 implies that, for all J1. .) is a ~o-transcendence basisof.1'('r/l, ... ,'r//J,d'r/l, ... ,d'r//J)Ll.o over.1'('r/l, ... ,'r//J)LI.. If

K = ~o- trdeg .1'('r/l, ... , 'r/n, d'r/lJ ... , d'r/v)Ll.o/.1'('r/l, ... , 'r/v, d'r/l, ... , d'r/v)Ll.o,

then we can assume that ('r/V+l, ... , 'r/I<) (II::; K ::; n) is a ~o-transcendence basis of the corresponding extension. Consider now the tower of ~o-fields


It is easy to see that the system (771,···,77"., d771'.··' d77>., 77 ... +1, ... ,771() is a Llo-transcendence basis of F (771, ... , 77n, d771, ... , d77 ... ) 6 0 over F.

We introduce now a ranking on the set of Ll-indeterminates (Yl, ... , Yn). Let To denote the free commutative semigroup generated by the elements of Llo, and let a ranking on the set consisting of one Llo-indeterminate z be given. Let the order on To be such that (h > {J2 ({Jl,{J2 E To) is equivalent to {JIZ > (J2Z. Set q(j) = 1, if 1 ~ j ~ /I and q(j) = 2, if /I < j ~ n. We shall order the derivatives of the form {JodkYj ({Jo E T, 1 ~ j ~ n) lexicographically with respect to (q(j), k,j,Oo). Clearly, this order is a ranking on the set of Ll-indeterminates (Yl, ... , Yn). We shall call it Ll-ranking. The order induced on the set of derivatives

{{JOYj}90 ETo • l$jSn U {{JOdY;}90ETo. 19$ ...

is a ranking of the family of Llo-indeterminates (Yl, ... , Yn, dYl, ... , dy ... ). We shall call it Llo-ranking.

Let Ao be a characteristic set of the ideal 1'0 with respect to the Llo-ranking. The leader UA of any A E Ao may have one of the following three forms:

(1) {JoYj, where {Jo E To, I' < j ~ /I; (2) {JodYj, where {Jo E To, A < j ~ /I; (3) {JoYj, where {Jo E To, II: < j ~ n.

Indeed, it was shown that (771, ... , 77n, d(77d, ... , d(77 ... )) is the general (algebraic) zero ofthe ideal 1'0. Let A E Ao and UA = {JoYj, where 1 ~ j ~ 1'. By Lemma5.3.12, IA rt 1'0, therefore, (Jo77j is algebraic over F(({J77i)9ETo. i$j) (since in this case q(j) = 1 and k = 1 by definition of all-ranking), and this contradicts the Llo-independence of (771, ... ,77".) over F. Let UA = (JoYj, where /I < j ~ 11:. In this case q(j) = 2, and (Jo77j is algebraic over F(({J77i,{Jd77r)9ETo. rS"', i<j), and this contradicts the 6.o-independence of the family (77 ... +1, ... ,771() over F(771, ... , 77 ... , d77l, ... , d77 ... ) 6 0 •

Similarly, the case UA = dYj, where 1 ~ j ~ A, is also impossible. We shall prove now the following key fact: if A E Ao has a leader UA of the form

(1), then there exists B E Ao such that either dUA is a proper 6.o-derivative ofuB, or dUA = UB and deguB B = 1.

Indeed, suppose that for A E Ao there exists no B with given properties. Applying d to A, we obtain dA = SAduA + Q, where Q = L-1J<uJoA/ov)dv + Ad, and Ad is the result of applying d to the coefficients of A considered as an usual polynomial in variables {JYj ({J E T, 1 ~ j ~ n). Since dUA = {JodYj, j ~ /I, we have dA E F {Yl, ... , Yn, dYl, ... , dYII} 6 0 = Ro. By Lemma 5.3.6, there exist u and v E Ro 6.o-reduced with respect to Ao and of rank no higher than dA such that

{PSA:::U .

, (mod [Ao]) in Ro, were P = TIFEA Ii! S~. Since SA rt 1'0, we have PQ::: v 0

U rt 1'0· By assumption dUA is not a proper derivative of UB for any B E Ao, and if dUA = UB, then deguB B ~ 1. Therefore, U . dUA + v is a nonzero element of Ro which is Llo-reduced with respect to Ao. By Lemma 5.3.12, UdUA + v rt 1'0. At the same time, dA E 1', hence, SAduA + Q E I' n Ro = 1'0 and u . dUA + v E 1'0. This contradiction proves the key fact.

Let now B denote the set of all B E Ao such that UB = dUA for some A E Ao and deguB B = 1. We are going to show that A = Ao \ B is a 6.-autoreduced subset of F {Yl, ... , Yn} (i.e. reduced with respect to the Ll-ranking).


Suppose that A, C E A and A-rk A < A-rk C. Since the set Ao is Aoautoreduced, we have deguA C < deguA A and C is free of any proper Ao-derivative of UA. Since C E Ro, C is free of any derivative of the form (JodkuA' where k> 1. Suppose that deg90 duA C > 0 «(Jo E To). Since C E Ro, the leader UA has the form (1) (recall that K, ~ v) and by the key fact either dUA is a proper Ao-derivative of UB, or dUA = UB and deguB B = 1. Since C, B E Ao, C is free of proper Ao-derivatives of UB, therefore, the first case is impossible and (Jo = 1. Now, deguB C ~ 1 and deguB B = 1, hence, C = B. At the same time B E B, and C f/. B. This contradiction shows that the set A is A-autoreduced.

We want to show that A is a characteristic set of p. Let F E Ro, and let AF denote the set of derivatives (JA «(J E T, A E A) such

that A-rk(JA ~ A-rkF. We shall prove that there exists a polynomial Fo E Ro which is A-reduced with respect to A and such that

II S~A I~A F == Fo (mod (AF)) (5.6.2) AEA

and F E Po if and only if Fo = O. The key fact implies that F E Ro is A-reduced with respect to A iff F is Ao

reduced with respect to Ao. The second part of the statement follows from this immediately.

If F is Ao-reduced with respect to Ao, we may set Fo = F, integers SA = iA = 0 and (5.6.2), obviously, holds.

Suppose that F is not Ao-reduced with respect to Ao. Let B E Ao and (Jo E To be such that (JOUB has the highest rank among all Ao-derivative of the leaders UB' of B' E Ao satisfying the condition: there exists (J~ E To such that deg8' U , F > 0 o B and either (J~ 1= 1, or deguB , F > deguB, B'. We denote (JOUB by v(F). It is easy to see that v(F) has the highest rank among the A-derivatives (JUA «(J E T) present in F of the leaders of elements A E A such that either (J 1= 1 or deguA F > deguA A. We shall use induction on the rank of v(F).

Let e(F) = degl/(F) F. Consider three cases. 1) Suppose that B E A and (Jo 1= 1. Then (JoB = SBv(F) + Q E Ro, where

A-rkQ < A-rkv(F). Dividing F by (JoB, we find a A-polynomial G E Ro such that S~(F) F == G (mod (JoB), and either G is A-reduced with respect to A, or rk v( G) < rk v(F). By inductive hypothesis,

II S~A I~AG == Go (mod (Aa)) AEA

for some integers tA, iA and a polynomial Go A-reduced with respect to A. Since Aa ~ AF, we have

II S~A I~A S~(F) F == Go (mod (AF)). AEA

2) Suppose that B E A, (Jo = 1 (i.e. v(F) = UB) and e(F) ~ b = deguB B. Similarly to the case 1), we obtain I~(F)-b+1 F == G (mod B), where G E Ro and rk v(G) < rk v(F). The remaining part of the proof is the same as in 1).


3) Let now B rt A (i.e. B E 8). Then UB = dUA for some A E Ao and deguB B = 1. The leader UA must have the form (1), therefore A rt 8, i.e. A E A n F {Yl. ... , Yv h~o, hence dA E Ro n l' = Po. Applying (Jo to dA, we obtain (JodA = SAv(F) + Q E Ro, where A-rkQ < A-rkv(F). Dividing F by (JodA, we obtain S~(F) F == G (mod (JodA), where G E Ro and either G is A-reduced with respect to A, or rk v( G) < rk v(F). The remaining part of the proof is the same as in 1).

Now we shall show that the set A is coherent and the ideal (A) : H'A is prime and contains no elements A-reduced with respect to A. By Theorem 5.5.6, A is then a characteristic set of a A-ideal q. By Proposition 5.5.9, q = [A] : H'A.

Indeed, let A 'I C, A, C E A and (JUA = (J'uc ((J,O' E T). According to the three cases considered, we have either (J, 0' E To, or one of the leaders, say UA, may be written in the form UA = (JOYi ((Jo E To) and I-' < j ~ II, and the second leader may be written as Uc = O~dYi ((J~ E To); in this case (J E Tod and (J' E To. In both cases F = ScOA - SAO'C E l' n Ro = Po. Therefore, (5.6.2) holds with Fo = o. According to Definition 5.5.1, the set A is coherent.

By Lemma 5.5.5, in order to prove that the ideal (A) : H'A is prime and contains no elements reduced with respect to A, it is sufficient to prove these statements in F[U], where U is a set containing all derivatives (JYi present in elements of A (for example, Ro). If FE (A) : H'A in Ro is A-reduced with respect to A then F E Po and F is Ao-reduced with respect to Ao, therefore, F = O. Let Roo C Ro denote the algebra F[V], where V is the least with respect to inclusion of all admissible sets (i.e. OYj E V, iff deg8y; A 'I 0 for some A E A). We shall show that the ideal (A) : H'A C Roo is prime in F[V]. Let F· G E Roo, F, G rt (A) : H'A. Then F and G are free of proper derivatives of UA (A E A). Therefore, there exist integers i A, iA and A-polynomials Fo, Go A-reduced with respect to A and such that

II I~A F == Fa (mod (A)) A€A

II I~AG == Go (mod (A)) A€A

Since F, G rt (A) : H'A, we have Fa 'I 0, Go 'I 0, therefore, Fa rt Po, Go rt Po, hence, FoGo fi. Po, i.e. FoGo rt (A) : H'A in Roo. Thus, FG fi. (A) : H'A.

By Theorem 5.5.6, A is a characteristic set ofa prime A-ideal q ~ F{Yl, ... , Yn}. By Proposition 5.5.9, q = [A] : H'A.

Clearly, Po C q ~ p. Since l' is a component of {Po} in F{yI. ... , Yn}, we have l' = q. Therefore, A is a characteristic set of p.

Using the fact that the differential dimension of l' is equal to 0, we prove now that .x = 0 and I'i. = II. Indeed, if .x ~ 1, then J.l ~ 1 and I'i. ~ 1, hence, the leaders of the forms (1)-(3) of elements of A C Ao are free of A-derivatives of Yl. Since A is a characteristic set of 1', Theorems 5.4.1 and 2.2.5 imply am(p) 'I o. If I'i. ~ 11+ 1, then ordY"+l F = -00 (i.e., F is free of the variable yv+d for all F E A, and we have again a contradiction.

Thus, (171, ... ,17,.) is a Ao-transcendence basis of F(171, ... ,17n,d17IJ ... ,d17v}Ao over F. In particular, any 17j for II < i ~ n is Ao-algebraic over F(171 , ... , 17,.}Ao •


By induction on k it is easy to prove that :F«dk7Jih<i<n O<k<r)Ao is ~o-algebraic over :F(7JI, ... , 7JJJ)Ao· Thus, we proved that ~o-trdegg/:F~ It S v. 0

5.6.5. THEOREM. Let ell ... , en EN, let E be a subset of:F[(BYi)BET(ej), l$j$n]' and lJ be a component of {E} in :F{Yl, ... , Yn}. If the differential type oflJ is equal to m - 1, then the ~-dimension oflJ does not exceed el + ... + en.

PROOF. This theorem follows immediately from Theorem 5.6.3 and Lemma 5.6.4. 0

For some systems one can prove a better estimate than that of Theorem 5.6.5, (see Section 5.8).

E. Kolchin at Moscow International Mathematical Congress [KoI66] formulated several conjectures concerning estimates of the leading coefficient of differential dimension polynomial. The main of them is Jacobi conjecture (see Section 5.8). The other are the following:

(1) a conjecture concerning an estimate of typical differential dimension, generalizing Theorem 5.6.5. Under the hypothesis of Theorem 5.6.5, let T be the differential type of lJ. Then

() "" (ej + m - T - 1) aT Wp S L...J '

1< ·<n m - T _3_

(5.6.3)

where aT(wp) is the T-coefficient of the polynomial wp; (2) let E C :F {Yl, ... , Yn}, Card E = nand lJ be a prime component of the ideal

{E}. If Ll-type of lJ is less than m - 1, then wp = o. If all polynomials in E are linear, then conjecture (2) is the exact formulation of

the Janet conjecture [Jan21]. For linear systems it was proved by J. Johnson, (see [Jo78]).

If T = m, then the conjecture (1) is true, since am(wp) S n by Theorems 5.4.1 and 2.2.5. If T = m - 1, then the conjecture (1) is also true, since am -l (wp) S 2::;=1 ej (by Theorem 5.6.5).

We shall show now that if T < m - 1, then the estimate (5.6.3) does not hold. Let n = 1, m = 2, E = {d~y, d~y}, lJ = [E] (lJ is a prime ideal, because the system E is linear). Then el = 2, wp(s) = W(~~) (s) = 4. On the other hand, according to

(5.6.3) we should have ao(wp) S (etl) = 3. If :F is a field of constants, n = 1 and all equations in E are linear, then the

Bezout theorem [Laz83] holds:

One can conjecture that

aT(w[El) S (5.6.4)

If :F is not a field of constants, then this conjecture does not hold. Inequality (5.6.4) may be wrong even if n = 1 and T < m - 2. Actually, if:F = C(Xl' ... , xm ),

9 = :F(1}1l ... , 1}n), then the module of differentials 0.9 /:;: is isomorphic to a factormodule ofthe ring oflinear differential operators D = g[~] by a left ideal. Similarly to the Weyl algebra, any ideal in D is generated by two elements (see [Sta78]). Therefore the Bezout theorem may be not valid in this case.


5.6.6. EXAMPLE. Let

Then ng /7: = D/ J, where a Grobner basis of J consists of elements d~ + x2d~, d~,

d~, d2d~. Therefore, W[l:] = WE, where E = (~~~) . It is easy to calculate that 004

WE = 12, hence, D.-type T of [E] is equal to 0, and the typical dimension aT = 12; on the other hand, according to (5.6.2) we should have aT ~ 23 = 8.

However, in case T = m - 2, conjecture (5.6.4) seems verisimilar. We shall prove it for linear systems when n = 1.

5.6.7. THEOREM. Let g be a D.-extension of a D.-field :F defined by a system of linear D.-equations

{ FICP = ° FkCP = 0,

where FI, ... , Fk E :F[dl , ... ,dm ], bi = ord Fi, bl ~ b2 ~ ... ~ bk and the D.-type ofg over:F is equal to m - 2. Then the typical D.-dimension ofg over:F does not exceed bl b2 •

PROOF. Let D = g[d l , ... ,dm ] be the ring of linear differential operators over g. We have the exact sequence of D-modules:

(5.6.5)

where k

H = LD(Fj + DFI), f(r + DFd = rdcp. j=2

If D = U.El'i D. is the standard filtration, then D/ DFI and its submodule H have induced filtrations

{D/DFd. = D. mod D.-b1FI and H. = H n {D/DFd •.

The sequence (5.6.5) is an exact sequence of filtered D-modules, therefore,

(t + m) (t + m - bl ) wcp/:F(t) = m - m - WH{t), (5.6.6)

where WH(S) = dimg H. for all sufficiently large s. By Theorem 4.3.40, the filtration H = U.El'i H. is excellent. Consider another filtration on H (connected with a choice of generators Fj , 2 ~ j ~ k, and "shifted" according to the order of generators) :

k

iI. = LD.-bj(Fj + DFd· j=2

5.7. MINIMAL DIFFERENTIAL DIMENSION POLYNOMIAL 267

This filtration is also excellent and iI, ~ H,; therefore, wjf(t) ::; WH(t). Since iI'+b2 = E j D.hj for all sufficiently large s and some hj E H (recall that b2 ~ b3 ~ ... ~ bk), we have wjf(t + b2) E W (see Section 2.4). Since U.EN H. and U.EN iI. are two excellent filtrations on H, the polynomials W Hand W ii have the same degree and the leading coefficients. Therefore, the condition am (W",/F) = am-1(w",/F) = 0 and formula (5.6.6) imply that am(wjf) = 0, am-1(wjf) = b1. Since wjf(t+b2) E W, by Corollary 2.4.8 we have wjf(t + b2) ~ e:,m) - e+:-b1 ), hence,

and am -2(w",/F) ::; b1b2. 0

The proof of conjecture (2) for linear system is based on the following result by Goodearl (see [Go075], and [Go078]):

5.6.8. THEOREM. Let a 6.-ring R be semisimple and left and right Noetherian. Suppose that M is a nonzero left 6.-R-module, finitely generated as a R-module. Let DR denote the ring of linear differential operators over R. Then pdDR M = Card(6.) + pdR M.

5.6.9. THEOREM. (See [J078].) Let F be a differential field, 6. its basic set of derivation operators, ~ a system of m linear differential equations in m indeterminates. If the 6.-type of the ideal [~] is less than m, then W[l:] = O.

PROOF. By Johnson's theorem 5.2.5, W[l:) = XOa/:F' where nO/ F is the module of differentials of 6.-extension 9 over F, given by the system ~. The hypotheses of the theorem imply the existence of the following exact sequence of Do-modules (where Do denotes the ring of linear differential operators over g):

(5.6.7)

(here Dg is a free Do-module with m generators). Suppose that nO/ F 1= 0 and the differential type of nO/ F does not exceed m - 2.

By Theorem 5.6.3(2), we can suppose that nO/ F is a finitely generated 6.1-vector space over g, where 6.1 = {dl, ... ,dm- 2 }. Let R be the subring of Do, generated by 9 and d1 , ... , dm - 2 . Since R is the ring of linear 6.1-operators over g, nO/ F

is a finitely generated left R-module. At the same time R is a differential 6.2-

ring, where 6.2 = {dm- 1, dm }, if the derivation operators dm- 1, dm act by the rule: d ·( dr, drm - 2 ) - d·( ) J7'1 drm - 2 & • - 1 9 d N • a 1 ... m-2 - I a a1 .. , m-2 lor, - m- , m, a E an Tl, ... , Tm-2 E 1'1.

Applying Theorem 5.6.8 to Rand nO/F, we see that pdDa nO/F = 2+pdR nO/F. At the same time the exact sequence (5.6.7) implies pdDa nO/F ::; 1. This contradiction proves Theorem 5.6.9. 0

5.7. Minimal Differential Dimension Polynomial

Let F be a 6.-field, 9 = F(1/1,"" 1/n) a finitely generated 6.-extension of F. By Theorem 5.4.1, the Kolchin differential dimension polynomial Wl)l .... ,I) .. (t) is equal


to E'l:l WEi (t), where WEi belong to the set of all Kolchin polynomials W, see Section 2.4. Proposition 2.4.13 shows that this polynomial also belongs to W. By Proposition 2.4.14, W is well-ordered with respect to the order introduced in Definition 2.4.1.

5.7.1. DEFINITION. A polynomial wTJ1 •...• TJ,,/:F(t) is called minimal differential dimension polynomial for a L\-extension 9 = :F(7]I, ... , 7]n), if for any system of L\-generators 9 = :F("Ij;l,.·., "lj;k) we have WV>l •...• V>k/:F(t) ~ wTJ1 •...• TJ,,/:F(t). In this case the polynomial wTJ1 •...• TJ,,/:F(t) will be denoted by wg/:F(t).

5.7.2. THEOREM. For any finitely generated L\-extension 9 ofa L\-field:F there exists a minimal differential dimension polynomial, which will be denoted by Wg /:F.

PROOF. The theorem follows from Propositions 2.4.13 and 2.4.14. 0

In Section 5.6 it was noted that the differential dimension polynomial is not a L\birational invariant of an extension of a L\-field. Therefore the notion of a minimal differential dimension polynomial is not trivial.

5.7.3. EXAMPLE. Let 9 = :F(7]I, ... ,7]k), L\ = {dl, ... ,dm } and ~ij = diT/j (i = 1, ... , m, j = 1, ... , n). Then 9 = F(~ij)1~i:::::;:', and it is easy to see that

the filtration g. = :F(T.~ii ){~i:::::::. of 9 associated with generators eij, is obtained from g. = F(T(s)7]j)j=I •.... n with the help the shift by 1, therefore, Wf,i,/:F(t) = wTJ1 •...• TJ,,/:F(t + 1) and if degwTJ1 •...• TJ,,(t) > 0 then W{ij(t) > wTJ1 •...• TJ,,(t).

Usually, it is a difficult problem to prove that a polynomial wTJ1 •...• TJ,,/:F(t) is minimal for an extension 9 = :F(7]I, ... , 7]n). We prove the following result.

5.7.4. THEOREM. [MP801 Let the cardinality IL\I = m be fixed. Consider L\extensions of the type m -1 and of typical L\-dimension r. The polynomial e"!;.m) -e+:- r ) is minimal in the set of L\-dimension polynomials for such extensions.

PROOF. Let wE W, degw = m - 1, am-dw) = r. By Corollary 2.4.6, we may assume that W = WE for some set E C Nm . Applying formula (2.2.3) to E and (rl, .. . ,rm ), where rj = minl9Sneij, we have

(t+m) (t+m-r) (~) w(t) = m - m + WH t - f;t rj ,

therefore, w(t) ~ e"!:') - e+:- r ). 0

In this section we consider linear differential equations. If E C :F {YI, ... , Yn} is a system of such equations, and [E] ::J F {Yl, ... , Yn} then, by Proposition 3.2.28, the ideal [E1 is prime, therefore, it defines a L\-extension 9 of :F. As before, let Dg denote the ring of linear L\-operators over g. In this case, the module of differentials Og/:F is a factor-module of a free Dg-module with generators OYI, ... , 0Yn with re-spect to the submodule H, generated by of (F E E), of = F(OYI, ... , 0Yn), where F is obtained from F by omitting its constant term (linear differential equations without a constant term will be called homogeneous). Let G be a Grobner basis of Hand Ej C Nm, 1::; j ::; n be the matrix whose rows are the m-tuples (il, ... , i m )


such that d~1 ..... d!;;'tSYj is the leader of an element from G. Then, by Theorems 4.3.5 and 5.2.5, W[E)(t) = 'L.;=l WEj(t). . The simplest case when we can prove the minimality of a differential dimension

polynomial is an extension given by a single linear differential equation for one generator.

5.7.5. EXAMPLE. Wave equation:

Here the submodule H is generated by a single element with the leader (the standard ranking is used) d~tp, therefore,

( t + 4) (t + 2) (t + 3) (t + 2) w'P/:F(t) = W2.0,O,O(t) = 4 - 4 = 2 3 - 2 .

By Theorem 5.7.4, this polynomial is minimal for the A-extension 9 = :F(tp).

Consider now examples where a differential dimension polynomial WI/I ..... I/ .. (t) computed for some generators is not minimal.

5.7.6. EXAMPLE. Let:F = C(XI,X2), A = {dl = fJ/fJxI,d2 = fJ/fJx2}, 9 be a A-extension of:F given by the following system:

{ dltpl + d2tp2 = 0,

d l tp2 = o.

If tp2 < tpl, then these equations form a Grabner basis, therefore,

W'PI'P./:F(t) = WlO(t) + WlO(t) = 2(t + 1).

If tp2 > tpl then we obtain a Grabner basis after adjunction the equation d~tpl = 0 to the original ones, therefore,

Consider the element 1/; = tpl + XI(tp2 + d2tp2) E 9 = :F(tpl, tp2)' It is easy to see that d l 1/; = tp2, therefore,1/; forms a A-basis of 9 over Y. Since d~1/; = 0, we have w.p/:F(t) = W(20)UE(t) for a matrix E (may be empty), whence, w.p/:F(t) ~ W20(t) = 2t + 1. On the other hand, by Theorem 5.7.4 we have w.p/:F(t) 2: 2t + 1. Hence, w.p/:F(t) = wQ/:F(t) = 2t + 1.

5.7.7. EXAMPLE. Let :F be the same A-field as in Example 5.7.6, and the A-extension 9 be given by the system

{ dld~tp = 0, XIX2d~d2tp - xld~tp + (1 + x2)dl d2tp - dltp = O.

These equations form a Grabner basis, therefore, w'P/:F(t) = We~) (t) = 2(t + 1).

Let 1/; = tp - XIX2(dl tp - X2dld2tp). This change of variables is invertible, since


<p = (1 + X1X2d1)¢. However, d1d2¢ = 0, hence, W",/F(t) = WQ/F(t) = 2t + 1 (the proof is similar to the one of Example 5.7.6).

Note that, by Shanuellemma (see [Ka69, p. 167]), the left ideal J, generated in the ring of differential operators D = g[d1 , d2] by the left-hand sides of the equations in Example 5.7.7, is projective (since D/J ~ D/Dd1d2 ). At the same time, J is not a free module, because otherwise its minimal Grabner basis consists of a single element and Kolchin dimension polynomial Wcp/F(t) is equal to e12) - G) = 2t + l. Note that the minimal differential dimension polynomial may have a form different from given in Theorem 5.7.4.

5.7.8. PROPOSITION. IfwQ/F(t) = e-;;,m) - e+:-r ), where r EN, m> 1, then Ll-module OQ/F is isomorphic to the Ll-module D/ D)" for some).. E D. (Here D is the ring of linear differential operators over g.)

PROOF. Let WQ/F(t) = w'll, ... ,'ln (t). Then

~ (t+m) (t+m-r) WQ/F(t) = L.."WEj(t) = m - m . )=1

Setting ej to be the vector of minimal elements of rows of Ej, and applying formula (2.2.3) to each matrix Ej, we obtain

where r = 2::j=l rj, and w(t) = 2::j=l W Hj (t - rj) for some matrices Hj. Since W Hj

is Kolchin polynomial of degree :S m - 1, W ~ 0 and deg W < m - l. It is easy to see that if rj =J. 0, rj =J. 0 for some i =J. j, then

therefore, we can assume that WQ/F(t) = w'l(t), i.e. n = l. Consider the D-module homomorphism D -+ OQ/F -+ 0 such that 1 -+ !ST}. It is clear that for the kernel J of this mapping we have XJ(t) = C+:- r ) (where XJ is the characteristic polynomial of J); it is possible only if its minimal Grabner basis consists of a single element, i.e. J = D)". 0

Thus, the hypothesis of Proposition 5.7.8 implies that the projective dimension (p.d.) of OQ / F is equal to l. If m > 1, then there exist nonprojective ideals in D;

they define extensions for which WOOI:F (t) =J. c-;;,m) - e+:- r ).

5.7.9. EXAMPLE. Let a Ll-extension of:F be given by the system of differential equations

(m = 2).

Then wOOlF (t) = 2t + 2.


PROOF. It is easy to see that wtp/:F(t) = W (~ ~) (t) = 2t + 2. The inequality

wOOlF (t) 2: 2t + 1 follows from Theorem 5.7.4. Suppose that WOOlF (t) = 2t + 1. Let 'l/J = dld20tp E ng/:F, H = D'l/J be the ~-submodule of the module of differentials generated by 'l/J. A filtration on ng/:F associated with a system of generators induces an excellent filtration on H (see Theorem 4.3.40). Since dl'l/J = d2'l/J = 0, we have XH(t) == 1. Then XO/H(t) = (2t + 1) - 1 E Wand this contradicts Proposition 2.4.10. 0

More interesting is the case when ~-type of a ~-extension 9 over :F is equal to m - 1, p.d. ng/:F = 1, but WOOFF i- C"!:') - C+:- r ).

5.7.10. EXERCISE. Let a ~-extension 9 be given by the system

Then wOQIF (t) = 2t + 2.

[Hint: If w1/J/:F(t) = 2t + 1, then by Proposition 5.7.8 >..o'l/J = 0 for some>.. E D, deg>.. = 2. Let o'l/J = Hlotpl + H2otp2, then >"Hl E Ddl , >"H2 E Dd2, therefore, >.. = d1d2+" deg, S; 1. Find a contradiction with the fact that o'l/J is a ~-generator of ng/:F.]

Exercise 5.7.10 gives us an example of a projective ideal in the ring of differential operators D which is not similar to a free one (i.e. DIJ i:. DID>" for any>.. ED). Actually, if we take C(Xl,X2) as an original field, dl = /JjfJx l , d2 = /JI/JX2' then by Theorem 5.3.15 (~-theorem on a primitive element), we have 9 = :F('l/J) for some'l/J E g, therefore, ng/:F ~ DI J ~ (DI Ddt) EB (DI Dd2). At the same time, if D I J ~ DID>" for some >.. E D then the minimal polynomial Wg /:F were equal to 2t + 1.

Note, that if E is a system of linear homogeneous differential equations then, using Algorithm 4.2.5 or Algorithm 4.2.6, we can compute a Grabner basis for this system. Then, with the help of methods given in Sections 2.3 and 4.3, we can find the differential dimension polynomial W[E). Similar algorithms for computation of minimal differential dimension polynomials are now unknown. Contrary to the problem of Grabner basis computation, here the structure of the field :F plays an essential role. We shall prove only, that the set of minimal differential dimension polynomials is large enough: every Kolchin dimension polynomial (an element of W) is the minimal differential dimension polynomial for some ~-extension of a ~-field:F. In the proof we use the ~-theorem on a primitive element (Theorem 5.3.15) and Theorem 4.3.40 asserting that any submodule of an excellently filtered module has an excellent filtration.

Let wmin denote the set of minimal differential dimension polynomials of finitely generated extensions of ~-fields, Card(~) = 1,2,3,.... By Proposition 2.4.13, wmin C W.

5.7.11. LEMMA. Let D be a ring of linear differential operators over a field, J and I be left ideals of D such that I = Ja for some a E D. Then XI(S) = XJ(s - dega) (see Definition 5.1.12; the ideals are considered with the filtration associated with the standard filtration on D).


PROOF. Is = J.-clega· a for all sufficiently large s. 0

5.7.12. THEOREM. W = wmin.

PROOF. By induction on m, for any set (bm , ... , b1) of non-negative integers we shall prove the following statement: let:Fm = C(X1, ... ,Xm), d; = a/ax;, ~ = {d1, ... , dm}, and 9m = :Fm('P1, ... , 'Pm) be a ~-extension of :Fm defined by the following system of linear differential equations

{ d~i'P; = 0 i = 1.,2, .... , m,

dj'Pj = 0 1:S Z < J :S m.

Then there exists '1/1 = 2:::7:::1 Cj'Pj, Cj E :Fm such that 9m = :Fm('I/1), wQm/Fm W1/J/Fm and b(w1/J/ Fm) = (bm, ... , b1).

If m = 1 then WQdF, = w1/JdF = b1. Let m > 1, then by inductive hypothesis, we have :Fm ('PI, ... , 'Pm-I) = :Fm ('1/11)'

where '1/11 = 2:::7:::~1 Ci'Pi, Cj E :Fm-1 and b(w1/JdFm) = (bm- 1, ... , b1). Since 9m = :Fm('Pm, '1/11) and ~-type of9m over:Fm is less than m, by ~-theorem on a primitive element (Theorem 5.3.15), we have 9m = :Fm('I/1), where '1/1 = 'Pm + c'I/11, C E :Fm . Let us prove that b(W1/J/Fm) = (bm, ... ,bI). Indeed, W1/J/Fm(t) = C:m) - XJ(t), where J = {j E D I jtS'I/1 = O}, D is the ring of ~-operators over 9m. It is easy to see, that J = Dd~m n Hc-I, where H = {h E D I M'I/11 = O} and D = Dd~mc + H. Let I = Jd;;,b m = DnHc- 1d;;,bm (recall that D is an Ore ring, see Corollary 5.1.16)' and '1/12 = dt;;; C'I/11. The condition )..'1/12 = 0 for)" E D is equivalent to the condition )"dt;;;c E H, i.e. ).. E I. Therefore, W1/J2/Fm(t) = e;:) - XJ(t), and since C; E :Fm- 1, i = 1, ... , m - 1, we have dm'l/11 = 0, hence, '1/12 = d~m(c) . '1/11. Since the filtration associated with '1/11 coincides with the one associated with '1/12, W1/J2/Fm = W1/JdFm' Using Lemma 5.7.11, we obtain

(t + m) (t + m) W1/J/Fm(t) = m -XJ(t) = m -X/(t-bm)

_ (t + m) (t + m - bm ) (b ) - m - m +w1/J2/Fm t- m

(t + m) (t + m - bm ) = m - m +Wt/JdFm(t-bm).

By Definition 2.4.9, we have b(Wt/J/Fm) = (bm , ... ,b1). It remains only to prove that wQm/Fm(t) = Wt/J/Fm(t). Let 9m = :Fm((h, ... '{h). By Theorem 5.3.15, we have 9m = :Fm((J), (J = a1(J1 + ... + ak(Jk, aj E 9 for all j = 1, ... , k, therefore, W8" ... ,e./Fm(t) = W8/Fm(t) and wQm/Fm = W8/Fm· Suppose that W8/Fm < Wt/J/Fm.

k t+' Then W8/Fm - Wt/J/Fm = 2:::;=0 9;( ; '), 0 :S k < m - 1 and 9k < O. Let 0 1 and O2 denote the sub modules of 0 = Ogm/Fm, generated by tS'Pk+2, ... , tS'Pm and tS'P1, ... , tS'Pk+1, respectively. The generator tS(J of 0 determines an induced filtration on O2 , which is excellent by Theorem 4.3.40, therefore, the degree of the characteristic Hilbert polynomial Xn2 is equal to k, and its leading coefficient is equal to bk+1. Hence, the k-minimizing coefficient of Xn/n2 = W8/Fm - Xn2 is equal to 9k < 0, but by Proposition 2.4.10, b(xn/n2 ) E Nm, since the filtration on 0/02

is excellent. This contradiction proves our statement, hence, Theorem 5.7.12. 0

5.8. JACOBI'S BOUND FOR ALGEBRAIC DIFFERENTIAL EQUATIONS 273

5.7.13. THEOREM. Supposethatwg/:F(t)=de~m),m=CardA. Then for any family 1] = (1]1, ... , 1]n) which A-generates 9 over:F, we have wf//:F(t) = wg/:F(t) iff trdeg:F :F(1]l, ... , 1]n) = d. Defining differential ideal for such point 1] is generated over :F by differential polynomial of order zero. In particular, 9 is an algebraic extension of a purely transcendental differential extension of A-dimension dover :F.

PROOF. The theorem follows from Theorem 5.4.13. 0

Using Theorem 5.4.12, we can give an equivalent definition of wg/:F(t). Let 1] differentially generate 9 over :F and let 1]0 be a subfamily of 1], which forms a Atranscendence basis of 9 over :F, and 1/1 = 1/ \ 1]0. Proposition 5.4.10 and Theorem 5.4.12 show that

and if we define (3c/:F(t) = minf//:F(t), then

( t+m) wC/:F(t) = d m + (3c/:F(t).

5.S. Jacobi's Bound for a System of Algebraic Differential Equations

In Section 5.6 we proved the following estimate: let E C :F {Y1, ... , Yn}, P be a prime component of the A-ideal {E} and differential dimension of p be equal to O. Then am -1(p) ~ EJ=l maxPEE ordYi F. The Jacobi conjecture consists in a refinement of this estimate. In [Ja890] it was stated the conjecture that if the number of ordinary differential equations is equal to the number of indeterminates then some number h (which we shall call the weak Jacobi number for this system) is an upper bound for the "number of arbitrary constants in a general solution of this system". The weak Jacobi number is defined in the following way: let eij be the order of ith equation in jth indeterminate, (eij = 0, if ith equation is free of any derivatives of Yj)' then

h = max(e1a(1) + ... + ena(n)), aES"

where Sn is the symmetrical group of order n. In terms of differential algebra, the Jacobi conjecture was formulated by Ritt

[Rit35], and was generalized to the case of algebraic partial differential equations without the condition of equality of the number of equations and the number of indeterminates by Tomasovic [To76].

Let E = (eij )iEI, j=l, ... ,n be a matrix with entries in the set N U{ -oo}, let

J(E) = sup (t e7r(j)j) EN U {-oo, oo}, 7r j=l

(5.8.1)

where 11' run over the set of all injective mappings from Nn = {I, ... , n} into I. In particular, if E is a square n x n-matrix, then

J(E) = max(el7r(l) + ... + en7r(n)). 7rES"


Let ~ be a set of ~-polynomials over a ~-field .1', and let E* (~) = (eh )FEE, j=l, ... ,n,

where ej... = { ordYi F, if some ~-derivative of Yj is present in F,

J 0, otherwise.

Futhermore, let E(~) = (eFj)FEE,j=l,.,n, where

{ ordy; F, if some ~-derivative of Yj is present in F,

eFj = . -00, otherwIse.

The Jacobi conjecture (the weak Jacobi conjecture, respectively) consists in the following.

Let ~ C .1'{Yl, ... , Yn}A, and p be a prime component of the ~-ideal {~}. If the differential type of p is equal to m - 1, then the typical ~-dimension of p does not exceed J(E(~)) (J(E*(~)), respectively).

The Jacobi conjecture was proved by Ritt [Rit35] in the case when Card ~ = n for ordinary linear differential equations and for n = 2. Lando [Land70] has proved the weak Jacobi conjecture for ordinary differential equations in the case when eij :::; 1. It is known, that any system of ~-equations is equivalent to a system of equations of the order not higher than 1 (see the proof of Lemma 5.6.4)' however, by this transition the weak Jacobi nubmer may increase (contrary to J(E(~))). Lando has proved also that the Jacobi bound cannot be improved, i.e. for any matrix E with J(E) = h, there exists a system ~ C .1'{Yl, ... , Yn} such that E(~) = E, am(p) = 0 and am-l(p) = h. Tomasovic has proved [To76] the Jacobi conjecture for any m, Card ~ in the cases: 1) of linear systems and 2) n = 2. In the general case the conjecture up to now is not proved, and it seems a hard problem, since even a more weak statement, namely, the dimension conjecture, is not proved.

We prove here the Jacobi conjecture in the following special case: m = I~I is arbitrary, Card ~ = n, and all equations in ~ are linear.

5.8.1. LEMMA. If Fl, ... ,Fk E .1'{Yl, ... ,Yn}A, k> n and all ~-polynomials Fj (j = 1, ... , k) are linear homogeneous then there exist ).1, ... ,).k E D not all equal to zero, such that ).lFl + ... + ).kFk = 0 (here D is the ring of ~-operators over the field F).

PROOF. Since the polynomials Fl , ... , Fk are linear and homogeneous, they can be viewed as elements of a free D-module with generators Yl, ... , Yn. Suppose, that the lemma does not hold, then in a free D-module with n generators there exist k > n elements independent over D. Since in D the property of invariance of the basic number (IBN, see [Co71]) holds, it is impossible. 0

5.8.2. LEMMA. Let ~ C .1'{Yl, ... , Yn} be a set of linear differential polynomials over a ~-field .1', Card ~ = n and let the ~-type of the ideal [~] be equal to m - 1, where m = Card~. Then J(E(~)) =f. -00.

PROOF. Suppose that there exists ~, satisfying the conditions of the lemma and such that J(E(~)) = -00. Since am([~]) = 0, there exists F fi. .1', F E ~; therefore, we can find r E fir, r =f. 0, indices 1 :::; iI < h < ... < jr :::; n and injective mapping 7r : {ii, ... , jr} --t ~ such that e"(it)it + e"(h)h + ... + e"(ir)jr =f. -00. Among all possible r pick up maximal and fix r,7r,iI, .. . ,jr. Since J(E(~)) = -00, we have


r =F n. If we change the order of ~-indeterminates then J(E(E)) does not change; therefore, we can assume that it = 1, ... , ir = r. Set Fj = 1l'U) U = 1, ... , r) and ejk = ordYk Fj. Let T' be the subset of E, consisting of F1 , ... , Fr and Til = E \ T'. Then by the choice of r, we have ell + e22 + ... + err> -00 and ordyr+! F = -00

for any ~-polynomial F E Til. By assumption, the ideal [E] is of ~-dimension 0; therefore, there exists FEE

in which a derivative of Yr+1 is present, since otherwise the characteristic set of [E] contains no element with the leader 9Yr+1, 9 E T and by Theorem 5.4.1 degw[I:] = m. Let Eo denote the subset of T', consisting of elements in which derivatives of Yr+l are present; it is clear, that ordYr+1 F = -00 for any FEE \ Eo.

Let us consider now the subsets EI ~ E, satisfying the following three conrlitions:

(1) EI ~ T'; (2) ordYr+1 F = -00 for any FEE \ E1 ;

(3) for any k such that Fk E EI there exist h ~ 1 and distinct indices 1 < iI,·· ·,ih $ r such that ih = k, Fj; EEl for i = 1, .. . ,h -1 and

a) ordYr+1 Fil =F -00; b) ordyi; Fj;+l =F -00 for all i = 1, ... , h - 1.

It is obvious that Eo satisfies conditions (1)-(3) (with h = 1 in (3)); therefore, there exists a set E' =F 0 maximal (with respect to the cardinality) among the sets satisfying these conditions.

We are going to show that for any k such that Fk E E' wt:. .:a.ve ordYk F = -00

for all ~-polynomials FEE \ E'. Suppose the contrary, i.e. Fk E E', however, there exists FEE \ E' such that

ordYk F =F -00. In this case condition (3) holds; therefore, there exists a sequence Fi>, ... , Fjh = Fk such that eilr+l + eilii + ... + ekjh_l > -00.

If F rt. T', then (ordYk F =F -00) ei!r+l + ej,jl + ... + ekjh_l + ordYk F > -00.

Consider a mapping ), : {I, ... , r, r + I} -+ E such that

{ ),(~l) = Fj" ... , ),(jh-l) = Fjh = Fk ),(Jh) = F, ),(r + 1) = Fjl

),U) = Fj otherwise

The mapping), is injective, and since eli > -00 for all i = 1, ... , r we have

e>.(l)l + ... + e>'(r)r + e>'(r+l)r+1 > -00,

that contradicts the maximality of r. Therefore, F E T' . Thus, F = Fj for some i, where 1 $ i $ rand i =F it, ... , ih (since we supposed

that Fj rt. El, and Fjp ... , Fjh E E' by (3)). Since Fj E T', the set E' U {Fj} satisfy the conditions (1)-(3) (in (3) we have to take ii, ... , ih, i as the set of indices, a) and b) follow from the assumption ordYk Fj =F -00). That contradicts the maximality ofE'.

Thus, for any k such that Fk E E', any ~-polynomial in E \ E' is free of any derivative of Yk.

Let Card E' = t, E' = (Fill"" Fi,). Then for any FEE \ E', we have ordy;, F = -00 for i = 1, ... , t and ordYr+1 F = -00 (by (2)); therefore, we can


assume that after reordering of ~-indeterminates, E \ E' C .1'{YI, ... , Yn-t-!l; at the same time Card(E \ E') = n - t. By Lemma 5.8.1, E is ~-independent over .1', and by Proposition 5.2.12, am ([E)) ::j:. O. 0

5.8.3. REMARK. The first problem on the way to the Jacobi bound is a proof of the fact that J(E(E)) is finite. Therefore, we can formulate the following conjecture:

Let E be a subset of .1'{YI, ... , Yn}. If {E} has a component of ~-dimension 0, then J(E(E)) > -00.

Tomasovic [To76] have proved that for some classes of systems this conjecture is equivalent to the dimension conjecture of R.M. Cohn:

Let E be a subset of .1'{YI, ... , Yn}. Any component of {E} is of differential dimension not less than n - Card E.

Ritt proved the first of this conjectures for linear differential polynomials, Tomasovic proved the second one for n = 2 and for linear systems. The conjecture of R. Cohn is of interest, since its negative solution will give the negative solution of the Jacobi conjecture.

Let E = (eij) be a square n x n-matrix with elements in IZ U {-oo}. Any sum of the form e".(I}1 + ... + e".(n}n, where rr E Sn will be called diagonal sum. In these terms the Jacobi number J(E) of E is equal to its maximal diagonal sum J(E) = max".ESn E~=I e".(i}i· (see (5.8.1)).

Consider the following transformations of E:

(1) interchange two rows; (2) interchange two columns; (3) add an integer to every element of a column.

Obviously, the transformations of the form (1) and (2) commute with (3), thus we can assume that we interchange rows and columns, and then transform E into an nxn-matrix E' = (e~j)' where e:j = eij + aj, aj E IZ.

5.8.4. LEMMA [To76]. Let E = (eij) be an nxn-matrix and J(E) ::j:. -00. Using transformations (1)-(3) we can reduce E to the form where the diagonal entry in every row will be maximal, i.e. eii ~ eij for all 1 ~ i, j ~ n.

PROOF. First of all note, that interchanging rows we can achieve that J(E) = ell + e22 + ... + enn · Indeed, let J(E) = e"'(I}1 + e"'(2}2 + ... + e".(n}n. After interchanging of the first row of E with 1I"(1)th, the second with rr(2)th, ... , nth with rr(n)th, we obtain E whose sum ell + e22 + ... + enn is maximal among all its diagonal sums. By assumption eii ::j:. -00 for i = 1, ... , n.

We shall prove the lemma by induction on n. For n = 1 the original matrix E has the required form.

If n = 2 then in E = (ell e 12 ) we have ell + e22 > eI2 + e21. If eI2 = -00 e2I e22 -

and e2I > e22, then it suffices to add a2 = e2I - e22 to the second column. If eI2 ::j:. -00 and eI2 > ell, then it suffices to add al = eI2 - ell to the first column (e22 ~ e2I + (e12 - ell) and ell + al ~ eI2).

Finally, let eI2 ::j:. -00, eI2 ~ ell, and e21 > e22. Add a2 = e2I - e22 to the second column; since ell ~ eI2 + a2 the obtained matrix has the required form.

Let n > 2 and the lemma hold for square matrices of order less than n. By inductive assumption we can using transformations of the form (1)-(3) over the


first n - 1 rows and n - 1 columns to achive that eii = maxI ~j ~ n - d eij} for i = 1, ... , n - 1. Now pick up a number a ? ekn - ekk for all k = 1, ... , n -_1 and add this a to all (n - 1) first columns. Then iii = ei; + a ? eii + ein - eii = tin for 1 < i < n-1j therefore, in the new matrix eii = maX1~j::;n{eij} for i = 1, ... , n-1. -We~upposed that J(E) = ell + ... + enn in the original matrix. This property

is stable under transformations of the form (3) because if E' = (eij) = (eij + aj) is an n x n-matrix, and

J(E') = e,,(1)1 + ... + e,,(n)n

= e,,(l)l + al + err (2)2 + az + ... + Crr(n)n + an

> e"'(1)1 + i"'(2)2 + ... + e".(n)n

= ell + ... + enn + al + ... + an,

then contrary to our assumption we have e".(1)1 + ... +e11'(n)n > e11 + .. ·+enn · Since transformations (1)-(2) commute with (3) we have eii ~ eij, i = 1, .. . ,n -1, j = 1, ... , n, and J(E) ::::: e"'(1)1 + ... + e11'(n-1)n-1 + enn , 1'( E Sn-1 (since the last row and column was not interchanged). Now we build E, whose first row coincides with the 1'((l)th row of E, ... , (n - l)th row coincides with the 1'((n - l)th row of E, and nth of E coincides with nth row of E. Thus, we transformed E into the form where Cii ? eij for i = 1, ... , n - 1, j = 1, ... ,n and J (E) = ell + ... + enn .

Let 0:; = ej; - ein (i = 1, ... , n - 1). If O:i ::::: -00 for all 1 :S i :S n - 1, then ein ::::: -00 for i ::::: 1, ... , n - 1; therefore, to obtain the required form of E it suffices to add a number greater then enk - enn (1 :S k :S n - 1) to the last column of E. Let b ::::: min1<i<n_da; 10:; f -oo}, b E IZ. Add b to the last column, then fin = ein + b :S tin + ai ::::: ein + ej; - ein = eii for i ::::: 1, ... , n - 1 and fkn = ekn + b ::::: ekk for some 1 :S k :S n - 1. Interchange the kth column with the first one, and the first row with the kth one. We obtain E such that

{ ei;:::::maxl5j::;n{eij} (i::::: 1, ... ,n-l)

ell = eln

J(E) ::::: ell + e22 + ... + enn

From this conditions and the inequality

eln + enl + en + ... + en -l,n-1 :S J(E) = ell + en + ... + enn

(the left-hand side is a diagonal sum for E) it follows that

(5.8.2)

(5.8.3)

If all the elements en 2, ... , en n-l do not exceed enn , then the matrix E has the form required. '

Suppose that at least one of the elements enj (2 :S j :S n - 1) is greater than enn · Let Tn (E) denote the quantity"" "2<'< -1 .> (en]' - enn ). By induction WJ. _J_n ,enJ_e nn

on Tn(E) we shall show that E may be transformed to the form required. If Tn (E) = 0, then E has the required form.


Let Tn{E) > 0 and the lemma hold for matrices satisfying (5.8.2) with Tn < Tn{E). Since Tn{E) > 0, there exists j (2 ~ j ~ n - 1) such that enj > enn . Interchange the jth column with the 2nd and the jth row with the 2nd. Thus, we suppose that we have (5.8.2), (5.8.3) and also

(5.8.4)

Since e2I + ein + en2 + e33 + ... + en-I,n-I ~ en + e22 + ... + enn , ein = en and e2n + en2 + ell + e33 + ... + en-l,n-I ~ en + e22 + ... + enn the inequality (5.8.4) implies e2I ~ e22 + (enn - en2) < e22, e2n ~ en + (enn - en2) < e22

Add (-I) to the second column of E:

C' e12 -1 eI3 'In) e21 e22 -1 e23 e2n

E'= .

eni en2 -1 en3 enn

Consider several cases. 1. If e2j < e22 for j = 3, ... , n -1, then E' satisfies (5.8.2) and Tn{E') < Tn(E).

By inductive assumption in this case the lemma holds. 2. Suppose that E' does not satisfy (5.8.2); in this case some elements among

e2j (3 ~ j ~ n - 1) are equal to e22. Interchanging columns (and rows) we can achieve that e23 = ... = e2p = e22, e2,p+1 < e22, ... , e2,n-1 < e22 (3 ~ p ~ n - 1) and the maximal diagonal sum is, as before, ell + e22 + ... + enn . In this case a new matrix E" is obtained by adding (-I) to the columns with indices 3, ... ,p:

ell eI2 - 1 e13 - 1 elp - 1 el,p+l ein e21 e22 - 1 e23 -1 e2p - 1 e2,p+l e2n

E" = epl ep2 - 1 ep3 -1 epp - 1 ep,p+1 epn ep+I,1 ep+l,2 - 1 epH,3 - 1 ep+l,p - 1 ep+I,p+1 ep+I,n

enl en2 - 1 en3 -1 enp - 1 en,p+1 enn

If 3 ~ i ~ p, then we have e2i = e22, eit + ein + en2 + e2i + e33 + ... + ei-I,i-I + ei+l,i+1 + ... + en-I,n-l ~ J{E) = en + e22 + ... + enn , ein = enn and ein + en2 + e2i+en +e33+·· +ei-I.i-I +ei+l.iH + .. ·+en-I.n-I ~ J{E) = en +e22+·· +enn ; therefore, eit ~ e;i + (enn - en2) < eij, ein ~ ei; + (enn - en2) < eij, for i = 3, ... , p.

Again we have two possibilities. 2.1. If eij < ei; for all i = 3, ... , p, j = p + 1, ... , n - 1, then E" satisfies (5.8.2)

and Tn{E") < Tn{E). By inductive assumption in this case the lemma holds. 2.2. Suppose that the conditions of 2.1 are not satisfied. After transformations

of the form (1)-(2) we obtain a matrix, where these conditions do not satisfied for the columns with indices p+ 1, ... , q, i.e. for every column with index p+ 1 ~ j ~ q there exists i such that 3 ~ i ~ p and eii = eij. Add (-I) to the columns of E"


with indices p + 1, ... , q. We obtain

e"p+1-1 '2,p+l -1 e3,p+l -1

el,q+l

e2,q+l

e3,q+l

epl ep2-1 e p3-1 epp-l ep,p+l-1 epq-l ep,q+l epn

EIII = 'p+',' 'p+l,2- 1 'p+l,3- 1 ... 'p+l,p-1 'p+l,p+1-1 .. 'p+l,q-1 'p+',q+' ... 'p+l,n

eqi eq2-1 eq3-1 ' .. eqp-l eq,p+l-l ... eqq-l e q,q+l e qn

eq+l,l eQ+l,2-1 eq+l,3-1 ". eq+l,p-l eq+l,p+l-l ", eq+l,q-l e q +l,q+l ... eq+l,n

Again we prove that

(5.8.5)

for all k = p + 1, ... , q. Indeed, if p + 1 ::; k ::; q, then find i = i( k) such that 3 ::; i ::; p and eii = eik.

This equality together with e1n = ell, e2i = en and

ek1 + el n + en 2 + e2i + eik + e33 + ... + eii + ... + ekk + ... + enn

::; ell + e22 + ... + enn ;

ekn + en 2 + e2i + eik + ell + e33 + ... + eii + ... + ekk + ... + enn ::; J(E)

where e means that the term e is omitted, give us (5.8.5) (since k fI. {I, 2, i, n}). We have two possibilities again. 2.2.1. If eij < eji for all i = p + 1, ... , q, j = q + 1, ... , n - 1, then we use the

inductive assumption. 2.2.2. If some inequalities in 2.2.1 do not hold, then we use the same scheme,

i.e. we transform them in equalities in the columns with indices q + 1, ... , r, where q + 1 ::; r ::; n - 1. We add (-1) to each of these columns. Comleting the sum ejl + eln + en 2 + e2i + eik + ekj to a diagonal one (q + 1 ::; j ::; r, ekj = ekk) we obtain ejl < ejj and ejn < ejj. After a finite number of steps we obtain a matrix satisfying conditions (5.8.2) and such that Tn < Tn(E). By inductive assumption the lemma is proved. 0

5.8.5. COROLLARY. Let E be an n x n-matrix with entries in N U {-oo} and J(E) f. -00. Then there exist integers al, ... , an E N and a matrix E' = (eij),

obtained from E by interchanging of rows and columns such that eij + aj ::; eii + ai

for all 1 ::; i, j ::; n.

PROOF. By Lemma 5.8.4 there exist integers bl , ... , bn E ;z. and a matrix E' satisfying the condition of the corollary. In order to achieve that the transormations (3) were with nonnegative parameters it suffices to replace b1 , .•• , bn by a1 = b1 + lal, ... , an = bn + lal, where a = minl:::;i:::;n bi. 0


5.8.6. THEOREM (JACOBI BOUND FOR LINEAR SYSTEMS). Let

be a set of linear t:l.-polynomials over a t:l.-field :F, Card t:l. > 1, and the differential dimension of[E] be equal to O. Then am-dW[E]) S J(E(E)).

PROOF. By Theorem 5.2.5, we have W[E] = XOQ / F , where r2g / F is represented as a D-factormodule of a free module F with n generators dYl, ... ,dYn with respect to the submodule H, generated by dFl , ... ,dFn (D is the ring of t:l.-operators over the field Q). We introduce another excellent filtration on F / H, which is not associated with a choice of generators dYl, ... , dYn. Consider the matrix E(E) of orders of E. By Lemma 5.8.2, we have J(E(E)) i- -00. After changing the order of the variables Yl, ... , Yn and the polynomials F l , ... , Fn, we can find (by Corollary 5.8.5) integers al, ... , an EN such that the n x n-matrix E(E) = (eij) will satisfy the condition eii + ai ~ eij + aj (1 S i, j S n). Let bi = eii + ai (i = 1, ... , n) and

It is clear, that F = U.F, is an excellent filtration on F, whose characteristic polynomial XF(t) is C+:- a ,) +- .. + (t+m

m- a .. ). On H we have the induced filtration H •. Consider the following filtration on H:

(Note, that am(XF/H) = 0 implies that H is a free submodule of F with generators dFl , ... , dFn.) We shall prove that fl, ~ H. for sufficiently large s. Indeed, let f E fl., then f = AldFl + ... + AndFn, where Aj ED, ord Aj S s - bj . Therefore, f = OldYl + ... + ondYn, where OJ = 2:7=1 Ai/ij, ord'Yij = eij· Hence, ordoj S max( ord Ai + eij) S s - bi + eij = s - eii - ai + eij S s - eij - aj + eij = s - aj,

and f E H •. The filtration fl. is excellent and since fl, C H" we have Xii S XH. By Proposition 5.2.9 (Theorem on t:l.-invariants of the characteristic polynomial of a differential module), we have

= (bl - ad + ... + (bn - an)

= ell + ... + enn

S J(E). 0

CHAPTER VI

DIMENSION POLYNOMIALS IN DIFFERENCE

AND DIFFERENCE-DIFFERENTIAL ALGEBRA

6.1. Characteristic Polynomials of Graded Difference Modules

Let R be a difference ring with a basic set u = {aI, ... , an} and let T = Ta be a free commutative semigroup generated by the elements aI, ... , an. As in Section 3.3, by the order of an element T = a~l ... a~n E T (k1l ... , kn EN) we shall mean the number ordT = 2:7=lk; and set Tr = {T E T I ordT = r}, T(r) = {T E T lord T ::; r} for any r EN. Furthermore, let U be a ring of difference (u-) operators over the ring R. As in Chapter 3, if f = 2:"'ET a ... T E U (a ... E R for any T E T and a ... = 0 for almost all T E T), then the number ord f = max{ ord T I a ... -:j:. O} will be called the order of the element f.

In this section the ring of difference operators U will be regarded as a graded one: U = ffiqEN U(q) where U(q) (q E N) is the R-module of all elements of the form 2: ... ET. a ... T. By a graded u-R-module (or by a graded difference R-module) we

shall mean a left graded U-module M = ffiqEN M(q).

6.1.1. REMARK. Let A be a ring (not necessarily commutative) and let M, N be two left A-modules. Let a be an injective endomorphism of the ring A and let 0: M ~ N be an additive mapping such that o(ax) = a(a)o(x) for any a E A, x E M. It is easy to see that Kero is an A-submodule of M, Imo is an a(A)-submodule of N, and if N' is an a(A)-submodule of N, then 6- 1(N') is an A-submodule of M. Therefore, if the mapping 6 is a bijection and N is irreducible as an a(A)-module, then M is irreducible as an A-module.

Let A be an Artinian ring and P a finitely generated A-module. In what follows, the length of the module P (i.e., the length of any its composition series) will be denoted by lA(P).

6.1.2. LEMMA. Let R be an Artinian commutative ring, a an injective endomorphism of R and let

O~K.2tM!..tN~O be an exact sequence of finitely generated R-modules, where 1 is an injection and 6 is a mapping satisfying the conditions of Remark 6.1.1. Then N is a finitely generated a(R)-module and

(6.1.1)

PROOF. If M = 2:;=1 Rx;, then N = 6(m) = 2::=1 a(R)6(xi), hence the elements 6(x1), ... ,6(x.) generate the a(R)-module N, if the elements X1, ... ,X.

281


282 VI. DIMENSION POLYNOMIALS IN DIFFERENCE-DIFFERENTIAL ALGEBRA

generate R-module M. Now, let K = Ko :::> Kl :::> ... :::> K t = 0 and N = No :::> Nl :::> .•. :::> Np = 0 be composition series of the R-module K and of the a(R)module N, respectively. We are going to show that

(6.1.2)

is a com position series of the R-mod ule M. At first, we note that 5- 1 (Nj) (1 ::; j ::; p) are R-submodules of M, (see Remark 6.1.1) and show that every module Lj = 5-l(Nj_d/5-1(Nj) is irreducible. Consider the mappings5j : Lj -t Nj_l/Nj (1::; j ::; p) such that 5j(~+5-l(Nj)) = 5(~)+Nj for any ~+5-l(Nj) E Lj (~ E Nj- 1 ). It is easy to see that the mappings 5j are correctly defined, additive and bijective, and that 5(a() = 5(a~) + Nj = a(a)5j (() for all (= ~ + 5- l (Nj) E Lj, a E R. Because a(R)-modules Nj_I/Nj are irreducible, Remark 6.1.1 shows that the modules Lj

(1::; j ::; p) are also irreducible. Furthermore, since each R-module z(Kr-d/z(Kr) (1 ::; r ::; p) is irreducible (because of the irreducibility of the R-module Kr-I/ Kr which is naturally isomorphic to z(Kr-dlz(Kr )), it follows that the chain (6.1.2) is a composition series of the R-module M, so lR(K) + la(R)(N) = lR(M).

6.1. 3. THEOREM. Let R be an Artinian difference ring with a basic set (j = {aI, ... , an} and let M = EeqEN M(q) be a finitely generated graded (j-R-module. Then the following statements hold:

(1) the length lR(M(q)) of every R-module M(q) (q E 'N) is finite; (2) there exists a numerical polynomial t.p(t) in one variable t such that t.p(q) =

lR(M(q)) for all sufficiently large q E 'N. The degree oft.p(t) is less than or equal to n - 1.

PROOF. (1) By Proposition 3.3.80, each module M(q) (q E 'N) is finitely generated over the ring R. Since R is an Artinian (hence, a Noetherian) ring, M(q) is an Artinian and Noetherian module, whence lR(M(q)) < 00.

(2) To prove the second statement of the theorem we will use induction on n = Card (j. If n = 0 then the ring of (j-operators U over the (j-ring R coincides with R. In this case, if Yl, ... , Ym is a system of homogeneous generators of an U -module M, then the degree of any nonzero homogeneous element of M is equal to the degree d(Yi) of some generator Yi (1 ::; i ::; m). Therefore, M(q) = 0 and lR(M(q)) = 0 for all q > maXl<i<m d(Yi), hence for n = 0 our statement is valid.

Now, let Card(j = n ((j = {~l~ ... ' an}) and suppose that the statement is valid for all (j-rings with Card (j = n - 1. Let (j' = {al,.·., a n - d, U' be the ring of (j'-operators over R (regarded as (j'-ring), (; the ring of (j-operators over the (j-ring an(R), and (;, the ring of (j'-operators over the (j'-ring an(R). We shall consider each of the rings U', (;, (;, as the graded one whose grading is of the same type as the grading of the ring of (j-operators U over the (j-ring R introduced at the end of Section 3.3.

Let B be the mapping of a finitely generated graded (j-R-module M = EeqEN M(q) into itself such that B(x) = anx for any x E M. Denote K = KerB, N = 1mB. Then K is a graded U-module and N is a graded (;-module: K = EeqEl\i K(q) and

N = EeqEN N(q), where K(q) = Ker(BIM(q)) (q E 'N) and N(q) = B(M(q)) (q E 'N) for all q E 'N. Since the sequence

o -t K(q) -4 M(q) !t N(q) -t 0

6.1. CHARACTERISTIC POLYNOMIALS OF GRADED DIFFERENCE MODULES 283

(q EN and II is an injection) satisfies the conditions of Lemma 6.1.2. (with 0 = (}), we have

(6.1.3)

Let L(q) = R(}(M(q») for every q E N. It is evident that L(q) ~ M(q+l) and

Clj(R)O(M(q») ~ Cli(R)O(M(q+1») ~ R(}(M(q+1») = L(q+1)

(q EN; i = 1, ... , n),

so we can consider the graded U-module L = EDqEN L(q). Let A(q) = M(q+l) / L(q)

(q EN). For every i = 1, ... , n and for every q E N the mapping A(q) -+ A(q+l) such that {+ L(q) I-t Cli{ + L(q+l) for any element {+ L(q) ({ E M(q+l») is correctly defined and may be regarded as an action of the element Cli of the basic set u on the graded u - R-module A = EDqEN A(q). Since M(q+1) (q EN) is a finitely generated R-module, it follows that the R-modules L(q) and A(q) are also finitely generated. Therefore, lR(L(q») < 00, LR(A(q») < 00 and

(6.1.4)

for all q E R Now, let us consider the graded V-module (}(L) = EDqEN Cln(R)(}2(M(q»). Let

Oq be the restriction of the mapping (J on the R-module L(q) and B(q) = Ker (Jq (q EN). Then B(q) ~ L(q) = RO(M(q») ~ M(q+1). Furthermore, and by Lemma 6.1.2, the exactness of the sequence

implies that the lengths of homogeneous components of the graded U -modules B = EDqEN B(q), L = EDqEN L(q) and of the graded V-module (J(L) satisfy the equation

IR(L(q») = IR(B(q») + lorn(R) (Cln (R)(J2(M(q»)). (6.1.5)

Letting C(q) denote the finitely generated Cln(R)-module

we see that the mapping C(q) t-t c(q+1) such that {+ Cln(R)(J2(M(q») t-t Cli{ + Cln(R)(J2(M(q+l») (1:::; i :::; n, q E N,{ E N(q») is correctly defined and may be regarded as an action of the element Cli on the graded V -module C = EDqEN C(q). Furthermore, the exactness of the canonical sequence of the finitely generated Cln(R)modules

implies that

(6.1.6)


Combining equalities (6.1.3.)-(6.1.6), we obtain

lR(M(q+1)) -IR(M(q)) =IR(L(q)) + lR(A(q)) -IR(J{(q)) -lan(R)(N(q))

;;;;::[R(B(q)) + (lan(R)(N(q)) -lan(R)(C(q))]

+ lR(A(q)) -IR(J{(q)) -lan(R)(N(q)),

whence

lR(M(q+1)) -IR(M(q)) =IR(A(q)) + lR(B(q)) -lan(R)(C(q)) -IR(J{(q)) (6.1.7)

Now, we note that the graded U -modules A, B, J{, and the graded 0 -module C are annihilated by the multiplication by O'n. Therefore, A, Band J{ are finitely generated graded U'-modules and C is a finitely generated graded O'-module. By the inductive hypothesis, there exist numerical polynomials ipdt), ip2(t), ip3(t), and ip4(t) in one variable t such that ip1(t) = lR(A(q)), ip2(t) = lR(B(q)), ip3(t) = lan(R)(C(q)), ip4(t) = IR(J{(q)) for all sufficiently large q E N, and deg ipi(t) :S n- 2 (i = 1,2,3,4). Let ipo (t) = ipdt)+ip2 (t) -ip3(t) -ip4 (t). Then deg ipo (t) :S n- 2 and equality (6.1.7) shows that ipo(q) = lR(M(q+1)) -IR(M(q)) for all sufficiently large q EN, i.e. for all qo EN, q ~ qo. Since

q-1 IR(M(q)) =IR(M(qo)) + L(lR(M(i+1)) -IR(M(i))]

q-1 = lR(M(qo)) + L ipo(i)

i=qo

for all q > qo, Proposition 2.1.5 shows that there exists a numerical polynomial ip(t) suchthat degip(t) :S n -1 and ip(q) = IR(M(q)) for all q E N,q ~ qo. This completes the proof of the theorem. 0

6.2. Dimension Polynomials of Difference Modules. Difference Dimension

Let R be a difference ring with a basic set (1' = {O'l' ... ,O'n} and let T = Ta , Tr ,

T(r) (1' EN) be the same as in the previous section. As above, we denote the ring of all (1'-operators over R by U and consider the ascending filtration (Ur )rElZ of the ring U such that Ur = {f E U lord f :S r} for l' ~ 0, and Ur = 0 for r < O. It is easy to see that the graded ring U with the grading introduced in Section 6.1 may be identified with the associated graded ring gr U of the filtered ring U with the filtration (Ur)rElZ. Below, treating the ring U as a filtered one we shall always suppose that the filtration of U is precisely (Ur )rElZ. If the contrary is not indicated, by a filtered (1'-R-module we shall mean a (1'-R-module M with the exhaustive and separated filtration of M as an U-module.

6.2.1. DEFINITION. Let M be a filtered (1'-R-module with a filtration (Mr)rElZ. If each R-module Mr (1' E ~) is finitely generated, then (Mr )rElZ is called a finite filtration of the u-R-modu/e M and M is called a finitely filtered u-R-module.

6.2. DIMENSION POLYNOMIALS OF DIFFERENCE MODULES 285

6.2.2. DEFINITION. A filtration (Mr )rEZ of a a-R-module M is called good (and the module M itself is called a good filtered a-R-module) if there exists a number r E Z such that M. = Us-rMr for all s E Z, s> r.

6.2.3. DEFINITION. A finite and good filtration of a a-R-module M is called excellent. In this case the a-R-module M is said to be an excellently filtered a-Rmodule.

6.2.4. REMARK. Let us consider the trivial filtration (Rr)rEZ of a ring R, i.e., the filtration for which Rr = R if r ~ 0 and Rr = 0 if r < O. Let P be a R-module and the non descending chain (Pr )rEZ of R-submodules of P such that UrEZPr = P and Pr = 0 for all sufficiently small r E Z be given. Then P may be regarded as a filtered R-module with the filtration (Pr )rEZ and the tensor product U 0R P can be naturally regarded as a left U-module. For every r E fl, let us consider the R-submodule (U @R P)r of U @R P which is generated over the ring R by the set of all elements u@x, where u E Ui, X E Pj and i + j = r. Then ((U @R P)r)rEZ is obviously a filtration of the a-R-module U @RP. In what follows, while considering U@RP as a filtered a-R-module (where P is an exhaustively and separately filtered module over the a-ring R with the trivial filtration), we shall always suppose that the filtration of U @R P has the above form.

6.2.5. THEOREM. Let R be an Artinian difference ring with a basic set a = {aI, ... , an} and let (Mr )rEZ be an excellent filtration of a a-R-module M. Then there exists a numerical polynomial 1lI(t) in one variable t such that 1lI(r) = IR(Mr ) for all sufficien tly large r E Z, and deg 1lI (t) ::; n.

PROOF. It is evident that IR(Mr ) = L:.<r IR(gr. M) (since we consider only separate filtrations, the sum is finite). Let us show that there exists a numerical polynomial ip(t) in one variable t such that degip(t) ::; n -1 and ip(s) = IR(gr. M) for all sufficiently large s E Z.

Since the filtration (Mr )rEZ is excellent, there exists an integer s E Z such that Ms = U.-rMr for all s E Z, s > r. By Remark 6.2.4, we may consider the Umodule U @R Mr as a filtered a-R-module with the filtration (U @R Mr )qEZ defined in the remark (we treat Mr (r E Z) as a filtered module over the ring R with the trivial filtration; the filtration ((Mr)q)qEZ of Mr is such that (Mr)q = Mr n Mq for all q E Z). It is easy to see that the mapping rr : U @R Mr ---+ M such that rr(u@x) = ux (u E U, x E Mr) is correctly defined; since rr((U0RMr )q) = Mq for all q E Z, q ~ r, this mapping is a surjective homomorphism of the filtered U -modules. By Proposition 1.3.41, the exactness of the sequence U 0R Mr ~ M ---+ 0 implies the exactness ofthe sequence gr( U @RMr ) ---+ gr M ---+ 0 of the graded gr U -modules. Taking into account the natural epimorphism

grU @RgrMr 4 gr(U @R Mr) ---+ 0

we obtain that the sequence of the graded gr U -modules (gr11")op

gr U @R gr Mr ) gr M ---+ 0

is exact. Since the gr U-module gr U @R gr Mr is finitely generated over gr U (because Mr is a finitely generated R-module), gr M is a finitely generated gr Umodule. As it was mentioned above, the graded rings gr U and U are isomorphic, hence gr M may be treated as a finitely generated graded U-module. By


Theorem 6.1.3, there exists a numerical polynomial !p(t) in one variable t such that deg !p(t) ::; n - 1 and !p(s) = IR(gr. M) for all sufficiently large s E Z (say for all s > So where So is some integer). By Proposition 2.1.5 there exists a numerical polynomial w(t) such that deg w(t) = deg !p(t) + 1 ::; nand W(r) = lR(Mr) = lR(M80 - l ) + 2::=80 lR(gr. M) for all r E Z, r > So. The theorem is proved. 0

6.2.6. DEFINITION. Let (Mr)rEZ be an excellent filtration of a O'-R-module M over an Artinian O'-ring R. Then the numerical polynomiaIW(t) whose existence is established by Theorem 6.2.5, is called the dimension polynomial or characteristic polynomial of the module M associated with the excellent filtration (Mr )rEZ,

6.2.7. EXAMPLE. Let R be a difference field with a basic set 0' = {al"'" an} and let U be the ring of O'-operators over R. Then the ring U may be considered as a filtered O'-R-module with the excellent filtration (Ur )rEZ defined above. Let Wu (t) be the characteristic polynomial of the O'-R-module U associated with the filtration (Ur )rEZ, If r EN, then the elements a~', ... , a~n, where kl'"'' kn E Nand 2:7=1 k i ::; r, form a basis of the vector R-space Ur . Therefore, IR(Ur ) = dimR UR = Card T(r) = p(n, r) = (r~n) (see Proposition 2.1.8). Thus, Wu(t) = c~n).

6.2.8. EXAMPLE. As in the previous example, let R be a difference field with a basic set 0' = {aI, ... , an} and let U be the ring of O'-operators over R. Let H be a free U-module of the rank k (k EN, k ~ 1) and Fk be the filtered O'-R-module

Fk with the filtration ((Fk)r )rEZ such that (Fk)r = 2:7=1 Ur-lii (It, ... , h are the free generators of the U -module Fk)' It is evident, that the filtration ((Fk)r) rEZ is

excellent. Since lR(Ur-l) = (r-~+n) for all r EN, r ~ I (see Example 6.2.7), the characteristic polynomial x(t) of the O'-R-module Fk associated with the filtration

((Fk)r )rEZ has the form X(t) = ke-~+n). Let R be a difference ring with a basic set 0' = {al, ... , an}, U the ring of 0'

operators over R, M a O'-R-module with a filtration (Mr )rEZ, and R[x] the ring of polynomials in one unknown x with the coefficients from R. Let fj denote the subring 2:rEZ Ur ®R Rxr of the ring U ®R R[x] and let M denote the left fj -module 2:rEZ Ur®RRxr (note, that the structure of a ring on the tensor product U®RR[x] is induced by the ring structure of U[x] with the help of the natural isomorphism of U-modules U ®R R[x]:::: U[x)).

The following lemma can be proved by analogy with the corresponding statement for the differential modules (see Lemma 5.1.13).

6.2.9. LEMMA. Let (Mr )rEZ be a finite filtration of a O'-R-module M. Then the following assertions are equivalent:

(1) (Mr)rEZ is a good filtration; (2) M is a finitely generated fj -module.

6.2.10. LEMMA. Let R be a Noetherian difference ring with a basic set 0' = {aI, ... , an} whose elements act on the ring R as automorphisms. Then the ring of O'-operators U and the ring fj constructed above are left Noetherian.


PROOF. First of all, note that the ring of O"-operators U and the ring of skew polynomials R[Zl, ... , Zn; al, ... , an) are isomorphic. By Theorem 1.5.19 the latter ring is left Noetherian, so the ring U is also left Noetherian.

Let us consider automorphisms f31, ... , f3n of the ring R[x) such that

m

= Lai(aj)Xj

j=O

for every polynomial L.t=o ajxj E R[x]. Furthermore, let dl (f3I), ... , dn(f3n) be skew derivations of the ring R[x) such that di(f3;)(f) = xf3i(f) - xl (1 ~ i ~ n) for every polynomial I E R[x) (it is easy to see that di(f3i) (1 ~ i ~ n) is a skew f3;-derivation of the ring R[x)) and let S denote the ring of skew polynomials R[ x)[ Zl, ... , Zn; d1 (f3d, ... , dn (f3n); f31 , ... , f3n). Then the rings U and S are isomorphic, the corresponding isomorphism 'P : U -+ S acts on the homogeneous components Ur r'im Rxr (r E Il) of the ring U as follows: if w Q9 axr = wa Q9 xr E Ur Q9 R Rxr (a E R,w E Ur ) and wa = Li" ... ,in ai, ... ina~' ... a~n (ai, ... i n E R for all indices iI, ... , in and the sum is finite), then 'P( wQ9axr ) = Li, , ... ,i n ai, .. i n Xr-(i, +···+i n ) (Zl + xli, ... (zn + x)i n • It is easy to verify that the mapping 'P is well-defined and bijec-tive (such a mapping is usually called a homogenization of the ring U). By Theorem 1.5.19, the ring S is left Noetherian, so the ring U is left Noetherian. 0

6.2.11. EXERCISE. Give an example of a Noetherian difference ring with a basic set 0" = {aI, ... , an} such that the corresponding ring of O"-operators U is not left Noetherian.

6.2.12. THEOREM. let R be a Noetherian difference ring with a basic set 0" = {aI, ... , an} whose elements act on the ring R as automorphisms. Let 1 : N -+ M be an injective homomorphism of filtered difference R-modules and the filtration of M is excellent. Then the filtration of N is also excellent.

PROOF. Let (Mr )rEZ and (Nr )rEZ be the given filtrations of the modules M and N, respectively. Since (Mr )rEZ is an excellent (hence, finite) filtration of M, every R-module Mr (r E Il) is finitely generated and, therefore, Noetherian (because the ring R is Noetherian). Since i is an injective homomorphism of filtered modules, every R-module Nr (r E Il) is isomorphic to a R-submodule of Mr (so that Nr is finitely generated R-module, and the filtration (Nr)rEz is finite). It remains to show that the filtration (Nr )rE:!: is a good one.

As above, let U denote the ring of O"-operators over R with the usual filtration (Ur )rE:!: and let the ring U = LrE:!: Ur Q9R Rx r , as well as the left U -modules

M = LrE:!: Mr Q9R Rxr and N = LrE:!: Nr Q9R Rxr, be the same as in Lemma 6.2.9. Since each Nr (r C Il) is isomorphic to a R-submodule of M r , the module N is isomorphic to a U-submodule of M. Furthermore, since the filtration (Mr)rE:!: is

good, Lemma 6.2.9 shows that M is a finitely generated U-module. By Lemma 6.2.10, the ring U is left Noetherian, hence M is a left Noetherian U-module. Therefore, N is a finitely generated U-module and we obtain (by Lemma 6.2.9) that the filtration (Nr )rE:!: is good. This completes the proof. 0


Theorems 6.2.5 and 6.2.12 imply the following result, which shows that if the basic set of a difference field consists of automorphisms, then the corresponding ring of difference operators is a left Ore ring.

6.2.13. COROLLARY. Let R be a difference field with a basic set of automorphisms u = {aI, ... , an} and let U be the ring of u-operators over R. Then U is a left Ore ring, i.e. for any elements u, v E U there exist nonzero elements I, g E U such that lu = gv.

PROOF. If u = 0, then our assertion is trivial. Let u =I 0, v =I 0 and let p :::: ord u, q = ord v. For every sEll, let U(s) denote the filtered u-R-module with the filtration (U(S)r)rEZ such that U(s)r = U.+r for all r E Il, where (Ur)rEZ is the usual filtration of the ring U. Then the map <p : U (-p) ffi U (-q) -* U such that (x, y) -* xu - yv for all x, y E U is a homomorphism of filtered Umodules. Let M = 1m <p and N = Ker <po Since the filtration of the u-R-module U( -p)ffi U( -q) is excellent (its r-th component has the form U(r - p) ffiU(r -q) for each r E Il), the induced filtrations of the modules M and N are also excellent (it is evident for the filtration of M, and for the filtration of N it is the consequence of Theorem 6.2.12). Let XM(t), XN(t) and X(t) be characteristic polynomials of the uR-modules M, N, and U(-p)ffiU(-q), respectively, associated with their excellent filtrations considered above. Then we have the exact sequence of excellently filtered u-R-modules

0-* N -* U(-p) ffi U(-q) -* M -* 0,

so that XN(r) + XM(r) ~ x(r) = (r+~-p) + (r+~-q) for all sufficiently large r E Il (see Example 6.2.8). Since Mr ~ Ur for all r Eiland dimR Ur = (r~n) for all sufficiently large r E Il (see Example 6.2.7)' we obtain that XN(t) ~ e+~-p) + e+~-q) - XM(t) ~ e+~-p) + e+~-q) - e~n). Therefore, deg Xn (t) ~ n hence N =I O. Now, if (f,g) is any nonzero element of N, then <p(f,g) = 0, so that lu = gv. 0

Let R be a difference field with a basic set u = {aI, ... , an}, let U be the ring of u-operators over R and let M be a u-R-module generated by a finite family of elements {Xl, ... , x m }. Then it easy to see that the vector R-spaces Mr = l:~l Ura; (r E Il) form an excellent filtration of M, and if (M:)rEZ is another excellent filtration of M, then there exists a number k E Il such that U. Mk = Mk+s and U.Mk = Mk+s for all sEN. Since each of the filtrations is exhaustive, there exists a number p E 'N such that Mk ~ Mk+p' Mf. ~ Mk+p and, consequently, Mr ~ M:+p, M: ~ Mr+p for all r E Il, r ~ k. Thus, if X(t) and Xl(t) are the characteristic polynomials of the module M associated with the filtrations (Mr )rEZ

and (M:)rEZ, respectively, then X(t) ~ Xl(t + p) and Xl(t) ~ X(t + p). Therefore, deg X(t) = deg Xl (t) and the leading coefficient of the polynomial Xl (t) is the same as that of X(t). Since degx(t) ~ n and the polynomial X(t) can be written in the

l:;."v1t l form X(t) = ~tn + o(t") (see (2.1.13)), ~nX(t) = ~nxdt). We obtain the following result.

6.2.14. PROPOSITION. Let R be a difference field with a basic set u = {al, ... , an}, let M be a u-R-module, and let X(t) be a characteristic polynomial associated with an excellent filtration of M. Then the integers ~nX(t), d = deg X(t)


and ~dX(t) do not depend on the choice of such filtration (therefore, these integers characterize the CT-R-module M itself).

This proposition justifies the following definition.

6.2.15. DEFINITION. Let R be a difference field with a basic set CT = {aI, ... , an}, M a finitely generated CT-R-module, and X(t) a characteristic polynomial, associated with an excellent filtration of M. Then the numbers ~nX(t), d = deg X(t), and ~nX(t) are called difference dimension, difference type, and typical difference dimension of M, respectively. These characteristics of the CT-Rmodule M are denoted by 8(M), t(M), and t8(M), respectively.

Now we intend to prove some properties of difference dimension which are similar to the suitable properties of differential dimension of finitely generated differential vector spaces considered in Section 5.1.

6.2.16. PROPOSITION. Let R be a difference field with a basic set of automorphisms CT = {al, ... , an} and let 0 -t N ~ M -4 P -t 0 be an exact sequence of finitely generated CT-R-modules. Then 8(N) + 8(P) = 8(M).

PROOF. Let (Mr)rEZ be an excellent filtration of the CT-R-module M and let Nr = z-l(z(N) n Mr), Pr = J(Mr) for all r E~. Clearly, the filtration (Pr)rEZ of the CT- R-module P is excellent; by Theorem 6.2.12 the filtration (Nr )rEZ of the CT-R-module N is also excellent. Let XN(t), XM(t), and Xp(t) be the characteristic polynomials of the modules N, M, and P, respectively, associated with our excellent filtrations. For any r E ~, the exactness of the sequence

o -t Nr -t Mr -t J(Mr) -t 0

implies the relation dimR Nr + dimR J(Mr) = dimR Mr, so that XN(t) + Xp(t) = XM(t). Therefore, 8(M) = ~nXM(t) = ~n(XN(t)+XP(t)) = ~nXN(t)+~nXp(t) = 8(N) + 8(P). 0

6.2.17. PROPOSITION. Let R be a difference field with a basic set of au tom orphisms CT = {aI, ... , an}, let U be the ring of CT-operators over R, and let M be a finitely generated CT-R-module. Then the difference dimension 8(M) is equal to the maximal number of the elements of the module M, which are linearly independent over U.

PROOF. First of all, let us show that 8(M) = 0 iff every element of M is linearly dependent over U. Let 8(M) = 0 and suppose that there exists an element x E M which is linearly independent over U. Then the map cp : U -t M, such that cp( u) = ux for every u E U, is an injective homomorphism of U-modules and we have the exact sequence of finitely generated CT-R-modules 0 -t U 4 M -t M/cp(U) -t O. By Proposition 6.2.16 we obtain that 8(U) = 8(M) - 8(M/cp(U)) :s 8(M) = 0, but Example 6.2.7 shows that 8(U) = ~n e!n) = 1. Thus, if 8(M) = 0, then each element of M is linearly dependent over U. Conversely, suppose that every element of M is linearly dependent over U. Let 6, ... ,ek be generators of M (so

that M = I:~=l Uei), let Ni = Uei (1 :s i:S k) and let 'ifJi : U -t Ni (1 :s i :s k) be the U -homomorphisms such that 'ifJi (u) = Uei for every u E U. Since every element ei is linearly dependent over the ring U, it follows that Ker'ifJi i- 0 for every


i = 1, ... , k. Furthermore, considering the canonical exact sequence of finitely generated U -modules 0 -+ Ker Wi -+ U -+ Ni -+ 0 (1 ::s: i ::s: k) we obtain that &(Ker W) + &(Ni) = &(U) = 1 (see Proposition 6.2.16). Since each nonzero element of Ker Wi is linearly independent over U (obviously, the ring U does not contain zero divisors), the above reasonings show that &(Ker Wi) =1= 0, hence &(Ker Wi) 2: 1 (1 ::s: i ::s: k). Therefore, 0 ::s: &(N;) = &(U) - &(Ker W;) = 1 - &(Ker W;) ::s: 0, so that &(N;) = 0 for every i = 1, ... ,k. Since M = L7=lNi, we have 0 ::s: &(M) = c5(L7=1 N;) ::s: L7=1 &(N;) = O. (Obviously, in order to prove the inequality &(L7=1 N;) ::s: L7=1 &(N;), it is sufficient to show that c5(Nl + N2) ::s: &(Nd +&(N2) for every finitely generated u-R-modules Nl and N2. Applying Proposition 6.2.16 to the canonical exact sequence 0 -+ Nl -+ M -+ M/Nl -+ 0 we obtain that &(M) = c5(Nd +&(Nl + N2/Nd = c5(Nl) +c5(N2/Nl n N2) ::s: &(Nd +c5(N2)') Thus, every element of M is linearly dependent over U iff &(M) = O.

Now, let p be the maximal number of elements of the u-R-module M which are linearly independent over the ring U. Let {Xl,"" xp} be any system of elements of M which are linearly independent over U, and let F = Lf=l U x;. Then (Lf=l UrX;)rEZ is an excellent filtration of F; the characteristic polynomial X(t) associated with this filtration has been determined in Example 6.2.8: X(t) = pe~n), so that &(F) = ~nX(t) = p. Furthermore, it follows from the maximality of the linearly independent over U system {Xl, ... , xp} that every element of finitely generated u-R-module M/F is linearly dependent over U, hence &(M/F) = O. Applying Proposition 6.2.16 to the exact sequence of finitely generated u-R-modules 0-+ F -+ M -+ M/ F -+ 0, we obtain that c5(M) = c5(F) + &(M/ F) = &(F) = p. This completes the proof. 0

6.3. Characteristic Polynomials of Inversive Difference Modules and their Invariants

Let R be an inversive difference ring with a basic set u = {al,"" an}, let E be the ring of u* -operators over R, and let (Er )rEZ be the filtration of E considered in Section 3.4.

6.3.1. DEFINITION. Let M be a u*-R-module over au-ring R. An ascending chain of its R-submodules (Mr) r E Z such that UrEzMr = M,Mr = 0 for all sufficiently small r E Z, and ErM. <; Mr+. for all r EN, s E Z, is called a filtration of M and is denoted by (Mr )rEZ, (Thus, by a filtration of a u* -R-module M we shall mean an exhaustive and separate filtration of M as a E-module.) If every R-module Mr (r E Z) is finitely generated, then (Mr )rEZ is called a finite filtration of M and the module M itself is called a finitely filtered u* -R-module.

6.3.2. DEFINITION. A filtration (Mr)rEZ of a u*-R-module M is called good (and the module M is said to be a good filtered u· -R-module) if there exists a number r E Z such that M. = E._rMr for all s E Z, s 2: r. A finite and good filtration of au· -R-module M is called excellent (in this case the u* -R-module M is said to be excellently filtered).

6.3.3. THEOREM. Let R be an Artinian u*-ring with a basic set of automorphisms u = {al, ... ,an } and let (Mr)rEZ be an excellent filtration of a u*-R-

6.3. CHARACTERISTIC POLYNOMIALS OF INVERSIVE MODULES 291

module M. Then there exists a numerical polynomial 'Ij!(t) in one variable t with the following properties:

(i) 'Ij!(r) = lr(Mr) for all sufficiently large r E IZ; (ii) deg 'Ij!(t) ~ n and the polynomial 'Ij!(t) can be written in the form 'Ij!(t)

2;!atn + o(tn), where a E IZ.

PROOF. Let gr e denote the graded ring associated with the usual filtration (er )rEZ of the ring of 0'* -operators over R, so that gr e = EDqEZ grq e, where grq e = eq/eq_ l for all q E IZ. Let Xl,"" x2n be the images of the elements al, ... , an, all, ... , a;;-l, respectively, in the ring gr e (so that Xi = ai + eo E eIleo = grl e for i = 1, ... , nand Xj = aj!n+eo E grl e for j = n+1, ... , 2n). It is easy to see that the elements Xl, ... , X2n generate the ring gr e over R, XiXj = XjX; for all i,j E {1, ... ,2n}, and X/eXn+/e = 0 for every k = 1, ... ,n. Furthermore, for any a E R, we have Xia = ai(a)xi if 1 ~ i ~ n, and Xia = a;!n (a)xi if n + 1 ~ i ~ 2n. A homogeneous component gr. e of the graded ring gr e is the R-module generated by the set of all monomials X~,' ... x~: such that kj E N (1 ~ j ~ n), 'L:;=l kj = sand iu - it! :j:. n for every u, v E {I, ... , n}. In what follows (until the end of the proof) we shall also denote the graded ring gr e by R{XI"'" X2n}.

Let gr M = ED.EZ gr. M be the graded gr e-module associated with the given excellent filtration (Mr)rEZ of the O'*-R-module M (so that gr. M = M./M._ I for every s E IZ). As in the proof of Theorem 6.2.5, we obtain that gr M is a finitely generated gd-module. Let M(') = gr. M (8 E IZ). Since lR(Mr ) = E.<r lR(M('), the theorem will be proved if we prove that there exists a numerical polynomial f(t) in one variable t with the following properties:

(a) f(s) = 'R(M(s)) for all sufficiently large s E IZ, say, for all s > So (so E N)j (b) the polynomial f(t) can be written in the form f(t) = (;:~)!tn-l +o(tn - l ),

where a E IZ.

Indeed, if the polynomial f(t) with such properties exists, then, by Proposition 2.1.5, there exists a numerical polynomial w(t) such that w(r) = Co + E:='o+l f(s) = lr(Mr) for all sufficiently large r E IZ (here Co = lr(M·o)). Furthermore, deg w(t) = deg f(t) + 1 and the leading coefficient of w(t) is equal to that of f(t) multiplied by lin. We intend to prove the existence of the polynomial f(t) with the properties (a), (b) by induction on n = Card 0'.

If n = 0, then gr M is a finitely generated module over the Artinian ring R, so that M(s) = 0 for all sufficiently large s E IZ and the polynomial f(t) = 0 has the desired properties.

Now, let n > 0 and let al,"" an be the elements of the basic set of an Artinian 0'* -ring R. Let us consider the exact sequence of finitely generated graded R{XI, ... , x2n}-modules

(6.3.1)

where On is the multiplication by Xn (i.e. On(Y) = XnY for any Y EM). It is evident, that On is an additive map and On (ay) = an(a)On(Y) for all y E M, a E R. Applying Lemma 6.1.2 with K = (Ker On)(s), M = M('), N = (xnM(')) (s E IZ), and a = an,


we obtain that IR((Ker 9n)('») + lR((xnM)(s») = lR(M('») for every s E Z. (If P is a graded module, then p(.) denotes its s-th homogeneous component.)

Since the modules Ker 9n and xnM are annihilated by the elements Xn and X2n, respectively, we can regard Ker 9n as a graded R{X1, ... , Xn-l, Xn+1, ... , X2n}-module and xnM as a graded R{X1, ... , x2n_d-module. (By R{X1, ... , Xn-l, Xn+l,···, X2n} and R{X1, ... , x2n-d we denote the graded subrings of the ring R{X1, ... , X2n} whose elements can be written in the form that does not contain Xn and X2n, respectively. The homogeneous components of each of these subrings are the intersections of the corresponding homogeneous components of R{ Xl, ... , X2n} with the subring.) By the inductive hypothesis, for every finitely generated graded R{xl, ... , Xn-1, Xn+l, ... , x2n_1}-module N = tBpEzN(p) (i.e. for every finitely generated gr l-module, where l is the ring of u* -operators over the inversive difference ring R with the basic set u = {a1, ... , an-d) there exists a numerical polynomial fN(t) in one variable t such that fN(p) = lR(N(p») for all sufficiently large P E Z. Furthermore, deg fN(t) ~ n - 2 and the polynomial fN(t) can be

written in the form fN(t) = (~~;t!tn-2 + o(tn- 2), where bE Z.

If L = tBqEzL(q) is a finitely generated graded module over R{X1, ... , Xn-1, Xn+l, ... ,X2n}, then the first and the last terms of the exact sequence of R-modules

(6.3.2)

(92n is the multiplication by X2n) are homogeneous components of finitely generated graded R{ Xl, ... ,Xn-l, Xn+1, ... , x2n}-modules. Dividing the sequence (6.3.2) into two short exact sequences and applying Lemma 6.1.2 (with a = a2n), we obtain that

lR(L(q+1») -IR(L(q») = IR(L(q+l) /X2nL(q») -lR((Ker 92n )(q»).

Now, the inductive hypothesis shows that there exists a numerical polynomial gL(t) = t:~;j!tn-2 + o(tn- 2) (b E Z) such that gL(q) = lR(L(q+1») - lR(L(q») for all sufficiently large q E Z, say, for all q ~ qo (qO E Z). Since

q

IR(L(q») = lR(L(qo») + 2: [lR(L('») - lR(L(·-l»)], .=qo+1

Proposition 2.1.5 shows, that there exists a numerical polynomial

2n - 1b hL(t) = tn- 1 + o(tn- 1)

(n - I)!

(b is the same integer as in the leading coefficient of gL(t)) such that hL(q) IR(L(q») for all sufficiently large q E Z. Clearly, the similar statement is valid for any finitely generated graded R{X1, ... , x2n-d-module.

Applying the above reasonings to the modules of the exact sequence (6.3.1), we

obtain that there exist numerical polynomials h(t) = c:=:1)ltn- 1 + o(tn- 1) and

f2(t) = (:~'1)~tn-1 + o(tn- 1) (a1,a2 E Z) in one variable t such that


for all sufficiently large 8 E Il, and a1, a2 E Il. Furthermore, by the exactness of sequence (6.3.1) we have

for all 8 E Il. Let us show that IR((xiM)('») = IR((xn+iM)(s») for all sEll and i = 1, ... , n

(in particular, IR((xnM)('») = IR((x2nM)('») for all 8 Ell). Since

(xn+iM)(s») = (ail Ms + M.)/M.,

we have IR((xn+iM)(S») = IR(a;l M. + M.) -IR(M.).

Applying Lemma 6.1.2 to the case when 8 is multiplication by Qi (which is a bijection), we obtain that

whence

Thus,

lR(M('») = lR((KerOn)('») + lR((xnM)('»)

= lR((KerOn)(s») + lR((x2n M)('»)

= 11(8) + 12(8)

for all sufficiently large 8 E Il. Consider the exact sequence of graded R{x!, ... , Xn-!, Xn+l,"" x2n}-modules

Its last term is a finitely generated R{X1"'" Xn-1, Xn+1, ... , x2n_d-module. By the inductive hypothesis, there exists a numerical polynomial fa(t) such that fa(8) = lR((Ker On/x2nM)('») for all sufficiently large 8 Eiland deg fa(t) :s n - 2. Since

for all sufficiently large 8 E Il,

h(t) = l1(t) - fa(t) = l1(t) + o(tn - 1).

Therefore, the leading coefficients of the polynomials 11 (t) and 12 (t) coincide, so that a1 = a2·

Now, letting I(t) = l1(t) + h(t) = 211(t) + o(tn- 1) = (~~a1vn-1 + o(tn- 1), we see that 1(8) = lR(M('») for all sufficiently large 8 E Il. This completes the proof. 0


6.3.4. DEFINITION. Let R be an Artinian u*-ring with a basic set u {a!, ... , an} and let (Mr )rez be an excellent filtration of a u* -R-module M. Then the polynomial w(t), whose existence is established by Theorem 6.3.3, is called the dimension polynomial or characteristic polynomial of the u* -R-modlJle M, associated with the excellent filtration (Mr )rez.

6.3.5. EXAMPLE. Let R be an inversive difference field with a basic set u = {aI, ... , an} and let £- be the ring of u* -operators over R. Then the ring £- can be regarded as a u*-R-module with the excellent filtration (£-r)rez considered above.

The corresponding characteristic polynomial will be denoted by wt)(t). Since for

every r E Z the elements 'Y = a~l, ... , a~" with ord'Y = E7=1 Ikil ::; r form a basis

of the vector R-space £-r, the number w~n\r) = IR(£-r) (r E N) is equal to the number of solutions (Xl, ... , Xn) E zn of the inequality IXll + ... + IXnl ::; r. By Proposition 2.1.8, the latter number (denoted by p(n, r» is as follows:

Therefore,

6.3.6. EXAMPLE. Let R be an inversive difference field with a basic set u = {al, ... , an}, let £- be the ring of u*-operators over R, and let Em (m E Z, m 2: 1) be a u* -R-module which is a free left £--module with free generators Yl,···, Ym. For every I E Z, let us consider the excellent filtration «E!")r )rez of Em such that (E!")r = E~l Cr-IYi for each r E Z. The filtered u*-R-module Em with the filtration «E!,.)r)rez will be denoted by E!,..

If w(t) is the characteristic polynomial of the module E!,., then

w(t) = mw~n)(t -I) = m ~2i (:) e ~ I) = m ~ (:) e + ! -l) = m ~( -1 t-i 2i (:) e + ! -l),

(6.3.4)

Let R be a u'-ring with a basic set u = {al, ... , an}, let £- be the ring of u'operators over R, and let R[x] be the ring of polynomials in one indeterminate X

over R. Let & denote the subring Erez £-r ®R Rxr of the ring £- ®R R[x]. If (Mr )rez is a filtration of a finitely generated u*-R-module M, then we denote the i-module Erez Mr ®R Rxr by M. The proof of the following lemma is similar to the proof of Lemma 5.1.13.


6.3.7. LEMMA. Let R be an inversive difference ring and let (Mr)rEZ be a finite filtration of an inversive difference R-module M. Then the following statements are equivalent:

(i) the filtration (Mr)rEZ is good; (ii) M is a finitely generated i-module.

6.3.8. LEMMA. Let R be a Noetherian u*-ring with a basic set 11' = {al, ... , an} and let C be the ring of u*-operators over R. Then the ring C is left Noetherian.

PROOF. Let I be any left ideal of C. As in Section 3.3, let U denote the ring of u-operators over R, which we consider as a subring of C. By Lemma 6.2.10, the ring U is left Noetherian, hence its left ideal In U has a finite system of generators Wi ... ,Wm , so that I n U = L:~l U Wj. Therefore, I = L:~l CWj. (Indeed, for any e E C, there exists an element "'fe = a~l, ... , a~n E r (k l , ... , kn E N), such that "'fee E U. Hence, if eEl, then "'fee E In U, so there exist elements Ul, ... , Um E U such that "'fee E L:~l eiwi and e = L:~l "'f;lejWi E L:~l CWi') Thus, every left ideal of C is finitely generated, so the ring C is left Noetherian. 0

Let C be the ring of 11'* -operators over a Noetherian 11'* -ring R with a basic set 11' = {ai, ... , an}, and let D be the ring of u-operators over the difference ring R with the basic set u = {ai, ... , an, an+l, ... , a2n} such that an+j(a) = ail (a) for all a E R; i = 1, ... , n. It is clear that if J denotes the ideal of D generated by all elements of the form ajan+i - 1 (1 ::; i ::; n), then the ring C is isomorphic to the residue ring of D modulo J.

Let f31, ... , f32n be automorphisms of the ring R[x] such that their restictions on R coincide with al, ... ,an,ai"l, ... ,a;;-l, respectively, and f3i(X) = x for any i = 1, ... , 2n. Let S denote the ring of skew polynomials

that was constructed in the proof of Lemma 6.2.10. Then the same reasoningd (and the same mapping) that was used in the proof of Lemma 6.2.10 show that & is isomorphic to the factor-ring of S by the ideal generated by all elements of the form

(Zj + X)((Zn+i + x) - x2 = ZjZn+i + XZj + XZn+j (i=l, ... ,n).

By Theorem 1.5.19, the ring S is left Noetherian, so that its factor-ring is also left Noetherian and we get the following result.

6.3.9. LEMMA. Let R be a Noetherian inversive difference ring with a basic set 11' and let C be the ring of 11'* -operators over R provided with the usual filtration (Cr )rEZ. Then the rings C and & = L:rEZ Cr ®R Rxr are left Noetherian.

The proof of the following theorem is similar to the proof of Theorem 6.2.12.

6.3.10. THEOREM. Let R be a Noetherian inversive difference ring with a basic set 11', let, : N -t M be an injective homomorphism of filtered u*-R-modules, and let the filtration of M be excellent. Then the filtration of N is also excellent.


6.3.11. EXERCISE. Show that the ring of (1"* -operators over an inversive difference field with a basic set (1" is a left Ore ring.

Let F be an inversive difference field with a basic set (1" = {a1, ... ,an }, let £ be the ring of (1"*-operators over F, and let M be a finitely generated (1"*-F-module with generators TIl, ... , TIm. Obviously, the vector F-spaces 0::::;:1 £rTl;)rEZ form an excellent filtration of M, so that every finitely generated inversive difference vector space may be provided with an excellent filtration. Let w(t) be a dimension polynomial associated with an excellent filtration (Mr)rEZ of a (1"*-F-module M and let d = deg w(t). Since w(t) is a numerical polynomial, its d-th finite difference ~dW(t) is a non-negative integer and ~nw(t) E &:.: (indeed, by Theorem 6.3.3, we have d ~ n, so that ~nw(t) is equal either to the integer ~dW(t), if d = n, or to the integer 0 if d < n). Furthermore, by formula (2.1.13), the polynomial w(t) can

be written in the form w(t) = Ad~(t)td + o(td). The formulation and the proof of the following proposition are analogous to those

of Proposition 6.2.14.

6.3.12. PROPOSITION. Let F be an inversive difference field with a basic set (1" and let (Mr )rEZ, (M:)rEZ be two excellent filtrations of a (1"* -F-module M. Let w(t) and WI (t) denote the dimension polynomials of M associated with the filtrations (Mr)rEZ and (M:)rEZ, respectively. Then

This proposition shows that the integers An2~(t), d = deg w(t), ~dW(t), as well as

the number Ad2~(t), do not depend on the choice of an excellent filtration (Mr )rEZ of a (1"*-F-module M.

6.3.13. DEFINITION. Let F be an inversive difference field with a basic set (1" = {a1,"" an}, let M be a finitely generated (1"*-F-module, and let w(t) be the dimension polynomial associated with an excellent filtration of M. Then the numbers ~nw(t)/2n, d = deg w(t), and ~dW(t)/2d are called the inversive difference dimension, the inversive difference type and the typical inversive difference dimension of the (1"*-F-module M, respectively. These numbers are denoted by ill"(M) , it(M), and till"(M), respectively.

Proposition 6.3.14 and 6.3.15 below can be proved precisely in the same way as the analogous statements about the difference vector spaces (see Propositions 6.2.16 and 6.2.17).

6.3.14. PROPOSITION. Let F be an inversive difference field and let

O-+N-+M-+P-+O

be an exact sequence offinitely generated inversive difference vector F -spaces. Then ill"(M) = ill"(N) + ill"(P).

Let F be a (1"* -field with a basic set (1" = {aI, ... , an}, let T be a free commutative semigroup generated by the elements aI, ... , an, and let r be a free commutative group generated by these elements. Let £ be the ring of (1"* -operators, and U the


ring of u' -operators over F. Elements Xl, ... , xp of au' -F-module M are said to be u* -linearly dependent over F, if they are linearly dependent over the ring £ (i.e. the equality 2:f=l WiXi = 0 holds for some elements WI, ... , Wp E £ not all of which equal zero). Otherwise, we say that elements Xl, ... , Xp E Mare u' -linearly independent over the u* -field F. It is easy to see, that elements Xl, ... ,xp E Mare u* -linearly dependent over a u' -field F iff the family {""(Xi I ""( E r, 1 ~ i ~ p} is linearly dependent over F. An equivalent condition: the family {TXj I T E T, 1 ~ i ~ p} is linearly dependent over the field F, i.e. the elements Xl, ... , Xp are linearly dependent over the ring U.

6.3.15. PROPOSITION. Let F be an inversive difference field with a basic set u = {aI, ... , an}, let £ be the ring of u' -operators over F, and let M be a finitely generated u' -F -module. Then the inversive difference dimension ic\'(M) is equal to the maximal number of elements of M which are u* -linearly independent over the field F.

6.3.16. EXERCISE. Let F be a u'-field with a basic set u = {al,a2}, let £ be the ring of u"-operators over F, and let M be a finitely generated u'-F-module with two generators X and y, connected by the defining relation alx + a2Y = 0 (i.e., M is the residue module of a free left £-module E2 with two free generators e1 and e2 modulo £(a1e1 + a2e2)). Determine the dimension polynomial of M, associated with its excellent filtration (Mr)rEZ such that Mr = £rx + CrY (r E Z). Find ic\'(M), it(M},and tic\'(M).

Now we are going to consider transformations of a basic set of an inversive difference ring. Let R be a u' -ring with a basic set u = {a1, ... , an} and let r 0' denote the free commutative group generated by the elements aI, ... , an. Obviously, if U1 = {T1, ... , Tn} is another system of free generators of the group r 0', then there exists a matrix K = (kijh$i,j$n E GL(n, Z) such that aj = T~il .. . T~'n (1 ~ i ~ n). Since T1, ... , Tn are pairwise commuting automorphisms of R, we can regard the ring R as a ui -ring (i.e. as an inversive difference ring with the basic set (1). For an arbitrary element 'Y = ail ... a~n = rfl ... T~" Era we set ordO' 'Y = 2:;=1 lirl, ordO'I ""( = 2:;=llirl and define the orders ordO' P and ordO'I P of an element P E £ as in Section 3.4 (clearly, the ring of u'-operators £ over the u" -ring R is also the ring of ui -operators over R when R is considered as an inversive difference ring with the basic set un. Below (until the end ofthe section) an inversive difference ring R with a basic set u = {al, ... , an} will be denoted by (R, u), the free commutative group generated by the elements of u will be denoted by r 0', and the ring of u'-operators over (R, u) will be denoted by £0' or by R(a1, ... , an}. If R is a u"-field and M is a finitely generated u*-R-module, then the inversive difference dimension, the inversive difference type and the typical inversive difference dimension will be denoted by ic\'O'(M), itO'(M) and tic\'O' (M), respectively.

6.3.17. DEFINITION. Let (R, u) and (R, ud be inversive difference rings with basic sets u = {a1, ... ,an} and U1 = {T1, ... , Tn}, respectively. The sets u and U1

are said to be equivalent if there exists a matrix K = (k ij h$i,j$n E G L(n, Z) such that Oij = Tfil ... T~iR (1 ~ i ~ n). In this case we write u ...., U1 and say that the

298 VI. DIMENSION POLYNOMIALS IN QIFFERENCE-DIFFERENTIAL ALGEBRA

transformation of the basic set u into the set Ul is an admissible transformation of u, and K is the matrix of this transformation.

It is evident, that if (R, u) and (R, u') are inversive difference rings such that u '" Ul, then r a = r a, and £a = £al' hence the notion of u*-R-module (respectively, of finitely generated u* -R-module) coincides with that of ui -R-module (respectively, of finitely generated ui-R-module). Furthermore, if R is a u*-field and M is a finitely generated u*-R-module, then Proposition 6.3.15 shows that each of the numbers ic5a (M), ic5u , (M) is equal to the maximal number of elements of M which are linearly independent over the ring £a, so that ic5a (M) = ic5al (M).

The following theorem shows that if R is a u* -field with a basic set u consisting of n elements and if M is a finitely generated u*-R-module such that ic5a (M) = 0, then there exists a set Ul consisting of (n - 1) automorphisms of R, such that M is a finitely generated ui-R-module. Moreover, the set ui can be obtained by the withdrawal of an element from some set u' of automorphisms of R such that u' '" u.

6.3.18. THEOREM. Let R be an inversive difference field with a basic set u = {al,"" an} and let M be a u*-R-module such that ic5a (M) = O. Then there exists a set Ul = {Tl, ... , Tn} of pairwise commuting automorphisms of R with the following properties:

(i) the sets u and Ul are equivalent; (ii) if U2 = {Tl' ... , Tn-d, then M is a finitely generated u;-R-module.

PROOF. Let £ = £a be the ring of u* -operators over R. We firstly consider the case when M is a cyclic left £-module with a generator "I: M = £"1. Since ic5a (M) = 0, Proposition 6.3.15 shows that "I is linearly dependent over £, so that PTJ = 0 for some nonzero element P E £. If PER, then M = £"1 = £(P- 1 PTJ) = 0 so that the statement of the theorem is trivial in this case.

Now, let P E £\R. Then the element P can be written in the form P = L:~1 ajai" ... a~Ri, where ai E R, aj # 0 (1 :S i:S m) and tki E IZ (1 :S k :S n, 1 :S i :S m). Clearly, we may suppose that the elements ,i = ai" ... a~ni (1 :S i :S m) do not have a common factor (if,i = ,,:, i = 1, ... ,m, for some, Era, then

m m

(,-1 P)TJ = ,-I (L ai/,DTJ = (L ,-I (aihDTJ = 0 i=1 i=1

and we can replace P by the element P' = ,-I P = L:~1 ,-I (ajh: for which P'TJ = 0 and ,L ... ,,:" do not have a common factor). It what follows we shall also suppose that tki ~ 0 for all k = 1, ... , n; i = 1, ... , m (if it is not so, then for every ak (1 :S k :S n) we may choose the minimal number jk E fir such that jk + tki ~ 0 for all i = 1, ... , m, and consider the element PI = a{1 ... a~n P = L:~1 a{' ... a~n(ai)o:{'+t" .. . o:~n+tni instead of P; for this element we also have P1TJ = 0). Since P ¢ R, there are nonzero integers among tkj (1 :S i :S m, 1 :S k :S n). Without less of generality we may suppose that tIi > 0 for some i (if tk; > 0 for some indices i and k where k > 1, then we may transform the basic set u into the set {,81, ... ,,8n} such that ,81 = O:k,,8k = 0: 1 and ,8j = 0: j for 1 < j :S n, j # k. Obviously, u' ...., u and the element P can be written in the form L:~1 aj,8r" ., .~ni, where Ulj E fir (1 :S j :S m, I:S I :S n) and UIi = tki > 0).


Let us consider the following partial ordering of the set of monomials aa~l ... a~n (O 'I- a E Rj k l , ... , kn E ~): a monomial aa~l ... a~n is said to be greater than a monomial bail ... a~n (this fact will be written as aa~l ... a~" >- bail ... a~), if the vector (kl' ... , kn ) is greater than (It, ... , In) with respect to the lexicographic order on ~n. Evidently, if E is a finite set of monomials no one of which can be obtained by the multiplication of some other monomial of E by an element of R, then the set E can be well-ordered with respect to the introduced ordering. Taking into account this remark, let us arrange the monomials aait; ... a~ni (1 ::; i ::; m) in the representation of the O'*-operator P in decreasing order, that is write P in the form

(6.3.5)

where b; E R, b; 'I- 0, k;j EN (1 ::; i ::; m, 1 ::; j ::; n) and

Let Aj be the maximal degree of the element OJ (1 ::; j ::; n), in representation (6.3.5), let d{P) = E'l::l Aj, and let p be some integer such that p > 2d{P). Let us consider the admissible transformation of the basic set 0' into the basic set 0'1 = {Tl, ... ,Tn } such that

(6.3.6)

(this transformation is admissible, since the determinant of the matrix of the transformation equals 1). Then our 0'* -operator P can be expressed in terms of 1'1, ... , Tn:

(6.3.7)

where lil = kil,

li2 = kilP + ki2' (1::; i ::; m).

lin = kilpn- I + ki2pn-2 + ... + kin

Since kil > 0 for some i (1 ::; i ::; m), it follows that kll > 0, hence, LIn > O. Let us show that lIn> L2n ~ 13n ~ ... ~ lmn. Indeed, we have bIa~ll ... a~ln >b2a~21 ... a~2n, so that there exists an index i (1 ::; i::; n) such that k1r = k2r for r k2i . Therefore,

n n

lIn -12n = ~)kIj - k 2j)pn- j > pn-i + L (k1j - k 2j )pn- j

j::1 ;::;+1

n 1 (n-; 1) ~ pn-i _ d(P) L pn- j > pn-i - 2'P P _ ~ > 0, j=i+l P


hence lIn> 12n. Similarly, one may obtain that lin ~ li+l,n for every i :::: 1, ... , m-1 (the equality holds iff lin:::: li+l,n :::: 0, that is k ij :::: ki+l,j :::: 0, for all j :::: 1, ... , n).

The equality P'fJ :::: (b i Ti" ... T~'n + ... + bm Tim' ... T~mn)'fJ :::: 0 implies

whence

T1'nn =[_T-1ll T-1"n-' (b-I)T- 1ll T-1"n-' (b )TI2 ,-l ll T I2 ,n-,-I"n-'l n'/ I'" n-I I I ... n-I 2 I ... n-I

X (T~2n'fJ) + ... + [_T-lll T-I"n-' (b-I)T-1ll T-1"n-' (b )T1m,-11l T1m,n-,-I"n-'l I ... n-I I I ... n-I mI'" n-I

X (T~m,n'fJ), (6.3.8)

The last equality shows that the element T~'n 'fJ can be written as a linear combination of the elements 'fJ, Tn'fJ, ... , T~'n-I'fJ with coefficients from R(TI, . .. , Tn-I).

Multiplying the both sides of equality (6.3.8) by the element Tn from the left, we obtain that

T~ln+I'fJ =[-Tn(b2)Ti21-lll ... T~2":'i-l-ll,n-l](T~2n+I'fJ) + ... + [-Tn(bm)Tim,-l ll ..• T~':.'i-l-ll,n-ll(T~mn+I'fJ),

where bi (1 ::; i ::; m) denote the elements

T- 1ll T-I"n-' (b-I)T-1ll T-l"n-' (b.) I ... n-I I I ... n-I , .

Therefore, the element T~'n+I'fJ can be also written as a linear combination of the elements 'fJ, Tn'fJ,···, T~ln-I'fJ with coefficients from R(TI, .. . , Tn-I). Let

(6.3.9)

where Vi E R(TI,.' .,Tn-I) (1::; i::; lIn -1). Multiplying the both sides of (6.3.9) by Tn from the left and applying (6.3.8), we obtain that the element T~,n+2'fJ can be also expressed as a linear combination of 'fJ, TnT, ... , T~ln-l'fJ with coefficients from R(TI"'" Tn-I). Continuing this process, we obtain that every element T~'fJ with s ~ lIn is a linear combination of the elements 'fJ, Tn'fJ, .. . , T~ln-I'fJ with coefficients from R(Tl"'" Tn_I)'

Now, let us consider the operator P written in the form (6.3.5). As above, let Aj :::: maxISiSm{kij} (1::; j::; n) and let

m

Q _ ~->., ~->'np _ "b.~kil->" ~k.n->'n - ~1 ... ~n - L...J '~1 ... ~1 .

i=1

Then Q'fJ :::: 0 and all degrees of CYi (1 ::; i ::; n) in the last representation of the u* -operator Q are nonpositive. We also note that the element CYI really appears in the representation of Q (otherwise, kil :::: Al :::: kll > 0 for every i :::: 1, ... , m,


so the monomials bia~i1 ... a~;" (1 ~ i ~ m) have the common factor a~ll that contradicts our suppositions). Let us write the 0-* -operator Q in the form

(6.3.10)

where Ci E R, Ci oF 0, Tij E Z, Tij ~ 0 (1 ~ i ~ m, 1 ~ j ~ n) and

(clearly, the vector (Cl, ... , cm) is obtained by the permutation of the coordinates of the vector (b 1 , ... ,bm )). If J1.j = maxl9~m ITiil (1 ~ j ~ n) and d(Q) = 2:.;=1 J1.j, then, obviously, J1.j = >'j and d(P) = d(Q).

Let us consider the transformation (6.3.6) of the basic set 0- (where p is any integer such that p > 2d(P) = 2d(Q)). Expressing the o-*-operator Q in terms of the elements 1'1, ... , Tn, we obtain that

(6.3.11)

where

and SIn < S2n ~ ... ~ Smn (this fact can be established by analogy with the proof of the inequalities hn > 12n ~ ... ~ lmn for the operator P written in the form (6.3.7)). Writing the equality Q'TJ = 0 in the form CITt" ... T~"''TJ = -c2Tt21 ... T~2n'TJ - ... - Cm Tt m' ... T~mn'TJ, we obtain the relationship

~"n.,., _[_1'-8" ~ -8" .. -, (c- 1 )1'8" ~-·"n-' (C ) 'n - 1 ... 'n-l 1 1 ... 'n 2 X 1'82,-8" r'2, .. -,-8""-'](r82 .. ") + 1 ... n-l n'" ..

+ [_1'-'" r -8',n_' (C- 1 )1"" r- 8 "n-' (C ) 1 ... n-l 1 1 ... n m

(6.3.12)

where SIn < S2n ~ ... ~ Smn ~ O. Repeatingly multiplying equality (6.3.12) by 1';1 and applying the same relationship (6.3.12), we obtain (as in the above considerations of the equality (6.3.8)) that every element T~"" (T E Z, T ~ SIn) is a linear combination of the elements "1,1';;1.,." ... , T~, .. +1.,., with coefficients from the ring R(Tb ... , Tn-I}.

Thus,

+00 M=£'TJ= L R(Tl, ... ,Tn-l}(T~'TJ)

i:-oo

', .. -I

L R{rt, ... , Tn-l)(T~'TJ). i=8, .. +1


Setting 0'2 = {Tl"'" Tn-d, we see that the elements (T~7]) (SIn + 1 ~ k ~ lIn -1) generate M as a O'i-R-module. Since the set 0'1 is obtained from 0' by the admissible transformation (6.3.6), the basic sets 0'1 and 0' are equivalent, so for a cyclic left £-module M our theorem is proved.

Now let M be a O'*-R-module which is generated by m elements 7]1,"" 7]m (i.e. M = E;';:,,1 £7]i) where m > 1. For each i = 1, ... , m we have a canonical exact sequence of 0'* -R-modules

0-+ £7]i -+ M -+ Mj£7]i -+ O.

By Proposition 6.3.14,

so that it5o (£7]i) = 0 (i = 1, ... , m). Now, as in the first part of the proof, we can conclude that there exist 0'* -operators Pi = Ej~1 aija~ij' ... a~ijn (1 ~ i ~ m) over the ring R, such that qijk ~ 0 (1 ~ i ~ m, 1 ~ j ~ ni, 1 ~ k ~ n), monomials of each Pi do not have common factor, and PI 7]1 = ... = Pm7]m = O. Let us choose a number p E iZ such that p > 2 maxl<i<m {d(Pi }} (as above, d(Pi ) = E~=1 (maXl~j~ni{qijk}) for every i = 1, ... , m) and let us consider the admissible transformation (6.3.6) with the chosen p, which transforms the basic set 0' into the equivalent set of automorphisms 0'1 = {Tl, ... , Tn}. By the above observations, every 0'* -R-module £7]i (1 ~ i ~ m) is finitely generated as a O'i-R-module where 0'2 = {Tl, ... , Tn-I}. Therefore, the O'· -R-module M is also finitely generated as a O'i-R-module. This completes the proof. 0

The following result may be considered as a generalization of Theorem 6.3.18.

6.3.19. THEOREM. Let R be an inversive difference field with a basic set 0' = {aI, ... , an}, let M be a finitely generated 0'* -R-module, and let k = ito(M). Then there exists a set 0'1 = {,Bl, ... , ,Bn} of pairwise commuting automorphisms of R with the following properties:

(i) the set 0'1 is obtained from 0' by an admissible transformation (so that 0'1 ~ O')j

(ii) if 0'2 = {,BlJ ... ,,Bk}, then M is a finitely generated O'i-R-module and iOo , (M) > o.

PROOF. Suppose that the set 0'0 = {Tl, . .. , Tn} of automorphisms of the field R is obtained from 0' by an admissible transformation such that OJ = Tfil ... T~i" andTi=aiil ... a~R (l~i~n, kij ,lijEiZforalli=I, ... ,n, j=I, ... ,n). Let q = maXl::::i~n{Ej=llkijl, Ej=l Ilijl}, let £ be the ring of 0'*- (and O'~-) operators over R (so that £ = £0 = £00), and let (£r)rEZ, (£~)rEZ be the usual filtrations of £ associated with the basic sets 0' and 0'0, respectively, so that £r = {w E £ I ordow ~ r}, £~ = {w E £ I ordoow ~ r} for all r E N, and £r = £~ = 0 for r E iZ, r < 0 (by ordo wand ordoo w we denote the orders of an element w E £ when it is considered as a O'· -operator and as a O'(j-operator, respectively). The filtered ring £ with the filtration (£r )rEZ will be denoted by £, and the filtered ring £ with the filtration (£~)rEZ will be denoted by £0.


Let Xl, ... , Xm be any finite system of generators of a u· -R-module M, so that M = :L7::1 £Xi. Let wo(t) be the dimension polynomial of M, associated with the excellent filtration (:L7::1 £rXi)rez (here we regard M as a filtered £-module), and let woo (t) be the dimension polynomial of M associated with the excellent filtration (:L~1 £~x;)rez (in this case we consider M as a filtered £o-module). Then degwo(t) = ito(M) = k, and degwoo(t) = itoo(M). Since

for every r E Z (because of the evident inclusions £r ~ £~q ~ £rq' for all r E Z), we have wo(r) ::; WOO (rq) ::; wo(rq2) for all sufficiently large r E Z, hence itoo(M) = ito(M) = k.

If iOo(M) = 0, then, by Theorem 6.3.18, we may choose the set Uo = {Tl, ... , Tn} such that M is a finitely generated uacR-module, where UOI = {TI, .. . ,Tn-d. Thus, M = :L~1 :L~=-p £001 (T~Xi) for some positive integer p, hence there exist elements Wijl, Vijl E £001 (1 ::; i ::; m, -p ::; j ::; p, 1 ::; I ::; m) such that

m p

T~+IXI = L L Wijl(~Xi)' i=1 j=-p m p

T;p-l XI = L L Vijl(T~Xi). i=1 j=-p

(6.3.13)

Let s = max1:5i,I~m;-p~j~p{ordool Wijl,Ordool Vijl} and let:F denote the ring of ua1-operators over the ua1-field R, provided with the ascending filtration (:Fr )rez such that :Fr = 0 for all r > 0, and if r EN, then :Fr is the vector R-space generated by the set of all uocoperators over R, whose order does not exceed r. Obviously,

(:L~1 I:~=-p :Fr(T~X;) tez is an excellent filtration of the ukR-module M and

m p m m p

L L :Fr(~x;) ~ L£~+pXi ~ L L :F(r+p).(~x;) i=1 j=-p i=1 i=1 j=-p

(the first inclusion is evident, because :FrT~ ~ £~+p for any j between -p and p, and the second inclusion is the direct consequence of (6.3.13)). Therefore, WOOl (r) ::; WOo(r + p) ::; WOOl ((r + p)s) for all sufficiently large r E Z (by WOOl (t) we denote the dimension polynomial associated with the excellent filtration

of the uo1-R-module M). These inequalities show that itoOl (M) = itoo(M), hence it"Ol (M) = it" (M) = k. If iOOOl (M) > 0 (i.e. k = n - 1), we set Ul = Uo and complete the proof. If ioo (M) = 0, then we may repeat the above reasonings


and obtain the set 0"02 = {Ill, ... , IIn-d of pairwise commuting automorphisms of R such that 0"02 '" 0"01 and M is a finitely generated 0"~3-R-module, where 0"03 = {1I1, ... ,lIn-2}. Furthermore, ita02(M) = itao,(M) = k. IficSao3 (M) > 0 (i.e. k = n-2), we set 0"1 = {/31, ... ,/3n}, where /3; = IIi for i = 1, ... ,n-1 and /3n = Tn, and complete the proof (in this case the admissible transformation of 0" into 0"1 is the composition of the transformation 0" -t 0"0 and of the admissible transformation 0"0 -t 0"1 which is obtained by the adjoining of the relation /3n = Tn to the admissible transformation 0"01 -t 0"02). As above, if icSao3 (M) = 0, (i.e. k < n-2), then we can use an admissible transformation to proceed to a new basic set and so on. On a certain stage (after a finite number of steps) we obtain a set 0"1 = {/31, ... , /3k, /3k+l, ... , /3n} of pairwise commuting automorphisms of R such that 0"1 '" 0", M is a finitely generated O";-R-module, where 0"2 = {/31, ... , /3d, and ita2(M) = k, that is icSa2 (M) > O. This completes the proof. 0

6.3.20. EXERCISE. Let R be an inversive difference field with a basic set 0" = { aI, a2}, let [ be the ring of 0"* -operators over R, and let M be a finitely generated O"*-R-module with two generators x, y and two defining relations alx + a2Y = 0, a2x + alY = 0 (i.e. M is a residue module of a free left [-module E2 with two free generators el and e2 modulo [(aIel + a2e2) + [(a2el + ale2)). Show that icSo- (M) = 0 and determine an admissible transformation of 0" into some new set 0"1 = {Tl, T2} of commuting automorphisms of R such that M is a finitely generated 0"2-R-module, where 0"2 = {Td·

6.4. Dimension Polynomials of Extensions of Difference and Inversive Difference fields

Throughout this section by a field we shall mean a field of zero characteristic. Let F be a difference field with a basic set of injective endomorphisms 0" =

{aI, ... , an}, which is also called a O"-field. By T (or by To-) we still denote the free commutative semigroup generated by the elements aI, ... , an. As above, if T = a~' ... a~n E T (kl , ... , kn EN), then the number ord T = I:~l k i is called the order of T. We also set T( r) = {T E T lord T ~ r} for any r EN.

6.4 .1. THEOREM. Let F be a difference field with a basic set 0" = {aI, ... , an} and let G = F(1]l, ... ,1].) be a O"-extension of F generated by a finite family 1] = (1]1, ... ,1].). Then there exists a numerical polynomial 'P'IIF(t) with the following properties.

(1) For all sufficiently large r EN,

'P'I\F(r) = trdegF F({T1]j IT E T(r), 1 ~ j ~ s}). (6.4.1)

(2) deg 'P7J\F :s n, so that the polynomial 'P7J\F(t) can be written in the form

'P'I\F(t) = I:7=oaieti), where aO,a1, ... ,an E;;Z (see Proposition 2.1.3). (3) The number an is equal to the O"-transcendence degree of Gover F. (4) Let P be the defining 0"* -ideal of 1] in the ring of O"-polynomials

F {Y1, ... , Y. } 0- and let A be a characteristic set of P with respect to some fixed orderly ranking of the set of terms {TYj I T E T, 1 :s j :s s}. For

6.4. DIMENSION POLYNOMIALS 305

every j = 1, ... , s, let Ej denote the set of all elements (k1, ... , kn ) E N n

for which a~l ... a~nYj is a leader of a CT-polynomial of A. Then

• CP'lIF(t) = LWE;(t), (6.4.2)

j=l

where WE; (t) (1 ::; j ::; s) is the Kolchin polynomial of the set Ej (see Definition 2.2.8)

PROOF. Let V denote the set of all elements TYj (T E T, 1 ::; j ::; s) that are not transforms of any UA where A E A (as in Section 3.3, the leader of aCT-polynomial A is denoted by UA), so that v E V iff v 'f:. T'UA for all T' E T, A EA. Furthermore, for any r E N, let V(r) denote the set {TYj E V I T E T(r), 1::; j ::; s}.

If A E A, then A(7]) = 0, hence the element UA(7]) is algebraic over the field F( {T7]j I T E T,I ::; j ::; s, TYj < UA}) (by the symbol "<" we denote the given ranking of the set of terms {TYj I T E T, 1 ::; j ::; s}). Therefore, for any r EN, the field G r = F({T7]j IT E T(r), 1 ::; j ::; s}) is an algebraic extension of the field F({v(7]) I v E V(r)}). By Proposition 3.3.40, the ideal P does not contain any nonzero CT-polynomial that is reduced with respect to A, hence the set V'I(r) = {v(7]) I v E V(r)} (r E N) is algebraically independent over F for every r EN. Thus, for every r E N the set V'I(r) is a transcendence basis of Gr over F, so that trdegF Gr = Card V'I(r) (r EN).

Now we note that for each Yj (1 ::; j ::; s) the number of terms a~l ... a~n Yj, which lie in V(r) (r E N) is equal to the number of elements (k1"'" kn ) E Nm with 2:~=1 ki ::; r that are not greater than or equal to any element of Ej with respect to the product order on Nm . It follows that if WE;(t) is the Kolchin dimension polynomial of the subset Ej of Nm (1 ::; j ::; s), then trdegF Gr = Card{ v(7]) I v E V(r)} = 2:;=1 wE;{r) for all sufficiently large r E N, so that the numerical polynomial CP'lIF (t) = 2:;=1 WE; (t) satisfies condition (6.4.1) and deg CP'lIF(t) ::; n.

It remains to show that if the polynomial CP'lIF(t) is represented in the form 2:7=0 ai Cii ), where ao, aI, ... , an E Il, then the coefficient an is equal to the CTtranscendence degree of Gover F. If d = u-trdegF G, then, without loss of generality, we may suppose that the elements 7]1,' .. , 7]d form a u-transcendence basis of the field G = F(7]l, ... , 7]d, 7]d+1,"" 7].)q over F (see Proposition 3.3.72). In this case the set {T7]j I T E T(r), 1 ::; j ::; d} is algebraically independent over F for every r E N, therefore, cp'lIF(r) 2: d· CardT(r) = d(r!n) for all sufficiently large r E N (see (2.1.22)), whence an 2: d.

Since each element 7]j (j = d+ 1, ... , s) is u-algebraic over the field F (7]1, ... , 7]d) q, there exists a monomial Vj = TjYj (Tj E T) such that the element Tj7]j is algebraic over the field F (7]1, ... , 7]d)" ( {T7]i I T E T, TYi < Vj, d + 1 ::; j ::; s}). Therefore, there exists a number pEN such that all elements Td+17]d+1, ... , T.7]. are algebraic over the field F( {T7]i I T E T(p), 1 ::; i ::; d} U {T1Jj I T E T, d + 1 ::; j ::; sand TYj < Vj}).

Now, we note that if rj = ordTj (d + 1 ::; j ::; s), a term T'Yj is a transform of Vj = TjYj and r' = ord T', then the element T'7]j is algebraic over the field F( {T7]; I T E T(p + r' - rj), 1 ::; i ::; d} U {T7]j I T E T, d + 1 ::; j ::; sand


rYj < r'Yj}). It follows that for all sufficiently large rEM (more precisely, for all r ~ max{rd+1, ... , r.}) each element of the field Gr = F( {r1/j IrE T(r), 1 ~ j ~ s}) is algebraic over the field

G~=F({r1/j IrET(r+p), l~i~d}U{r1/j IrET(r)\ U T(r-rj)rj}). d+1:Sj:S.

Hence,

IP'1IF(r) = trdegF Gr

~ trdegF F( {r1/j IrE T(p + r), 1 ~ j ~ d} U {r1/j IrE T(r), d + 1 ~ j ~ s}) •

= trdegF G~ ~ d· Card T(r + p) + L [Card T(r) - Card T(r - rj)] j=d+1

for all sufficiently large rEM. Since the last sum is a polynomial of r, whose degree is less than n, we have am ~ d. Thus, am = oo-trdegF G. 0

6.4.2. DEFINITION. The numerical polynomial IP'1IF(t) , whose existence is established by Theorem 6.4.1, is called the dimension polynomial of the finitely generated oo-extension G of the oo-field F, associated with the family of oo-generators 1/ = ('1/1, ... , '1/.).

6.4.3. EXAMPLE. Let F be a difference field with a basic set 00 = {O'l, ... , O'n} and let G = F(1/1, ... , 1/.)/7 be a oo-extension of the field F generated by a finite family of elements 1/ = ('1/1, ... ,1/.) which are oo-algebraically independent over F. Then the defining ideal of the point 1/ = (1/1, ... , 1/.) in the ring of oo-polynomials F {Y1, ... ,Y.}/7 is the zero one, so its characteristic set is empty. Therefore, each set Ej (1 ~ j ~ s) in (6.4.2) is empty, hence WEj (t) = e~n) (see Theorem 2.2.7) and IP'1IF(t) = sc~n).

6.4.4. PROPOSITION. Let F be a difference tield with a basic set 00 = {O'1' ... ,O'n} and let '1/ = (1/1, ... ,1/.), < = (6, ... , <p) be tinite families of elements of au-extension G of F such that F(1/1. ... ,1/.)/7 = F(6, ... , <p)/7. Let IP'1IF(t) = E~=o aj cti) and IPelF(t) = E~=o bj Ctj ) (aj, bj E IE for every i = 0, 1, ... , n) be the dimension polynomials of the oo-extension G of F, associated with the families of oo-generators 1/ = ('1/1. ... , 1/.) and < = (6, ... , <p), respectively. Then an = bn, the polynomials IP'1IF(t) and IPelF(t) have the same degree d, and ad = bd ·

PROOF. Since F('1/1, ... , 1/.)/7 = F«1,.·. ,<p)/7' for each i = 1, ... , s there exists a number qj E M such that '1/i E F({r<j IrE T(qj)' 1 ~ j ~ p}). Similarly, for each j = 1, ... , p there exists a number lj E M such that <j E F ( {r1/j IrE T(lj)' l~j~s}). Letq=max{q1, ... ,q.,l1, ... ,lp}. ThenforanylEMwehave the inclusions F({r1/j IrE T(l), 1 ~ j ~ s}) ~ F({r<j IrE T(l+q), 1 ~ j ~ p}) and F({r<j IrE T(l), 1 ~ j ~ p}) ~ F({r1/j IrE T(l+q), 1 ~ j ~ s}) Therefore, IP'1IF(l) ~ IPelF(l + q) and IPelF(l) ~ IP'1IF(l + q) for all sufficiently large I EM, hence


the degree and the leading coefficient of cp{IF(t) are the same as those of CPIJIF(t). This completes the proof. 0

Let G = F("'b' .. ,"'.)0 be a finitely generated u-extension of au-field F with the family of u-generators ", = ("'1, ... , ",.), and let

n (t + i) CPIJIF(t) = ~ aj i

be the corresponding dimension polynomial of this u-extension. Then Proposition 6.4.4 shows that the integers d = deg CPIJIF and ad do not depend on the choice of the family"" and, therefore, these numbers characterize the u-extension G 2 F itself. The integers d = deg CPIJIF and ad will be denoted by u-typeF G and u-t.trdegF G, respectively.

6.4.5. DEFINITION. Let G be a finitely generated u-extension of au-field F. Then the integers u-typeF G and u-t.trdegF G considered above are called the difference (or u-) type and the typical difference transcendence degree (or the typical u-transcendence degree) of the u-extension G of F, respectively.

6.4.6. THEOREM. Let F be a difference field with a basic set u = {at. ... , an} and let G be a finitely generated u-extension of F with a family of u-generators ", = ("'1, ... , "'s) such that elements "'l, ... ,"'d (1 ~ d ~ s) form au-transcendence basis of the u-field G = F ("'1, ... , "'.)0 over F. Then

CP(IJd+l , ... ,IJ.)IF(IJI , ... ,IJd) ... (t) ~ CPIJIF(t) - d C : n). (6.4.3)

PROOF. Let

and

Hl(r) = {T"'i IT E T(r), 1 ~ j ~ d},

H2(r) = {T"'i IT E T(r), d+ 1 ~ j ~ s},

H(r) = Hl(r) U H2(r) = {T"'i IT E T(r), 1 ~ j ~ d}

for every r EN. By the definition of the polynomial CP(IJd+l , ... ,IJ.)IF(IJI , ... ,IJd} ... (t) we have

(6.4.4)

for all sufficiently large r EN, so that

CP(IJd+l" .. ,IJ.)IF(lJl"",lJd) ... (t) ~ CPIJIF(t) - dC: n). 0


6.4.7. THEOREM. Let G = F(TJ1, ... ,TJ')f7 be a finitely generated difference extension of a difference field F wi th a basic set u and let c,of/IF (t) be the dimension polynomial of this u-extension, associated with the family of u-generators TJ (TJ1, ... , TJ.)· Then the following conditions are equivalent:

(i) c,of/IF(t) = dc~n); (ii) u-trdegF G = trdegF F(TJlo···, TJ.) = d.

PROOF. Let c,of/IF(t) = d('~n), then by Theorem 6.4.1(3) d = u-trdegFG. Furthermore, by Proposition 3.3.72 we may suppose that the elements TJ1, .. ·, T/d form a u-transcendence basis of Gover F so that c,o(f/d+l, ... ,f/.)iF(f/l, ... ,f/d)" (t) = 0 (see Theorem 6.4.6). Let P be the defining ideal of the point (TJd+1, ... , TJ.) in the ring of u-polynomials F(TJlo ... , TJd)f7{Y1, ... , Y,-d}f7 over F(TJ1, ... , TJd)f7 and let A be a characteristic set of the ideal P (with respect to some orderly ranking < of the set of terms {OYj I 0 E T, 1 :S j :S s - d}; for definiteness we shall suppose that Y1 < Y2 < ... < Ys-d). For each j = 1, ... , s - diet Ej denote the set of all elements (k1 , ... , kn ) E Nn such that the term Q~l ... Q~"Yj is a leader of some (clearly, the unique) u-polynomial of the set A. Then by Theorem 6.4.1(4) we

have W(f/d+l, ... ,f/.)IF(f/l, ... ,f/d),,(t) = L:;:~WEj(t) = 0, hence WEj(t) = 0 for every j = 1, ... , s - d, so that each set Ej (1 :S j :S s - d) consists of the single element (0, ... ,0) (see Theorem 2.2.7(3)). Since Yl < Yi for every i = 2, ... , s - d, a upolynomial from A with the leader Y1 is the usual polynomial in one indeterminate Y1 with coefficients from the field F (T/1, ... , TJd)f7. Hence, the element TJd+1 and all elements ofthe form rTJd+1 (r E T) are algebraic over F(TJl' ... ,TJd)f7. Furthermore,

trdegF(f/l , ... ,'1d)" F(TJ1, ... , TJd)f7 ({ rTJj IrE T(r), d + 2 :S j :S s})

= trdegF(f/l, ... ,f/d)" F(TJl, ... , TJd)f7( {rT/j Ir E T(r), d + 1 :S j :S s})

= c,o(f/d+l, ... ,f/.)iF(f/l, ... ,f/d),,(r) = 0

for all sufficiently large r E N, whence c,o('1d+2, ... ,f/.)iF('11, ... ,'1d),,(t) = 0 so we may repeat the above reasons and obtain that each element TJj (d + 1 :S j :S s) is algebraic over the field F(T/1, ... , TJd)f7.

It follows from (6.4.4) that there exists a number ro EN such that F(U:=l T{r)TJd is an algebraic extension of the field F(Uf=1T(r)TJd for any r E Z, r ? roo Let us suppose that T/d+1 is not algebraic over the field F{TJlo ... , TJd). Let P be the minimal number in the set of all q E N such that T/d+1 is algebraic over the field F{Uf=lT{q)TJi). Since TJd+1 is not algebraic over F(Uf=1T{P - 1)T/;), there exists an element v = roTJj (1 :S j :S d) such that ord ro = P and v is algebraic over the field F((Uf=1 T(p)TJi \ {v}) U {TJd+d). It is easy to see that if r' E T and ord r' = r ?: ro, then the element r'v = r'roTJj is algebraic over the field F((Uf=1T(r + p)T/i \ {r'v}) U {r'TJd+d). Furthermore, r'T/d+1 is algebraic over F{u1=1 T(r)1}i) and, therefore, over F(Uf=lT(r+p)T/;), so that the element r'v is algebraic over the field F{U1=1T{r +p)TJi \ {r'v}). It follows that the elements of the set Uf=1 T{r+ p)TJi are algebraically dependent over F, but this contradicts the u· -algebraic independence of TJ1, ... ,TJd over the field F. Thus, the element T/d+1 is algebraic over the field F{TJlo· .. ,TJd) and similarly each element T/d+2, ... , 1}. is algebraic over F(1}1, ... , T/d), so that d = u-trdegF G = trdegF F(TJlo ... , TJ.).


Conversely, suppose that the condition (ii) of the theorem holds, and let '111, ••. , 1)d be the u-transcendence basis of F(1)I, ... , "1.)<7 over F. Clearly, the elements 1)1, ... , "Id are algebraically independent over F and F ("11, ... , 1).) is the algebraic extension of F(1)1, ... , 1)d). It follows that if T E T(r) (r E N) and 1 ::; j ::; s, then the element T1)j is algebraic over the field F ( {T1)j I T E T( r), 1 ::; j ::; d}). Therefore, trdegF F( {T1)j I T E T(r), 1::; j ::; s}) = trdegF F( {T1)j I T E T(r), 1::; j ::; d}) = d(r~n) for all sufficiently large r EN, so that S01/!F (t) = de~n). This completes the proof. 0

Now, let F be an inversive difference field (as usual, we suppose Char F = 0) with a basic set of automorphisms u = {Ol, ... , On}. As in Section 3.4, we shall say that F is a u*-field and denote the set {Ol, ... , On, all, ... , 0;;1} by u*. The free commutative group generated by the elements 01, ... , on will be denoted by f (or by f (7), and by the order of an element I = O~l, ..• , o~" (k1' ... ' kn E :;E) we shall mean the number ord')' = 2:~=1 Ikil. Furthermore, for any r E N the set hE fiord')'::; r} will be denoted by f(r).

The following theorem on the dimension polynomial is similar to the corresponding results for differential and difference fields (see Theorems 5.4.1 and 6.4.1).

6.4.8. THEOREM. Let F be a u*-field with a basic set of automorphisms u = {Ol, ... ,On} and let G = F(1)l, ... ,1).) be a u*-extension of F generated by the finite family 1) = (1)1, ... ,1),). Then there exists a numerical polynomial W1/IF(t) in one variable t with the following properties.

(1) W1/IF(r) = trdegF F(h1)j II E f(r), 1::; j ::; s}) (6.4.5)

for all sufficiently large r EN. (2) deg W1/IF ::; n and the polynomialW1/!F(t) can be written in the form

(6.4.6)

where a E :;E, and o(tn ) is the polynomial with rational coefficients whose degree does not exceed n - 1.

(3) The integer a equals the u* -transcendence degree of the u* -field Gover F. (4) Let P be the defining ideal of the point 1) in the ring of u*-polynomials

F {Yl, ... , Y.} in u* -indeterminates Yl, ... , Y. over F. Let A be a characteristic set of P with respect to a permissible ordering of the set {IYj II E f, 1::; j ::; s} (in the sense of Definition 3.4.19 and Remark 3.4.20), and let £j = {(k1' ... ' kn ) E:;En I O~l, ... , o~"Yj is the leader ofsomeu*-polynomial A E A} (1::; j ::; s). Then

8

W1/IF(t) = L Wej(t), (6.4.7) j=l

where Wej(t) (1::; j ::; s) is the standard dimension polynomial of the set £j ~ :;En (see Definition 2.5.8).


PROOF. Let us consider a permissible ordering < of the set {"'tYj I "'t E f, 1 ~ j ~ s} and let V be the set of all terms "'tYj (I E f, 1 ~ j ~ s), that are not transforms of any leader UA, A E A. By the same arguments as in the proof of Theorem 6.4.1, we obtain that if

V(r) = bYj E V I ord"'t ~ r, 1 ~ j ~ s} (r EN),

then the set {v (17) I v E V (r)} is a transcendence basis of the field F (b17j I "'t E f(r), 1 ~ j ~ s}) over F. Furthermore, for every j = 1, ... , s and for any r EN, the number of terms "'tYj = 0'~1 ... O'~nYj in the set V(r) equals the number of elements (k1 , ... , kn) E u.:;n \ Ej with 2:?=llk.1 ~ r that are not greater than any element of Ej with respect to the order ~ on u.:;n introduced in Section 2.5. It follows that if wc;(t) is the standard dimension polynomial of the set Ej, then trdegF F(b17j I "'t E r(r), 1 ~ j ~ s}) = 2:;=1 wc;(t), for all sufficiently large r E N, so that the polynomial W'1IF(t) = 2:;=1 Wc;(t) satisfies conditions (6.4.5), (6.4.7), and degw'1IF ~ n.

It remains to show that ifw'1IF(t) is written in the form (6.4.6), then the coefficient a coincides with the (7"* -transcendence degree of the field Gover F (of course, this also means a E u.:;). Let d = (7"* -trdegF G. Without loss of generality we may suppose that the elements 171, ... , 17d form the (7"* -transcendence basis of the field G over F (see Proposition 3.4.39(1», so that

W'1IF(r) ~ trdegF F( b17j I "'t E r(r), 1 ~ j ~ d})

= d. Cardr(r) = dt(-lt-'2' (:) (r ~ i) .=0

(see formula (2.1.23» for all sufficiently large r EN. Therefore, a = ~n1lf2'JF(tl ~ d. To prove the inverse inequality let us (as in Section 3.4) denote the k-th ortant

of the set u.:;n by u.:;k (1 ~ k ~ 2n) and set

Since elements 17d+ 1, ... ,'T/. are B-algebraic over F (171, ... , 17d), for each j = d + 1, ... , s and for each k = 1, ... , 2n , there exists an element "'tY' E rk such that

"'tY ' 17j is algebraic over the field F(171, .. ·,17d)(b17j l"'t E rk, "'tYj < "'tY'Yj}).

Therefore, there exists a number ro EN such that "'tY ' 17; is algebraic over the field

F(b17' I "'t E f(ro), 1 ~ i ~ d} U b17; I "'t E r, "'tYj < "'tY'Y;})

for every j = d + 1, ... , s. If "'t'Y; (I' E rk, d + 1 ~ j ~ s) is a transform of "'tY' Y;

(in the sense of Definition 3.4.19) and ord"'t' = r' ~ ry' = ord"'tY', then "'t'17; is algebraic over the field

F(b17' I "'t E r(ro + r' - r)kl), 1 ~ i ~ d} U b17; I "'t E r,"'tYj < "'t'Yj})


Thus, for every sufficiently large r E N (precisely for r 2: maXj,dr?)}) any element of the field F( b1)j I '"'/ E f(r), 1::; j ::; s}) is algebraic over the field F( b1)i I '"'/ E

f(ro + r), 1::; i::; d} U b1)j 1'"'/ E r(r) \ U1~k9nfk(r - rjk)hY), d+ 1::; j::; s}), where fk(p) (p E N) denotes the set bE fk I ord,",/::; pl.

Now, the same arguments as in the last part of the proof of Theorem 6.4.1, show that

::; d· Cardf(r + ro) + L

for all sufficiently large r E N. Since Card fk (r - r)k») = p(r - ry), n) = (r-r;;l+n) (see formula (2.1.22))'

W'1IF(t) ::;d~(_1)n-i2i (7) C ~ i)+ + L L [(t:n) - c-rJ~)+n)l+o(tn).

d+1~j~.1~k~2n

It follows that a = L1 n;:..IF whence a = d = u* -trdegF G. This completes the proof. 0

6.4.9. DEFINITION. The numerical polynomial W'1W(t), whose existence is established by Theorem 6.4.8, is called the dimension polynomial of the finitely generated u* -extension G of F, associated with the family of u* -generators 1) = (1)1, ... ,1).).

6.4.10. EXAMPLE. Keeping the notations of Theorem 6.4.8 and supposing that the elements 1)1, ... , 1). are u* -algebraically independent over the u* -field F, we obtain that the defining ideal P of the point 1) = (1)1, ... , 1).) in the ring of u*polynomials F {Y1, ... , y.} is the zero one, hence its characteristic set is empty. Therefore, £j = 0 for every j = 1, ... , s, so that

(see formula (6.4.7)).

Let G be a u* -extension of a u* -field F (Char F = 0 with a basic set u = {a1, ... ,an } and let G = F(1)l, ... ,1).) for some finite family 1) = (1)1, ... ,1).)

of elements of G. By Proposition 3.4.46, the vector G-space DerF G of all Fderivations of the field G into itself can be considered as a vector u* -G-space such that a(D) = a-i. D . a for all elements a E u*, DE DerF G. Proposition 3.4.46 also shows that the dual vector G-space (DerF G)* = HomG(DerF G, G) and its vector G-subspace of differentials f2F( G) can be regarded as a vector u* -G-space such that a(ip)(D) = a(ip(a- 1 (D))) and a(d() = da(() for all ip E (DerF G)*, DE


DerF G, 0' E (1'*, ( E G. (Remember that if ( E G, then d( is the element of (DerF G)* such that (d()(D) = D(() for every D E DerF G. Furthermore, the set {de I (( E Gn generates nF(G) as a vector G-space). In what follows, for every r EN, we denote the field F(f(r)771U· . ·Uf(r)77s) and the vector G-subspace nF(G)r of nF(G), generated by the set {de I ( E Gr}, by Gr and nF(G)r, respectively. Furthermore, we set Gr = F and nF(G)r = 0 for every r E Z, r < O.

6.4.11. THEOREM. With the preceding notation the following statements hold:

(1) the sequence of vector G-spaces (nF(G)r )rEZ is an excellent filtration of the vector (1'* -G-space n F ( G) ;

(2) dimG nF (G)r = trdegF Gr for every r E Z; (3) the dimension polynomial of the vector (1'*-G-space nF(G), associated with

the excellent filtration (nF( G)r )rEZ, coincides with the dimension polynomialllt'lIF(t) of the (1'*-extension G 2 F, associated with the system of (1'* -generators 77 = (771, ... , 77.)·

PROOF. (i) The definition of the vector G-space nF(G)r (r E Z), the equality ad( = da(() (0' E (1'*, (E G), and the fact that Gr+p = Gr ( {-y(() I, E f(p), (E Gr}) for all r E Z, pEN, show that (nF ( G)r )rEZ is a filtration of the vector (1'*G-space nF(G) and £pnF(G)r = nF(G)r+p for all r E Z, pEN ((£p)PEZ denotes the usual ascending filtration of the ring of (1'*-operators £ over G). Since the finite family {de I ( E Uf=lf(r)1J;} generates the vector G-space np(G)r (r EN), (nF(G)r)rEZ is an excellent filtration of nF(G).

(ii) Let (1, ... , (I(r) be a transcendence basis of the field Gr (r E N) over F and let Fr = F((l, ... , (I(r)) for every r E N. By Proposition 1.5.8, any element d( with ( E Fr is a linear combination of the elements de!' ... , d(l(r) with coefficients from the field G. Furthermore, if ~ is arbitrary elements of Gr , then ~ is algebraic over the field Fr , so that there exists a polynomial

in one indeterminate x with coefficients from Fr such that Pe(~) = 0, but p{(~) =1= 0 (as usual, p{(~) denotes the value of the derivative of Pe(x) for x = ~). The equality Pe(~) = 0 implies

whence d~ = - p~(e) dla - ... - pJ~e) dlq (remind that P€(~) =1= 0). Since each of the

elements dla, ... , dlq can be written as a linear combination of de!' ... , d(l(r) with coefficients from G, the element d( can be also written as such linear combination. Thus, the elements d(l, ... , d(l(r) generate the vector G-space nF( G)r.

It remains to show that the elements d(l, ... , d(l(r) are linearly independent

over G (so we shall have l(r) = dimG nF(G)r = trdegF Gr ). Let E~~~ Aid(i = 0, where Ai E G (1 ~ i ~ l(r)). By Proposition 1.5.5, for every i = 1, .. . ,l(r) there

exists a F -derivation Di of the field Fr into G such that Di ((j) = ' ~ J. ~' { 0 ·f· =1= .

1, If J = L

Since all fields considered are of zero characteristic, for every i = 1, ... , l(r) and


for any q E N, q ~ r, there exists a unique F-derivation D;q) of the field Gq into G whose restriction on Fr coincides with Di (see Proposition 1.5.6). Clearly

D;q+1) IG q = D;q) (q = r, r+ 1, ... ) for every i = 1, ... , l(r) and for all q EN, q ~ r.

Now we can naturally define a derivation Di: G -+ G such that DilGq = D;q) for . - - { 0, if j "I i,

all q E N, q ~ r. In partIcular, DdFr = Di and Di((j) = .. .. Now, 1, If J = z.

applying the mapping I:~~{ Aid(i to the element Dj E DerF G (1 ~ j ~ l(r)), we

obtain that (I:~~{ Aid(i)Dj = I::~~ AiDj((;) = Aj = O. Therefore, the elements d(1, ... , d(l(r) are linearly independent over G, so these elements form a basis of the vector G-space nF(G)r. Thus, dimg nF(G)r = trdegF G r and this completes the proof of the theorem. 0

6.4.12. REMARK. Theorems 6.4.8 and 6.4.11 on the dimension polynomials of (J"* -fields extensions have been formulated for the fields of zero characteristic. However, Theorem 6.4.11 (and, consequently, Theorem 6.4.8) may be generalized for inversive difference fields of arbitrary characteristic if we introduce the notion of excellent and separable filtration of a (J"* -extension of a (J"* -field (just as it has been done for differential extensions in [J069a]). We leave the formulation and the proof of corresponding assertions to the reader.

The following result is the immediate consequence of Theorem 6.4.11 and Example 6.3.6.

6.4.13. COROLLARY. Let F be an inversive difference field with a basic set (J" = {Cl1, ... ,Cln }, let G = F(1]1, ... ,1].) be a (J"*-extension of F generated by a finite family 1] = (1]1, ... , 1].) and let IJI '1IF (t) be the dimension polynomial of this (J"* -extension, associated with the family 1]. Furthermore, let

dp _ 1 do ~". P r. (G) 0 o -+ Mp ~ Mp- 1 -+ ... ~ lVJO -'-+ UF -+

be a free resolution of the module of differentials nF(G), associated with the extension G 2 F. (Each Mj (0 ~ j ~ p) is a finitely generated filtered (J"* -G-module whose dimension polynomialIJlMj(t) is defined by formula (6.3.4). The homomorphisms p, do, ... , dp - l are compatible with the filtrations of the modules, where nF(G) is considered with the filtration described before Theorem 6.4.11.) Then

P

lJI'IIF(t) = })-I)jIJlMj(t). (6.4.8) j=O

In particular, it follows that the polynomiallJl'lIF(t) can be written in the form

n .(t+i) lJI'IIF(t) = t;ai 2' i ' (6.4.9)

where ao, al, ... , an E ~. (Expression (6.4.9) will be called the canonical form of the dimension polynomiallJl'lIF(t)).

The formulation and the proof of the following three statements are precisely analogous to those of the corresponding assertions on dimension polynomials of difference extensions (see Propositions 6.4.4, 6.4.6 and Theorem 6.4.7).


6.4.14. PROPOSITION. Let F be an inversive difference field with a basic set u = {a1, ... ,an} and let 7] = (7]1, ... ,7].), (= ((l, ... ,(p) be finite families of elements of a u* -extension G of F such that G = F (7]1, ... , 7].) = F «(1, ... , (p). Let WIJIF(t) = a~.r tn +o(tn) and W(IF(t) = b;!" tn +o(tn) be the dimension polynomials of the u* -extension G ;2 F, associated with the families of u* -generators 7] and (, respectively. Then a = b = u· -trdegF G, deg WIJIF(t) = deg W(IF(t) and the leading coefficient of the polynomial WIJIF(t) coincides with that ofW(IF(t).

6.4.15. DEFINITION. Let G = F(7]l, ... , 7].) be a finitely generated u·-extension of an inversive difference field F (Char F = 0) with a basic set u = {a 1, ... , an} and let

n i(t+i) WIJIF(t) = t;ai2 i

be the dimension polynomialofthis extension, written in the canonical form (6.4.9). Then the numbers d = deg W'IIF(t) and ad = D,dW'IIF(t)/2d are called, respectively, the inversive difference (or the inversive u· -) type and the typical inversive difference (or the typical u* -) transcendence degree of the u· -extension Gover F. These numbers will be denoted by u·-typeF G and u·-t. trdegF G, respectively.

6.4.16. PROPOSITION. Let F be an inversive difference field with a basic set u = {a1, ... , an}, let G = F(7]l, ... , 7].) be a u·-extension of F, and let WIJIF(t) be the dimension polynomial of this extension, associated with the system of u·generators 7] = (7]1, ... , 7].). Furthermore, let d = u· -trdegF G and let 7]1, ... , 7]d be the u· -transcendence basis of Gover F. Then

(6.4.10)

6.4.17. THEOREM. Let G = F(7]l, ... , 7].) be a finitely generated u·-extension of an inversive difference field F with a basic set u = {a1, ... , an} and let WIJIF(t) be the dimension polynomial of this extension, associated with the system of u*generators 7] = (7]1, ... ,7].). Then the following conditions are equivalent:

(1) WIJIF(t) = dL:~=o(-1)n-i2i(7)eti); (2) u· -trdegF G = trdegF F(7]l, ... ,7].) = d

Theorem 5.6.3 for extensions of differential fields might lead us to expect that if G is a finitely generated u· -extension of an inversive difference field F with basic set u = {a1, ... ,an} such that u·-trdegFG = 0, then there exists a set U1 = {PI, ... , Pn} of pairwise commuting automorphisms of G such that F is the ui-subfield of G, the set u and U1 are equivalent and G is a finitely generated u;extension of F where U2 = {PI, ... , Pn-d. The following example shows that this is not so.

6.4.18. EXAMPLE. Let R. be field of real numbers considered as an inversive difference field with the basic set u = {a} where a is the identical automorphism of the field R.. It is evident that R. can be regarded as a u· -su bring of the u· -ring


m (IT = {a}) of all functions f(x) (x E~) with real values which are defined for all x E ~, except, maybe, the finite number of points. The action of a on the ring m is defined by the condition af(x) = f(x + 1) for any f(x) Em.

Let 1) denote the function 22'" E m and let G = ~(1)) be a IT"-extension of ~ (which is, of course, a IT*-subfield of m). Since a(1)) = 22"'+1 = 1)2, we see that G = ~(1),.j1}, 111, ... ), where ..jii = 22"'-' = a- 1(1)), 111 = 22",-2 = a- 2 (1)), .... Furthermore, with the notation of Theorem 6.4.8 we have

for every r E N, so that W!)IJlt(t) = 1 and 1T*-trdegJltG = O. Because Card IT = 1, any basic set of automorphisms which is equivalent to IT coincides with IT. Therefore, in order to establish that the property mentioned before this example does not hold in our case, it suffices to show that the field G cannot be generated by any finite family of elements over F. But this is evident: if G is a finitely generated extension of JR, one could choose the system of generators of G over ~ from the set {1),..jii, 111, ... } (because G = F( 1), ..jii, 111, ... )), and obtain that G = F(1),..jii, ... , 2\f77) for some pEN. However, this is impossible, because

2 P+if77 E G, but 2 P+if77 (/. F(1),..jii, ... , 2\f77). Thus, IT"-trdeglll G = 0, but the field G is not a finitely generated extension of JR.

Using Theorem 6.3.19, we can obtain the following result that is a weak version of Theorem 5.6.3 in the difference case.

6.4.19. PROPOSITION. Let F be an inversive difference field with a basic set IT = {a1, ... , an}, G a finitely generated IT"-extension of F, and d = 1T"-typeF G. Then there exists a set 1T1 = {PI, . .. , Pn} of pairwise commuting automorphisms of G and a finite family ( = ((1, ... , (p) of elements of G such that 1T1 '" IT and if 1T2 = {PI, ... , Pd}, then G is an algebraic extension of the field H = F ((1, ... , (p)o; (i.e. of the 1T2-extension of the 1T2-field F generated by a finite system ().

PROOF. By Theorem 6.4.11, the module of differentials nF(G) associated with the extension G ;2 F is a finitely generated vector IT* -G-space and IT" -type nF (G) = d. By Theorem 6.3.19, there exist a set 1T1 = {PI, ... , Pn} of pairwise commuting automorphisms of G and a finite family of elements ( = ((1, ... , (p) ((i E G for every i = 1, ... , p) such that ITI '" IT and the elements d( 1, ... ,d(p generate n F (G) as a 1T2-G-module where 1T2 = {PI, ... , Pd}. If H = F((I, ... , (p)o; and 1) E G, then the element d1) E nF (G) is a linear combination of the elements d(l, ... ,d(p with coefficients from the ring of IT:;-operators over G, i.e. d1) is a linear combination of the finite number of elements d(-Yij(i) with coefficients from G (1 ::; i ::; p, 1::; j ::; ki for some kl, ... ,kp E N), 'Yij E r 02 for all i, j, so that each lij can be written in the form P~' ... ridd for some 11, ... ,Id E Z. By Proposition 1.5.9, the element 1) is algebraic over the field F( bij(i I 1 ::; i ::; p, 1 ::; j ::; ki}), hence it is algebraic over the field H. This completes the proof. 0

Let F be an inversive difference field with a basic set IT = {Q1, ... , Qn} and let G be a IT* -extension of F that has finite systems of IT* -generators. Let W; (G, F) denote the set of all dimension polynomials of the IT" -extension G ;2 F, associated with various finite systems of IT* -generators of this extension. By Theorems


6.4.8(4) and 2.5.5(3), W*(F, G) is a subset of the set W of all Kolchin numerical polynomials of subsets of Nn . Since the latter set is well-ordered (see Proposition 2.4.14), there exists a system of u* -generators 7] = (7]1, ... ,7].) of Gover F such that the corresponding dimension polynomial qi'llF(t) is the minimal element of W; (G, F) (with respect to the ordering of the set of all numerical polynomials that was introduced in Definition 2.4.1). The polynomial qi'lIF(t) is called the minimal polynomial of the u*-extension G of F and is denoted by qiGIF(t).

6.4.20. EXERCISE. Let F be an inversive difference field with a basic set u = {aI, a2}, let F {Yl, Y2} be the ring of u* -polynomials in u* -indeterminates Yl, Y2 over F and let P = [alYl + a2Y2] be the u"-ideal of the ring F{Yl' Y2} generated by the linear u" -polynomial alYl + a2Y2. Let G denote the quotient u* -field of the u"-ring F{Yl,Y2}/P. Show that u"-ideal P is prime, and determine the dimension polynomial of the u* -extension G 2 F associated with the system of u" -generators (:iiI , jh) where Yi is the canonical image of the u" -indeterminate Yi (i = 1, 2) in the field G.

6.4.21. REMARK. In what follows, we concentrate our attention on dimension polynomials of inversive difference structures. This is connected with the fact that the main results in the "difference" case, much more often than in the "inversive difference" one, are similar to the corresponding statements of Differential Algebra. In particular, formula (6.4.2) for the dimension polynomial of a finitely generated difference field extension is similar to the formula of Theorem 5.4.1 for the differential dimension polynomial. It follows that the set W of all differential dimension polynomials associated with different finitely generated differential field extensions coincides with the set of all dimension polynomials of a finitely generated difference field extensions. In particular, if Wo (G, F) is a set of all dimension polynomials of finitely generated difference (u-) extension G of a difference field F with a basic set u = {al, ... , an} (associated with various finite systems of u-generators of Gover F), then the set Wo(G, F) contains the minimal element 'PGIF(t). The polynomial 't'GIF(t) is called the minimal polynomial of the u-extension G 2 F.

Theorem 6.4.8 allows to associate a certain numerical polynomial with any prime u" -ideal of a ring of u" -polynomials over a u" -field F.

6.4.22. DEFINITION. Let F be an inversive difference field with a basic set u = { aI, ... , an}, let F {Yl , ... , Ys} be the ring of u" -polynomials in u" -indeterminates Yl, ... ,Y. over F, and let P be a prime u* -ideal of the ring F {Yl, ... , Y. }. Let "'I = (7]1, ... , 7]s) be a generic zero of the ideal P (see Definition 3.4.11). Then the dimension polynomial qi'llF(t) of the u* -extension F("'Il,··., "'Is) of F, associated with the family of u" -generators 7] = (7]1, ... , 7].), is called the dimension polynomial of the prime u*-ideal P and is denoted by qip(t).

Remark that, by Proposition 3.4.12(3), any two generic zero of a prime u*-ideal P of the ring of u" -polynomials F {Yl, ... , Yn} are u· -equivalent, so that the notion of the dimension polynomial of the ideal P is well-defined.

6.4.23. DEFINITION. Let P be a prime u·-ideal of a ring of u·-polynomials F {Yl , ... , Ys} over a u" -field F, let 7] = (771, ... , 7].) be a generic zero of the ideal P, and let qip(t) = 2::7=0 ai2i e-;i) be the dimension polynomial of P written in the


canonical form. Then the coefficient an (that is equal to (T*-trdegF F(TlI, ... , TIs») is called the inversive difference (or (T"-) dimension of the prime (T"-ideal P and is denoted by (T" -dim P. The type d = deg W P (t) and the typical (T" -transcendence degree ad of the (T* -extension F (TIl, ... , TI.) of F is called the (T" -type and the typical inversive difference (or the typical (T* -) dimension of P, respectively. These characteristics of the ideal P are denoted by (T* -type P and (T* -t.dim P, respectively.

6.4.24. PROPOSITION. Let F be an inversive difference field with a basic set (T = {al,"" a} and let F {YI, ... ,Y.} be the ring of (T" -polynomials in (T*indeterminates YI, ... ,Y. over F. Let PI and P2 be prime (T* -ideals of the ring F{YI, .. ·,Y.} such that PI ~ P2· Then WPI > WP2 •

PROOF. Let Fr = F[{-yYj 11:S j:S s,,,/ E [(r)}) and Pir = PinFr (i = 1,2) for any r EN. Obviously Plr and P2r are the prime ideals of the ring Fr (r E N) and W P; (r) = trdegF Q(Fr / Pir) (i = 1,2) for all sufficiently large r E N (by Q(Fr / Plr ) and Q(Fr/P2r ) we denote the quotient (T"-fields of the (T"-rings of residue classes of Fr with respect to Plr and P2r , respectively). Furthermore, since PI _~ P2, we have Plr ~ P2r for all sufficiently large r E N hence trdegFQ(Fr/Plr ) > trdegF Q(Fr / P2r) for such r, so that W PI (t) > W P2 (t). 0

It is easy to see that one may apply the above scheme to introduce the notions of dimension polynomial, (T-type and typical difference ((T-) dimension for any prime reflexive ideal of a ring of difference ((T-) polynomials over a (T-field. We leave the corresponding formulations, as well as the formulation of the analogue of Proposition 6.4.24, to the reader.

Let F be an inversive difference field with a basic set (T = {al,.'" an}, let r be the free commutative group generated by the elements al, ... , an, and let F {YI, ... , Yn} be the ring of (T* -polynomials in (T" -indeterminates YI, ... ,Yn over F. Let ~ = {J>.(YI, ... , Y.) I A E A} be any set of (T* -polynomials of F{YI' ... , Ys} and let TI = (TIl, ... , TIs) be a point whose coordinates lie in some (T* -extension of F. In accordance with Definition 3.3.17 we shall say that the point TI is a solution of the set ~ or of the system of (T* -equations

(6.4.11)

if~ ~ KerWry, where Wry is the natural (T*-homomorphism

considered in Section 3.4. By the Basis Theorem (see Theorem 3.4.16)' system (6.4.11) is equivalent to some its finite subsystem, i.e. there exists a finite family of (T" -polynomials ~l ~ ~ whose set of solutions coincides with that of the system ~. Thus, while studying systems of algebraic (T" -equations, it suffices to consider finite systems of the form

(6.4.12)

where h,.·., fp E F{YI,"" Ys}.


6.4.25. DEFINITION. A system of algebraic 17* -equations (6.4.12) is called prime, if the 17* -ideal of F {Yl, ... , Y.} generated by the 17* -polynomials

is prime.

Note that any linear homogeneous system of 17* -equations, (i.e. a system of 17* -equations of the form E;=1 WjjYj = 0 (1 ~ i ~ p), where Wjj (1 ~ i ~ p, 1 ~ j ~ s) are u*-operators over a u*-field F) is prime. Indeed, the corresponding u*-ideal p = [E;=1 WljYj, ... , E;=1 WpjYj 1 is generated by the linear 17* -polynomials, so this ideal is prime (see Proposition 3.2.28).

6.4.26. DEFINITION. Let F be an inversive difference field with a basic set 17 = {a!, ... , an} and let £ = £F be the ring of 17* -operators over F. A 17* -operator W E £ is called symmetric iff the following condition holds: if the irreducible form of the operator W (i.e. the expression of W in the form W = al a~l1 ... a~ln + ... +ara~r' ... a~r .. , where aj E F, aj =I- 0 and (kil, ... ,kjn) =I- (kjl, ... ,kjn) for i =I- j (1 ~ i ~ r, 1 ~ j ~ r)) contains a term aai' .. . a~n (a E F, a =I- 0), then this form also contains all (2n - 1) monomials of the type aat" ... a~'n (a =I- 0) for various combinations of signs before 11, ... , In.

6.4.27. REMARK. With the preceding notation, we have ord(uv) = ord(vu) = ord u + ord v for any 17* -operator u E £ F and for any symmetric 17* -operator v E £ F.

Indeed, let u = ala~l1 ... a~'n + ... +ara~rl ... a~rn (ai =I- 0 for i = 1, ... , r) be the irreducible form of an element u E £F, and let the terms in this form are arranged according to the following condition: if i > j, then the monomial aja~il ... a~in is less than aja~;' ... a~;n, that is the vector (E;=1 Ikiql, IkilL ... , IkinD is less than or equal to (E;=1 Ikjql, Ikjd, ... , IkjnD with respect to the lexicographic ordering of Wn+1. Let us write the 17* -operator v in the form

where sign{lli} = sign{kli} (1 ~ i ~ n) and all monomials in the irreducible form of VI are less than b1 ai" ... a~ n. Then the monomial

a ~k11 ~kln (b )~k11 +111 ~kln+I'n 1~1 ···~n 1~1 ···~n

in the expression of uv, which is obtained by the multiplication of the above irreducible forms of u and v, cannot be canceled, hence

n n

ord(uv) = L Ikli + IIi I = L(lkIiI + Ihd) = ord u + ord v. i=1 ;=1

Similarly ord(vu) = ord u + ord v.

6.4. DIMENSION POLYNOMIALS

6.4.28. DEFINITION. A system oflinear O"*-equations of the form

• LWijYj = bj , where Wij E CF, bi E F (1:S; i::S; p, 1:S; j:S; s), j=1

319

is said to be symmetric if all 0"* -operators Wij (1 :s; i :s; p, 1 :s; j ::s; s) are symmetric.

It follows from Definition 6.4.25 that with any prime system of algebraic 0"* -equations of the form (6.4.12) one may associate a certain 0"* -ideal

P = [1I(Yl, ... , y.), ... , f p (Yl, ... , y.)]

of the ring F {Yl, ... , Ys}. Let G be the quotient 0"* -field of the integral domain F {Yl, ... , Ys} / P, and let 1]1, ... , 1]. be the images of the 0"* -indeterminates Yl, ... , Ys, respectively, under the natural epimorphism

F{Yl, ... , Y.} -+ F{Yl, ... , Ys}/ P r; G.

Clearly, G is the finitely generated 0"* -extension of F with the system of 0"* -generators 1] = (1]1, ... ,1].), and the system of 0"* -equations (6.4.12) is the defining system of equations on the generators, that is the 0"* -polynomials

of this system generate the kernel of the natural 0"* -homomorphism

F{Yl, ... , Y.} -+ F(1]lJ···, 1].) (Yi t-+ 1]i for i = 1, ... , s)

as a O"*-ideal of F{Yl' ... ' Ys}. Let W"fJIF(t) denote the dimension polynomial of the 0"* -extension G 2 F, associated with the system of 0"* -generators 1]. Then the polynomial W"fJIF(t) will be called the dimension polynomial of the prime system of algebraic O"*-equations (6.4.12).

Now, let us consider the notion of strength of a system of equations in finite differences. We intend to show that the natural analogue of the "measure of strength" of a prime system of algebraic 0"* -equations is the dimension polynomial of this system.

The strength of a system of equations in finite differences over a field of functions of several real variables is defined as follows (by analogy with the similar notion for systems of differential equation introduced by A. Einstein [Ei53]). Let us consider a system of equations in finite differences with respect to s unknow grid functions II, ... , f. of n real variables with coefficients from a functional field F. Suppose that the difference grid, whose nodes form the domain of the considered functions, has equal cells of dimension hI x ... x hn (hI, ... , hn E lR) and fills all space lRn. Furthermore, let us fix some node P of our grid.

A node Q of the grid is called a node of order i (with respect to the node P), if the shortest route between P and Q passing along the edges of the grid consists of precisely i steps (i E N). (By a step we mean a path from a node of the grid to a neighbour node passing along the edge between these nodes.) For example,


the order of nodes of two-dimensional grid are as follows (a number near a node designates the order of this node):

12 11 12 13 14

1 0 1 2 3 P

2 1 2 3 4

Now, let us consider the values of the grid functions h, ... , f. at the nodes whose order does not exceed i (i E N). If the unknown functions should not satisfy any system of equations, their values at the nodes of any order may be chosen arbitrary. Because of the system of equations in finite differences (and because of the equations that are obtained from ones of the system by the transformations of the type

/j(Xl' ... ,xn) ~ fj(Xl + Tlh l , ... ,Xn + Tnhn)

for some Tl, ... , Tn E :l, j = 1, ... , s), the number of independent values of the functions h, ... , f. at the nodes, whose order does not exceed i, decreases. This number, that will be denoted by Ai, expresses the "weakness" (by A. Einstein) of the system and also may be regarded as its "measure of strength" .

Suppose that the field of coefficients F satisfies the following condition: the mappings aj : F ~ F (1 ::S j ::S n) such that

for any function f(Xl' ... , xn) E F, are automorphisms ofthe field F. (For example, F is the field consisting of the zero function and of the grid functions of the form PIQ, where P and Q are grid functions defined all over the grid and vanishing at a finite number of nodes.) Then we may consider F as an inversive difference field with the basic set u = {al, ... , an}. Furthermore, suppose that the replacement of the unknown functions of the given system of equations in finite differences by the abstract indeterminates leads to the prime system of u* -equations over F. As above, we may define the dimension polynomial w(t) of the latter system; this polynomial is called the dimension polynomial of the given system in finite differences. Clearly, W(i) = Ai for all sufficiently large i EN. We see that the dimension polynomial of a prime system of algebraic u* -equations can be regarded as the difference-algebraic analogue of the "measure of strength" of a system of equations in finite differences.

6.5. Linear u*-Ideals and their Dimension Polynomials

Let F be an inversive difference field with a basic set u = {al, ... , an}, r a free commutative group generated by al, ... , an, and F{Yl, ... , Y.} the ring of u*polynomials in u* -indeterminates Yl, ... ,Y. over F (here and below we keep the notation and conventions introduced in Section 3.4).

6.5.l. DEFINITION. A u* -ideal I of F {Yl, ... , Y.} is called linear, if I is generated (as a u* -ideal) by linear u* -polynomials.

Proposition 3.2.28 implies that every proper linear u* -ideal is prime.

6.5. LINEAR a*-IDEALS AND THEIR DIMENSION POLYNOMIALS 321

Below we shall assume that the set of terms Y = {/Yj I I E r, 1 ~ j ~ s} is ordered by the order < introduced in Section 3.4: u = a~' ... a~nYi < v = a~l .. . a~nYj if and only if the vector (I:~=I Ikvl, i, kl"'" kn ) is less than the vector (I:~=I Ilv I, j, ll, ... , In) with respect to the lexicographic order on ~n+2.

6.5.2. DEFINITION. An autoreduced set A = {AI, ... , Ap}, consisting of linear u* -polynomials from F {YI, ... , Y.} is called coherent if the following conditions hold:

(1) if A E A and I E r, then IA is reducible to zero with respect to A; (2) if A, BE A and v = IllA = I'UB is a common transform of the leaders UA

and UB, then the u*-polynomial (I'IB)(lA) - (lIA)(J' B) is reducible to zero with respect to A.

6.5.3. THEOREM. Any characteristic set ofa linear u* -ideal of F{YI, ... , Ys} is a coherent autoreduced set. Conversely, if A is a coherent autoreduced set consisting of linear u* -polynomials in F {Yl, ... , Ys}, then A is a characteristic set of the linear u*-ideal [A).

PROOF. Let A be a characteristic set of a linear u* -ideal P of F {YI, ... , Ys}. Then P contains no nonzero u* -polynomials, reduced with respect to A (see Proposition 3.4.32)' hence, A is a coherent autoreduced set.

Conversely, let A be a coherent autoreduced set, consisting of linear u* -polynomials in F {YI, ... , Ys}. In order to prove that A is a characteristic set of the u* -ideal [AJ, we are going to show that [A) contains no nonzero u* -polynomials reduced with respect to A. Suppose that there exists a u* -polynomial 0 =I- M = I:f=ICi/jAi (Ai E A, Ci E F{YI, ... ,Ys}, Ii E r for i = 1, ... ,p) reduced with respect to A. Since the set A is coherent, without loss of generality we can assume that u"y. A • = liUA. for all i = 1, ... , p. Let v be the heighest term in the set {II UA" ... , IP UA p}. Without loss of generality we can assume that lillA. < v for 1 ~ i < q and liuA. = v for q ~ i ~ p for some q EN, 2 ~ q ~ p. Denoting the leading coefficient IpIp of the u* -polynomial/pAp by ap, we obtain the relationship:

q-I p-l P

apM = I>pCibi A ) + L Ci (apli Ai - (J;IA.hpAp) + L(JiIA.)Ci/pAp, (6.5.1) i=1 i=q i=q

in which the second sum is a u*-polynomial free of v. Since the set A is coherent, u*-polynomials apliA; - (J;IA.hpAp for i = q, ... ,p - 1 can be reduced to zero. Therefore equation (6.5.1) implies that apM can be expressed in the form

PI

a M = "dl) .. ,(l) AU) + C(1)", A p L...J. i,. P iP p, (6.5.2)

i=1

h C(I) { } (I) (I) A ( . ) C(I) { } were i E F YI,···, Ys , Ii E r, Ai E 1 ~ z ~ PI, P E F YI,···, Ys

and u ... ,f)A. = I~I)UA~I) < V = IpUA p' Since M is reduced with respect to A, M

is free of the leader U-YpAp, hence, this leader is present in some u* -polynomials


cP) (1 :s i :s pd· Therefore, there exist (T*-polynomials C~2), ... , d~) reduced with respect to A and an integer kEN, k f. 1 such that

(i=I, ... ,p).

Multiplying both sides of (6.5.2) by a;, we obtain

PI

a;M == L Cl2)'Yl 1) AP) (mod (/pAp)), ;=1

i.e. a;M = L:f~l Cl2)'Yl1) A~l) (since M and d2)'YP) AP) (1 :s i :s PI) are free of 'YpUAp ).

Let Vl be the greatest term among the terms 'YPlUA(I), ... ''Y~!)UA(I) (with re-I PI

spect to the fixed order on the set of terms). Obviously, VI < V and we can apply

the previous considerations to the (T*-polynomial Ml = L:f~l C;(2\P) AP) (since Ml is reduced with respect to A). We see that there exists a (T*-polynomial M2 = "1:'2 d 3).,P) A~2) reduced with respect to A and such that A~2) E A ... P) E r L..",=1 , I, , , , I, ,

Cl3) E F{Yl, ... , Y.} and u..,(3)A(2) = 'YP)UA (2) < Vl (P2 EN, PI ~ 1, 1 :s i :s P2). Continuing in the same way, ~e come to the contradiction: there exists a (T*polynomial N = C ·'YA reduced with respect to A and such that C E F{Yl, ... , Y.}, 'Y E r, A E A and U..,A = 'YUA. The theorem is proved. 0

Let F be an inversive difference field with a basic set (T = {O:I, ... ,O:n}, r the free commutative group generated by 0:1, ... , O:n, and F {Yl, ... , Y.} the ring of (T* -polynomials in (T* -indeterminates Y1, ... ,Y. over F. Consider a preorder ~ on F{Yl' ... , Y.} such that A ~ B (A, B E F{Yl, ... , Y.}) if and only if the leader UB

of B is a transform of the leader UA of A (in the sense of Definition 3.4.19). Under these assumptions Theorem 6.5.3 implies the following statement.

6.5.4. COROLLARY. If A is a linear (T*-polynomial in F{Yl, ... , Y.} and A fi F then the set of all minimal (with respect to ~) elements of {fA I 'Y E r} is a characteristic set of the (T* -ideal P = [A] .

PROOF. The set A of all minimal elements of {fA I 'Y E r} is, obviously, autoreduced. Besides, this set is coherent, since the conditions of Definition 6.5.2 are, obviously, fulfilled. Applying Theorem 6.5.3 we obtain that A is a characteristic set of P. 0

Theorem 6.5.3 allows to suggest the following scheme of constructing a characteristic set of a linear (T* -ideal determined by a finite set of linear generators. Namely, let P be a nontrivial linear (T*-ideal of F{Yl, ... , Y.} generated by a finite set of linear (T. -polynomials E = {Al' ... , Ap} (if the contrary is not said explicitly, we assume that the elements of E are written in ascending order). According to Theorem 6.5.3 the problem of constructing a characteristic set of P is reduced to the problem of finding a coherent autoreduced set ell ~ F{Yl, ... , Y.} such that [ell] = P. The process of constructing of such a set contains the procedure of computing the remainder of a linear (T* -polynomial B with respect to a linear (T* -polynomial A,

6.5. LINEAR u·-IDEALS AND THEIR DIMENSION POLYNOMIALS 323

i.e., the procedure of finding a linear /T. -polynomial E, which is reduced with respect to A and satisfies the condition E == aB (mod [AD for some a E F. By virtue of Theorem 3.4.27, such E always exists and can be found in the following way: if B is reducible with respect to A, then some transforms of the leader UA of A are present in B. We eliminate them, beginning with the greatest transform 'YUA. On the first step we obtain a /T·-polynomial Bl = b1A)B - b'YA, where b is the coefficient at U..,A in B. Then we apply the same procedure to Bl and A and so on. After a finite number of steps we obtain the desired /T·-polynomial E. The transition from B to E is called the reduction of the /T. -polynomial B with respect to A.

The construction of a coherent auto reduced set cJ> ~ F {Yl, ... , Y.} such that [cJ>] = [E] = P is carried out in two stages.

(1) The construction of an autoreduced set E' of linear /T·-polynomials which is equivalent to E (i.e. [E/] = [E]).

Consider an arbitrary pair Ai, Aj E E (1 ~ i < j ~ p). If Aj is reducible with respect to Ai, then we replace Aj by its remainder Aj with respect to Ai. We obtain a new set t, sort its elements in ascending order and apply the same procedure to this set, and so on. After each step the number of /T. -polynomials in the set does not increase, and the rank of each of /T. -polynomial does not increases either, so the process terminates after a finite number of steps, and we obtain an autoreduced set E' of linear /T.

polynomials such that [E/] = [E]. (2) The construction of a coherent autoreduced set cJ> of linear /T. -polynomials

such that [cJ>] = [E]. We set Eo = E and subsequently build sets of linear /T. -polynomials

E l , E2,"" each of which is equivalent to E, in the following way: if Ei is a noncoherent autoreduced set, then in order to obtain Ei+l we add /T. -polynomials of the following type to Ei:

(a) /T. -polynomials bl1B, b2B2-b21BJ'YlBl that are constructed for every pair B 1 , B2 E Ei such that the leaders UB, and UB. has a common transform v = 'YIUB, = 'Y2UB. and the /T·-polynomial bl1B,b2B2 -b21B2blBl is not reducible to zero;

(b) 'YA (A E Ei,'Y E r), if'YA is not reducible to zero. Then we reduce the system of /T. -polynomials obtained (i.e. we apply

the procedure of stage 1) and denote the result by Ei+l' Since Ei+l is of lower rank than Ei (see Definition 3.4.29), Proposition

3.4.30 implies that the process of constructing the sets Ei terminates after a finite number of steps. Obviously, the autoreduced set cJ> such that cJ> = Ek = Ek+l = ... for some kEN, is coherent, hence, cJ> is a characteristic set of[E], and [cJ>] = [E].

It follows from this considerations that one of the methods of computing the dimension polynomial of a linear /T. -ideal P of F {Yl, ... , Y.} can be described as the construction of a characteristic set of P by the scheme suggested and the application of formula (6.4.7). The second method is based on Corollary 6.4.13; in order to obtain the dimesion polynomial, we apply formula (6.4.8) to a free resolution of /T·-G-module of differentials Op(G), where G = F(1]I,"" 71.) is a


finitely generated u· -extension of F such that 1/ = (1/1, ... , 1/.) is a generic zero of P. (We can consider the quotient field of u· -ring F {Yl, ... , y.} / P as this u· -field extension G = F(1/1' ... ' 1/.), where 1/i = Yi + PEG for i = 1, ... , s.) Recall, that a free resolution of the u·-G-module np(G) has the form

(6.5.3)

where Mi (0 SiS p) are free filtered u· -G-modules, i.e. finite direct sums of filtered £a-modules of the form E!,. (m E N,l E~) described in Example 6.3.6 (£a is the ring of u·-operators over the u·-field G). It is known, that resolution (6.5.3) can be constructed, using the short exact sequences

0-+ Ker p ~ Mo -4 np(G) -+ 0,

0-+ Ker 7ro ~ Ml ~ Ker p -+ 0,

o -+ Ker 7rl ~ M2 ~ Ker 7ro -+ 0,

.,

where io, iI, i2 ... are inclusions, and 7ro, 7rl, 7r2, ... are natural epimorphisms of filtered £a-modules. (The construction ofthe resolution is illustrated by the following well-known diagram:

(6.5.4)

where do = io1l"o, d1 = il7rl, d2 = i27r2 and so on.) Thus, in order to construct modules Mo, Ml' ... ' we first have to construct modules Ker p, Ker 7ro, ... and homomorphisms p, do, d1 , ....

The following proposition shows the form of the u· -G-module Ker p in the case when u·-extension G = F(1/1, ... , 1/.) of F is generated by a generic zero 1/ = (1/1, ... , 1/.) of a linear u· -ideal P of F {Yl, ... , Y.} and p is a u· -epimorphism of a free filtered £a-module E~ with free generators h, ... , f. onto np(G) such that

• L Wdi -4 L wid1/i i=1 i=1

for any u· -operators WI, ... , w. E £a (such epimorphism p will be called natural).


6.5.5. PROPOSITION. Let F be an inversive difference field with a basic set 0' == {al"'" an}, F {Yl, ... , Y.} the ring of 0'. -polynomials in 0'. -indeterminates Y1 , ... , Y. over F and G == F (T}1, ... , 1/.) a 0'. -extension of F, generated by a system of 0'. -generators T} == (1/1, ... , T}.). Furthermore, suppose that the system of 0'''

equations on this generators that determines the extension has the form

{ ~1.1~1.+.W.12~2. ~ '.' '.+W16Ys == 0,

WplYI + Wp2Y2 + ... + wp.Y. == 0,

(6.5.5)

where wii E £p for i == 1, ... , p, j = 1, ... , s. In other words, T} is a generic zero of the prime 0'. -ideal of F{Y1, ... , Y.} generated by the O'''-polynomials E;=1 wiiYi (1 ~ i ~ pl. If p : Mo -+ np(G) is the natural epimorphism of the free filtered £G-module Mo == E~ with the free generators II,· .. , f. onto the £G-module np(G) = E:=1 £GdT}i' then Ker p is a £G-submodule of M o, generated by the elements g1 == E;=1 w1ih, ... , gp == E;=1 wpjfj·

PROOF. Since d(ax) == adx and d(,x) == ,dx for any a E F" E r,x E G (see the definition of the action of r on np(G) introduced in Section 6.4), we have p(g;) = P(Ej=1 wiih) == E;=1 Wii d1/i == d(Ej=1 Wij1/i) = dO == 0 for any i == 1, ... , p. Therefore, Ef=1 £Ggi ~ Ker p.

Conversely, let us show that Ker p ~ Ef=1 £agi. First of all, let us introduce the following terminology. If an element W E £a is written in the irreducible form

W = E~(~~ aklk, where l(w) E~, 0 # ak E G, Ik E r (1 ~ k ~ l(w)) and Ii # Ii for i # j, then l(w) will be called the length of w. If 9 = E%=1 wklk E Mo, then by the length of 9 we shall mean the number l(g) == E%=I'(Wk). Suppose that the inclusion Ker p ~ Ef=1 £agi does not hold. Then we can take an element 9 of minimal length among the elements of Ker p \ Ef=1 £Ggi. Let 9 == E~=1 Hkfk, where Hk == E~=1 Cki'Yki E £a (Cki E G, 'Yki E r for k == 1, ... , s). By the

assumption, p(g) == E~=1 E:~1 Cki'Yki d1/k == E~=1 E:~1 ckid(Tki1/k) = 0, i.e. the family (d(TkiT}k)h~k~'.199k is linearly dependent over G. Applying Proposition 1.5.9, we obtain that the family "'11/ == (Tki1/kh<k<. l<i<lk is algebraically dependent over F, i.e. there exists a 0'. -polynomial H E -F {y~,~ .~ , Y.}, which is a polynomial in the indeterminates (TkiYkh<k<. l<i<lk' such that H vanishes at the family 11/. Obviously, we can assume that-H-hM' the least total degree among all polynomials in the indeterminates (TkiYkh<k<. l<i<lk which vanish at IT} (in particular, we can - - , --assume that

for some k, i (1 ~ k ~ s, 1 ~ i ~ lk))' Since H(T1/) = 0, the O''''-polynomial H lies in the ideal P == [Ei=lw1iYi, ... ,E.i=IWpiYj] of F{Yl, ... ,Y.}. Let Wur = E~~l buri'Y~rj' where tur E~, burj E F, I~rj E r (1 ~ u ~ p, 1 ~ r ~ s, 1 ~ j ~


t"r). Then H can be written in the form

p qu S tur

H = LLh"v'Y"v(LL>"rn~rjYr) = u=1v=1 r=1j=1

s qu stu r

= LLhuv LL'Yuv(burj ) ·'Yuv'Y~rjYr, (6.5.6) u=1v=1 r=1j=1

where q" E N, huv E F{Y1'···' Ys}, 'Yuv E f (1 :S u :S p, 1 :S v :S q,,). Let ct> = h~bYa I 1 :S a :S s, 1 :S b :S La} be the set of all elements of the form 'YYi (/ E f, 1 :S i:S s) present in H or in the right-hand side of the equation (6.5.6) (obviously, 'YkiYk E ct> for any k = 1, ... , s; i = 1, .. . Lk) and let H~b = 8( 8,H ) (/1J)

'abYa for any a = 1, ... , Sj b = 1, ... , La (it is easy to see that if 'Y~bYa coincides with none of 'YkiYk (1 :S k :S s, 1 :S i :S Lk), then H~b = 0). The equality H(/1J) = 0 implies dH(/1J) = 0, i.e.

Let us show that

s 10 s Ik

go = L L H~b 'Y~bla = L L H~;'Ykdk E L £Gg;. a=1b=1 k=1 j=1 ;=1

Indeed, by (6.5.6) we have

p qu S tur

= L L L L huv ('Y1J)'Yuv (burj )'Yuv'Y~rj Ir u=1v=1r=1j=1

p qu S

= LLhuv{,1J)'Yuv LWurlr u=1v=1 r=1

p qu p

= L L huv (/1J)'Yuvgu E L £Ggj. u=1 v=1 ;=1

6.5. LINEAR u*-IDEALS AND THEIR DIMENSION POLYNOMIALS 327

It has been already mentioned that not all Hki (1 ~ k ~ s, 1 ~ i ~ lk) are equal to zero. Let, for instance, H~1 '1= O. Since 0 E Ker p \ L:~=1 faOu and go E L:~=l faou, we have 9 - cll(H~I)-lg0 E Ker p \ L:~=1 fagu, and I(gCll(H~1)-100) < I(g) (since the irreducible form of go contains only those elements of the form ,Ii ("{ E r, 1 ~ i ~ s) which are present in 0, and the number of elements of the form ,Ii in g-Cll (H~1)-1g0 is less than the number of such elements in 0). This contradiction with the choice of 0 shows that Ker p = L:f=1 faOi . The proposition is proved. 0

In each of the following examples we apply this proposition to the computation of the difference dimension polynomial of a system of linear u* -equations of the form (6.5.5) (i.e. of linear u* -ideal of the ring of u* -polynomials F {Yl, ... ,Y.} generated by the left-hand sides of this system). The method of computation is based on the construction of a free resolution of the corresponding module of differentials.

6.5.6. EXAMPLE. Linear homogeneous symmetric u*-equation. Let F be an inversive difference field of zero characteristic with a basic set u = {01,"" on}. Consider a u* -equation of the form

WIYl + ... +w.Y. = 0, (6.5.7)

where Yl,"" y, are u*-indeterminates over F, WI,'" ,W, E fF, and u*-operators WI, .. ' ,W. are symmetric (see Definition 6.4.26). Denote by G the quotient u*

field of the u* -ring F {Yl, ... ,Y.} / P, where P = [L::=1 WiY;] is a prime u* -ideal of F{Y1, ... , Y.}, set 17i = Yi+P E G (1 ~ i ~ s) and consider (6.5.7) as a determining u* -equation for the system of u* -generators 17 = (171,"" 17.) of the u* -estension G = F(17l' .. ·,17.) ;2 F (then the dimension polynomial \I1'1IF(t) of this u* -extension is the dimension polynomial of the u*-equation (6.5.7) and of the linear u*-ideal Pl. As it was mentioned in Section 6.4, in this case the module of differentials nF(G) has the form nF(G) = E:=1 fad17i. If p : E~ -+ nF(G) is the natural epimorphism of the free filtered fa-module E~ with the free generators It, ... , I. onto nF(G), then by Proposition 6.5.5 we have Ker p = fag, where 9 = E:=1 Wdi. The induced filtration on Ker p (with respect to which the inclusion (3 : Ker p -+ E~ is a homomorphism of filtered fa-modules) has the form (Ker pn L::=1 (fa)rf;)rEZ (by Theorem 6.3.10 this filtration is excellent). We shall show that for any r E ~ we have

• Ker p n ~)fa)rfi = (fa)r-k9, (6.5.8)

i=1

where k = maxl:::;i:::;.{ordw;}. Indeed, the inclusion

• (fa)r-k9 ~ Ker p n ~)fa)rfi

i=1

is obvious: if 0 E Ker p and u E (fa)r-k, then

ord(uw;) = ord u + ordw; ~ (r - k) + k = r


for any i = 1, ... , s (by virtue of symmetry of the operators Wi, see Remark 6.4.27), whence ug = E:=l (uwdli E E:=l (tG)rk Conversely, let HE Ker pnE:=l(tG)r' Ii. Then

i=l i=l

where Vi E (tG)r (1 :S i :S s), W E tG. Since /l, ... ,1s is a basis of the free tG-module E~, this equality implies Vi = WWi (1 :S i:S s). Since u"-operators Wi

are symmetric, we have ordw = ord Vi - ordwi for all i = 1, ... , s. In particular, ordw = ord Vi - k :S r - k for some i (1 :S i :S s), whence, W E (tG)r-k. Thus, Ker p n E:=l (tG)rf;) ~ (tG)r-kg, so that equality (6.5.8) is proved.

Let us consider the mapping 11" : Et -t Ker p such that >'h -t >.g for any >. E tG (h denotes the element of the basis of the tG-module Ef). It follows from (6.5.8) that 11" is a homomorphism of filtered tG-modules and the sequence of filtered tG-

modules Ef ~ Ker p .!!...t E~ is exact in Ker p. Furthermore, Ker 11" = O. (Indeed, if >'h E Ken (>. E tG), then >.g = E:=l )..Wdi = 0, therefore >'w; = 0 (1 :S i :S s) whence>. = 0). Thus, we obtain the following free resolution of the module Op(G):

(6.5.9)

where do = (3o 11". Applying Corollary 6.4.13 and relationship (6.3.4) to this resolution, we obtain the dimension polynomial W1)W(t) of u"-equation (6.5.7):

Formula (6.5.10) allows to find the invariants u"-typepG, u"-trdegFG and u"t.trdegpG of the extension G;2 F, which coincide with u"-typeP, u"-dimP and u" -t. dim P for the ideal P = [E:=l WiY;]: if s > 1, then u" -type p G = u" -type P = n, u"-trdegp G = u"-dimP = u"-t. trdegp G = u"-t. dimP = s - 1; if s = 1 and k :2: 1 (the case k = 0 is trivial), then u"-typeF G = u"-typeP = n - 1, u"_trdegF G = u"-dimP = 0, u"_t. trdegF G = u"-t. dimP = 2k.

Let F be a field of functions of n real variables. Let us consider mappings ai : F -t F (1 :S i :S n) such that (ad)(xl,"" xn) = I(Xl,"" Xi-I, Xi + hi, Xi+l, ... , xn) for any I E F, i = 1, ... , n; (hI, ... , hn are some fixed real numbers). Suppose that aI, ... , an are automorphisms of the field F. If F is viewed as an inversive difference field with the basic set u = {aI, ... ,an}, then the finite difference approximations of some classical differential equations give algebraic u*equations of the form (6.5.7). In what follows, we determine dimension polynomials for some of such equations.

a) Using five-point scheme for the finite difference approximation of two-dimensional Laplace equation (see [Hac87, Sect. 4.2]) we obtain a u"-equation of the form

6.5. LINEAR (To-IDEALS AND THEIR DIMENSION POLYNOMIALS 329

where O:f hl,h2 E JR. Applying (6.5.10) (with n = Cardu = 2), we see that the dimension polynomial for this equation is

Note, that dimension polynomials of some other algebraic u· -equations which result from the finite difference approximations of classic differential equations have the same form. For example, the dimension polynomial of the u· -equation

which results from the finite difference approximation of the simple diffusion equation by the scheme of Dufort and Frankel (see [RM67, Sect. 8.2, Tabl. 8.1]), is also equal to 4t.

b) The algebraic u· -equation, that is obtained by the finite difference approximation of two-dimensional Laplace equation using nine-point scheme (see [Hac87, Sect. 5.1]), has the form

[ 1 ( -1) 1 ( -1) h 2 Ql + Q1 - 2 + h2 Q2 + Q2 - 2 1 2

h~ + h~ ( -1)( -1)] + 12 . Ql + Q1 - 2 Q2 + Q2 - 2 Y = 0

Using (6.5.10), we find the dimension polynomial \lI(t) for this equation (here, as in the previous case, n = Card u = 2):

c) Consider the algebraic u·-equation, which is obtained by finite difference approximation of the Lorentz equation for the potentials of electromagnetic field (see [TS90, App. 2 to Ch. 5, Sect. 2]) if every partial derivative is replaced by the corresponding central difference. This equation has the form

3

'"' 1 ( -1) 1( -1) L...J -;; Qi - Qi Yi + ~ Q4 - Q4 Y4 = 0 ;=1 •

where (hI, h2' h3, T E JR). The dimension polynomial of the equation is

\lI(t) = 4 ~(-1)4-i2iG) C: i) -~(_1)4-i2i (:) C +! -1) =

= 48 C : 4) - 80 C ~ 3) + 40 C ~ 2) - 5.


In the two following examples we consider systems of algebraic u· -equations that are obtained from the well-known systems of differential equations (see, for example, [Ov82, Appendix, Sect. A4 and Sect. A15]) by finite difference approximations such that every partial derivative is replaced by the corresponding central difference. In the both examples F denotes an inversive difference field of coefficients, and OJ

(1 ~ i ~ 4) are the elements of the basic set u of F. The systems of algebraic u· -equations are viewed as determining systems of equations on the generators 1/1, ... ,1/$ of the u·-extension G = F(1/b···, 1/,) 2 F. In each example a free resolution of the module of differentials OF( G) is written, the dimension polynomial w(t) of the corresponding system is computed, and the invariants u·-typeF G, u·trdegF G, and u·-t. trdegF G of the extension G 2 F are found.

6.5.7. EXAMPLE. The finite difference approximation of the Dirac equation (with zero mass) gives the following system of linear u· -equations:

a4(04 - 04"1)Y1 - a3(03 - 0;1)Y3 - [ado1 - all) + a2(02 - ( 21)] Y4 = 0,

a4(04 - 04"1)Y2 - [a1(01 - all) - a2(02 - ( 21)] Y3 + a3(03 - 0;1)Y4 = 0,

a3(03 - 0;1)Y1 + [a1(01 - all) + a2(02 - ( 21)] Y2 - a4(04 - 04"1)Y3 = 0,

[ado1 - all) - a2(02 - ( 21)] Y1 - a3(03 - 0;1)Y2 - a4(04 - 04"1)Y4 = 0,

where the coefficients aj (1 ~ i ~ 4) belong to the field of constants C(F) of the u·-field F. Let p : E2 -t OF (G) be the natural epimorphism of the free filtered £a-module E2 with free generators /1, .. . ,14 onto OF( G) (here G = F(1/1 , ... ,1/4»). By Proposition 6.5.5, we have Ker p = L:=l £a9j where

91 = a4(04 - 04"1)/1 - a3(03 - O;l)fa - [a1(01 - all) + a2(02 - ( 21)] 14;

92 = a4(04 - 04"1)12 - [a1(01 - all) - a2(02 - ( 21)] fa + a3(03 - 0;1)/4;

93 = a3(03 - 0;1)/1 + [ado1 - all) + a2(02 - ( 21)] 12 - a4(04 - 04"1)/a;

94 = [a1(01 - all) - a2(02 - ( 21)] /1 - a3(03 - 0;1)12 - a4(04 - 04"1)f4.

Let us show that Ker p n L:=l (£a)r/; = L:=l (£a)r-19j. Indeed, the inclusion L:=l (£a)r-19j ~ Ker p n L:=l (£a)rf; is obvious (since u·-operators in the expressions of 9i (1 ~ i ~ 4), being linear combinations of elements of the basic, are symmetric, W9j E L:=l(£a)r/i (1 ~ j ~ 4) for any W E (£a)r-d. Conversely, let H = L:=l W;/i E Ker p n L:=l (£a)rfi, where Wi E (£a)r (1 ~ i ~ 4). Then there exist >'1, >'2, >'3, >'4 E £a such that L:=l w;/; = L;=l >'j9j. Comparing the coefficients at Ii (1 ~ i ~ 4) in the left- and right-hand sides of this equality, we obtain a system of equations which, after reducing to row-echelon form, can be written as follows:

>'la4(04 - 04"1) + >'3a3(03 - 0;1) + >'4 [a1(01 - all) + a2(02 - ( 21)] = WI,

>'2a4(04 - 04"1) + >'3 [ado1 - all) + a2(02 - (21)] - >'4a3(03 - 0;1) = W2,

>'3U = w~, >'4U = w~,

(6.5.11)


where

U =aHa2 - a2"1)2 + a~(a4 - a;1)2 - ai(a1 - all) - a~(a3 - ( 31);

W; =W3a4(a4 - a;l) - W1a3(a3 - ( 31) - w2 [ada1 - all) - a2(a2 - a2"l)] ;

W~ =w4a4(a4 - a;l) - WI [a1 (a1 - all) + a2(a2 - a2"l)] + w2a3(a3 - ( 31).

Since the u· -operator u is symmetric and w~, w~ E (£a )r+l, we have ord A4 ordw~ - ord u ~ r - 1, and similarly ord A3 ~ r - 1. Using these inequalities, we subsequently obtain from the first two equations of (6.5.11):

Therefore, AI, A2 E (£a)r-1, hence H E "L:=1 (£a)r-1gi. Thus, the equality Ker pn "L:=1 (£a)r/i = "L:=1 (£a)r-1gi is proved. It shows that if the mapping 7r : E~ ~ Ker p is such that "L:=1 uihi ~ "L:=1 Uigi (Ui E £a, hi are elements of the basis of the £a-module E~ (1 ~ i ~ 4)), then the sequence of the filtered £a-modules

El ~ Ker p ~ E~ (f3 is an injection) is exact in Ker p. If Q = "L:=ll-'ihi E Ker 7r (J1.i E £a for i = 1, ... ,4), then "L:=1I-'i9i = O. Replacing in this relation gi (1 ~ i ~ 4) by their expressions in terms of the basis II, .. . ,14 and equating the coefficients at every Ij to zero, we see that the u*-operators Pi (1 ~ j ~ 4) must satisfy a system of equations which, after reducing to row-echelon form, has the form (6.5.11) with WI = W2 = w~ = w~ = 0 (and with Pi = Ai (1 ~ i ~ 4)). Since such a system has the unique solution 1-'1 = ... = P4 = 0, we obtain, that Q = "L:=1 Pigi = 0, hence, Ker 7r = O.

Thus, the free resolution of f!F(G) has the form

o ~ E~ ~ E~ ~ f!F(G) ~ 0,

where do = f3 0 7r. Applying Corollary 6.4.13, we obtain the desired dimension polynomial:

W(t)=4~(-1)4-i2iG) x [e~i) -e+:- 1)]

= 64 e ~ 3) - 128 e ~ 2) + 96 e ~ 1) - 32.

In this case u*-trdegF G = 0, u*-typeF G = 3, u*-t. trdegF G = 8.

6.5.8. EXAMPLE. The system of linear u*-equations that is obtained by the


finite difference approximation of the system of Lame equations has the form:

[A - ~i (oi + 0 12 - 2)] Y1 - h11h2 (01 0 2 - 010:;1 + 0110:;1)Y2

1 -1 -1-1 - h1 h3 (01 0 3 - 01 0 3 + 0 1 0 3 )Y3 = 0,

- h11h2 (01 0 2 - 010:;1 + 0110:;1)Y1 + [A - ~i (oi + 0 12 - 2)] Y2

1 -1 -1-1 - h2 h3 (02 0 3 - 02 0 3 + O 2 0 3 )Y3 = 0,

1 -1 -1 -1 1 -1 -1-1 - h1 h3 (01 0 3 - 01 0 3 + 0 1 0 3 )Yl - h2 h3 (02 0 3 - 02 0 3 + O 2 0 3 )Y2

+[A- ~~(0~+032_2)]Y3=0, (6.5.12)

where A = -ir(0~+042-2)-aL:;=1 t:(0[+0;2-2); a,h i E C(F) (1:S i:S 4). Proceeding as in the previous example, we obtain that the free resolution of the

module of differentials in this case has the form

o -t E~ ~ E~ -4 Qp(G) -t O.

Thus, we arrive to the following expression for the dimension polynomial of the system of linear cr"-equations (6.5.12):

w(t) =3~(-1)4-i2iG)· [C~i) - C+~-2)]

=96C~3) -240C~2) +240C:1) -120.

The invariants of the cr" -extension G = F (TJ1, TJ2, TJ3) ::2 F have the following values: cr' -trdegp G = 0, cr" -typep G = 3, cr" -to trdegp G = 12.

6.6. Computation of Dimension Polynomials in the Case when the Basic Set Consists of Two Translations

Let F be an inversive difference field of zero characteristic with a basic set cr = {0,,8}, r a free commutative group generated by elements 0,,8, and

r(k) = b = oP,8q E r i ord, = ipi + iqi:S k} for every kEN.

Let us consider the ring of cr" -polynomials F {y} of one cr" -variable Y over the field F and let us associate with every cr"-polynomial A E F{y} the finite set r(A) of points (p, q) of the plane ]R2 such that p, q E ::z:: and a term oP ,8Qy appears in the irreducible representation of the polynomial A. Let 7r A denote the closed rectangle of the smallest area among all the rectangles which contain r(A) and whose sides are perpendicular to the vectors (1,1) and (1, -1). For example, if

6.6. COMPUTATION OF DIMENSION POLYNOMIALS 333

then r(A) = {(I, -1), (-1,1), (1,0), (0, -1), (2, 0), (0, On,

and the rectangle 11'A is shown in Fig. 6.6.1.

6.6.1. EXERCISE.

M4 -2

Fig. 6.6.1

a) Show that if M(Xl,X2) and M'(xLx~) are two adjacent vertices of the rectangle 11'A (A E F{y}), then \x~ - xli + \x~ - X2\ EN.

b) Show that if, for every term a P (3q y in the irreducible representation of a u*polynomial A E F{y}, the inequality \p\+\q\ ~ a holds for some nonnegative real number a, then \Xl\ + \X2\ ~ a for any point M(xl, X2) of the basic rectangle 7!' A .

Let us consider a metric p on the plane ~2 such that

for any two points M(Xl,X2) and M'(x~,x~).

6.6.2. DEFINITION. The rectangle 7!'A constructed above is called the basic rectangle of u* -polynomial A; its perimeter in metric p is called a u* -order of the polynomial A and is denoted by u* -ord A.

Note that the notion of u*-order (when Cardu = 2) is a natural generalization of the notion of the effective order in ordinary case (see [CoR65]).

6.6.3. LEMMA. With the above notation, u* -ord A = u* -ord(-yA) for every u*-polynomial A E F{y} and for every element IE f.

PROOF. Let "'( = aP (3Q, where p, q E Z. Then the basic rectangle 11'')'A is obtained by shifting of the rectangle 7!'A by the vector (p, q). Obviously, such shifting does not change the perimeter of the rectangle. 0

6.6.4. THEOREM. Let F be an inversive difference field with a basic set u = {a, (3} and F {y} the ring of u* -polynomials of one u* -indeterminate y over F. Let A be a linear u*-polynomial from the ring F{y}, A fI. F and P = [A] the u*-ideal of the ring F{y} generated by A. Then u*-type P = 1, u*-t.dimP = (u*-ordA)J4, and the dimension polynomial wp(t) of the linear u*-ideal P has the form wp(t) = (<1'-~rdA)t + b, where b E Z.

PROOF. By Corollary 6.5.4, we can choose a characteristic set A of the ideal P as the set of minimal elements of the set {IA \1 E f} with respect to the partial


order :9 on the ring F{y} introduced in Section 6.5. Let us represent 1E2 in the 2 4 - - - -

form IE = Uj=llEj, where lEI = N x N, 1E2 = IE_ x N, 1E3 = IE_ x IE_, 1E4 = N x IE_. Set fj = {aP{3q E f I (p,q) E IEj} and Yj = bY I , E fj} (1 ~ j ~ 4). Then f = UJ=1 fj and the set Y = by I, E r} can be represented as a union UJ=1 Yj. Without loss of generality, we may assume that the u* -polynomial A belongs to the characteristic set and has the following properties:

(1) the leader UA lies in the first quadrant (i.e. UA = aP {3q y, where (p, q) E N2);

(2) the side of the basic rectangle 'irA, which is perpendicular to vector (1, -1) is not shorter than any other side of this rectangle;

(3) the point (0,0) lies in the rectangle 'irA.

Let di denote the distance (in metric p) from the point (0,0) to ith side of the rectangle 'irA. We suppose that sides of this rectangle are numerated in counterclockwise order, beginning from the side which is perpendicular to the vector (1,1) and intersects the first quadrant. So, in Fig. 6.6.1 the sides of 'irA are numerated in the following order: M I M2, M 2M3, M3M4 M4Md. It easy to see, that u*-ordA = E:=l di (if UA = ai {3jy, then i + j = dt).

For every, = aP{3Q E f, the set T(-yA) is obtained by the shifting ofthe set T(A) by the vector (p, q). Therefore, in order to determine a characteristic set ofthe ideal [A] we can consider only those points of the set T(A) which lie on the sides of the basic rectangle 'irA. Actually, all leaders of u*-polynomials ,(A) (-y E f), that lie in the same quadrant, are defined by one of the monomials of the u· -polynomial A: leaders U...,A from the jth quadrant (1 ~ j ~ 4) can be obtained by multiplication of certain monomial of the u· -polynomial A by some elements " E f. Let us denote this monomial by Vj. Obviously, VI = UA and if, E f l , then U...,A = ,UA. It follows that ,Aft. A for all , E f I, , -I l.

In order to find elements of the characteristic set A, whose leaders lie in the second quadrant, we will consider polynomials ,A such that, E f 2 . For every subset A ~ f, let Jl(A) denote the lowest element of the set A (relative to an order on the set f for which a i {3j < a P {3Q iff the vector (Iii + iii, i, j) is lower than the vector (Ipl + Iql,p,q) with respect to the lexicographic order on the set 1E3 ). Let us construct a sequence of subsets Ao, AI, A2 , ... of the set f as follows:

Ao = f2,

{ 0,

A2k+1 = -IA a 2k

if UJ.I(A2k) E Y2 ,

otherwise;, (k = 1,2, ... ).

{ 0,

A2k = (3A2k - 1

if UJ.I(A2k) E Y2 ,

otherwise;

Note that Jl(A2k ) = a-k{3k and Jl(A2k+1) = a-k - 1{3k (k EN), so that ir,f = Jl(A!), (l EN), we can write ,f = a-[~l{3[~l (here [r] denotes the integer part of a real number r). Let m be the minimal integer for which Am -I 0. Then the polynomials ,~A (k = 0, ... , m) belong to the characteristic set A, the leader U",,:"A = ,:,. V2 lies in the second quadrant, and the leaders U...,;A (I = 0, ... , m - 1) lie in the first quadrant if 1 is even, and in the third quadrant if I is odd.

Similarly, if we replace a by {3, and {3 by a, we will find a number n E Nand elements ,~, ... ,,~ E f such that u· -polynomials ,i' A (i = 0, ... , n) lie in the


characteristic set A, and the leader u-y:: A = ,~ V4 of 0"* -polynomial,~ A lies in the fourth quadrant.

Let us now consider two cases. Case 1. At least one of the conditions max(m, n) > 1, V2 = V3, V3 = V4 holds.

Then A = biA I 0 ~ I ~ m} U b~A I 0 ~ k ~ n} is a characteristic set of the ideal P, and the set of leaders of 0"* -polynomials from A which we denote by UA, is as follows:

UA = {'~k VI I 0 ~ 2k < m} U {,~ VI I 0 ~ 21 < n}U

U b~P+IV3 I 0 ~ 2p+ 1 < m}U

U b~q+lV3 I 0 ~ 2q + 1 < n} U b~V4}'

(6.6.1)

Case 2. None of the conditions max(m, n) > 1, V2 = V3, V3 = V4 holds. Then the set

A = bi A I 0 ~ I ~ m} U b~ A I 0 ~ k ~ n} U {a -1 ,B-1 A},

is a characteristic set of the ideal P, and the set of leaders UA of the 0"* -polynomials from A is obtained by adjoining of the term a- l ,B-l v3 to set (6.6.1).

By Theorem 6.4.8, the dimension polynomial of the 0"* -ideal P coincides with the standard dimension polynomial w(t) of the set [; = {r(u) I u E UA} ~ ;Z2. In Case 1, when elements Vi (1 ~ i ~ 4) are pairwise distinct, the set [; does not contain points with nonzero coordinates. Setting [;j = [; n ;Zj, we obtain that

[;i={r(u)E[;lu=,Vj forsome ,Er} (l~i~4).

In accordance with Proposition 2.5.13, the leading coefficient of the dimension poly-4 . .

nomial We(t) = at + b is equal to I:j =l(e1 + e~) - 4, where

ej =( min {r(u)tJ, min {r(uh}) for j=I, ... ,4 u:T(u)Efj u:T(u)Efj

(hereinafter i-th coordinate of a point w E ;Z2 (i = 1,2) is denoted by w;). Note now that, if max(m, n) > 1, then

el = (Ir(Vlhl- [m; 1] ,Ir(vlhl- [n; 1]), e2 = (Irb~v2hl, Irb~v2hl),

e3 = (Ir(v3hl- [; + 1] , Ir(v3hl- [i + 1]), e4 = (Irb~v4hl, Irb~v4hl)·

If m = n = 1, then

e1 = (Ir(vt}11, Ir(vlhl)'

e2 = (Ir(v2hl + 1, Ir(v2hl),

e3 = (Ir(v3hl + 1, Ir(v3hl + 1),

e4 = (Ir(v4hl, Ir(v4hl + 1).


In both cases, for the leading coefficient of the polynomial we(t) we obtain the following expression: a = L:~=l d; = q·-~rdA.

Let Vl = V2, V3 ::j:. V4 and n > 1. Then the set C contains one point with zero coordinates. Applying Proposition 2.5.13, we obtain, that a = L:1=l (e{ + e~) - 3.

In this case e~ = IT(vlh 1- [mIl] -1, and all other elements e1 (1 ::; j ::; 4; i = 1,2) are the same as in the case of pairwise distinct Vj (1 ::; j ::; 4). As above, we obtain

a = L:~=l d; = u*-ordA/2. Similarly, we can consider any other case of coincidence of some of the terms

Vl, ... , V4. As a result, we obtain that the dimension polynomial of the u* -ideal P = [AJ has the form Wp(t) = q·-~rdAt+b, where b E~. Therefore u*-typeP = 1, u*-t.dimP = u*-ordA/4. The theorem is proved. 0

6.6.5. COROLLARY. Let F be an inversive difference field with a basic set u = {o, /1} and P a u* -ideal of the ring F{y}, generated by one linear u*-polynomial A E F{y} \ F. Let k be the minimal element of the set of all numbers lEN such that the ring F[hy I, E r(l)}] contains some element of the form ,'A, where " E r. Then u*-t.dimP < k.

PROOF. Since [AJ = [,' AJ for every element " E r, then without loss of generality we may assume that A E F[ hy I, E r( k)}]. Therefore, any point (p, q) of the basic rectangle 1I"A satisfies the condition Ipi + Iql ::; k, whence u*-ordA ::; 4k. Applying Theorem 6.6.4, we obtain that u*-t.dimP = u*-ordA/4::; k. 0

6.6.6. COROLLARY. Let F be an inversive difference field with a basic set u = {o, /1}, F {Yl, ... , Y.} the ring of u-polynomials in u-indeterminates Yl, ... , Y. over F and P a u* -ideal of this ring, generated by one linear u* -polynomial A E F {Yl, ... , Y.} \ F. Let B denote the u* -polynomial that is obtained by replacing all terms ,Yj (2 ::; j ::; s" E r) of the polynomial A bY'Yl. Then the dimension polynomialill' p (t) of P has the form

( t + 2) u* - ord B Wp(t)=4(s-l) 2 +( 2 -4n+4)(t+l)+b,

where b E~.

PROOF. By Corollary 6.5.4, the characteristic set A of the u* -ideal P = [AJ consists of the minimal elements of the set hA I , E r) with respect to the order ~ on the ring F {Yl, ... , Y.} such that A ~ B iff the leader UB of the u*polynomial B is a transform of the leader UA. For every j = 1, ... , s, we set Cj = {(p,q)} E ~2 I oP/1QYj is the leader of some u*-polynomial from the set A}. Then, as Theorem 6.4.8 shows, wp(t) = L:j=lWCj(t).

Now, let us represent each set Cj (1 ::; j ::; s) in the form £j = U~=I Cji where £ji is the family of all points of the set Cj that lie in i-th quadrant (1 ::; i ::; 4). It is easy to see (as in the appropriate part of the proof of Theorem 6.6.4) that all leaders oP/1QYj of u*-polynomials of A, for which the corresponding points (p,q) lie in the same quadrant, are determined by one of the terms of the u* -polynomial A. Therefore, if we fix i (1 ::; i ::; 4), then Cji is not empty for only one index j (1 ::; j ::; s). Thus, if Q is a u*-ideal of the ring F{YI, ... ,Y.} generated by the u*-polynomial B, then wp(t) = wQ(t) = L:j=IWCj(t), where C; = {(p,q) E ~2 I


term o:P j3q Yj is the leader of some u* -polynomial from the characteristic set of the u"-ideal Q} (1 ~ j ~ s). Since B E F{yt}, £; = 0 for j = 2, ... , s, hence (see

Theorem 2.5.7(4)) W£j = 4C12) - 4(t + 1) + 1 (2 ~ j ~ s). Furthermore, by

Theorem 6.4.4, W £~ (t) = G' - ~rdB (t + 1) + b', where b' E 2. Letting b = b' + 1, we obtain that

( t + 2) u* - ord B IVp(t)=4(s-1) 2 +( 2 -4s+4)(t+l)+b. 0

6.6.7. LEMMA. Let F be an ordinary inversive difference field with a basic set u = {o:}, F {Y1, ... ,Ys} the ring of u* -polynomials in u* -indeterminates Y1, ... , Ys over the field F and A a irreducible u* -polynomial from this ring. Let M = I::~=1 CiAi be a nonzero u* -polynomial from the u* -ideal [AJ (Ci E F {Y1, ... , Ys}, Ai = o:k'A (1 ~ i ~ I) for some k1, ... , kp E 2). Then there exists i E Pi, 1 ~ i ~ 1 such that degu• M 2: degu, Ai, where Ui is the leader of the u* -polynomial A;.

PROOF. Suppose, that the statement of the lemma is not true. Let m be the minimal positive integer for which there exists a u* -polynomial

(6.6.2)

from the u* -ideal [AJ such that degu• M < degu• Ai for i = 1, ... , m. (Here Ci E F{Y1, .. . , Ys}, A; = o:k. A for some ki E 2, and Ui is the leader of the u* -polynomial Ai (1 ~ i ~ m)). Every u*-polynomial Ai (1 ~ i ~ m) can be written in the form Ai = IiU1' + o( u1'), where di = degu, Ai, Ii is the leading coefficient of the u*

polynomial Ai, and o(u1') is a u* -polynomial such that degu• o(u1') < di . Without loss of generality we may assume that U1 < U2 < ... < U m and U m = o:kYj, where kEPi, (1 ~ j ~ s). Since Ui < U m for i = 1, ... , m - 1, the u*-polynomials A1 , •. . , Am - 1 do not contain term U m . Furthermore, without loss of generality we can assume that Am ACi for every i = 1, ... , m - 1. Since Am is irreducible, the resultant of the polynomials Ci and Am with respect to U m is a nonzero polynomial that does not contain _n. Thus, for every i = 1, ... , m - 1, there exists a number qi E Pi and a u* -polynomial c: E F {Y1, ... ,Y.} that is reduced with respect to Am and satisfied the condition

I~Ci = Cf (mod (Am)). (6.6.3)

Multiplying both sides of equality (6.6.2) a by suitable power of the leading coefficient 1m , we obtain that

IinM = C~A1 + ... + C:"_l (mod (Am)), (6.6.4)

for some q E Pi and some u* -polynomials C~, ... , C:"_l reduced with respect to Am. Since U m is not contained in 1m and degu~ M < degum Am we have

(6.6.5)


If the u·-polynomial fm does not contain U1, ... ,Um-1, then degu.(f:hM) < d; (1 ::; i ::; m-l), that contradicts the choice of M. Therefore, fm contains some of the leaders U1, ... ,Um-1. Let U r = a q• Yj (qr E W, 1 ::; j ::; s) be the greatest such leader in f m. Since the u* -polynomial A is irreducible and deg fm < deg Ar , the u·polynomials f:h and Ar are relatively prime, hence the resultant R1 = P1f:h +Q1Ar of the polynomials f:h and Ar with respect to the indeterminate U r is distinct from zero and does not contain the term Ur (here P1 , Q1 E F {Y1, ... , Y. }). Let up = aq" Yj (qp E W, 1 ::; j ::; s) be the greatest of the leaders Ul, ... , Ur-1,

contained in R1. Since R1 contains a term v that is not contained in neither Am, nor Ar (we can take the lowest term of the form al Yj in Ap as v), u· -polynomials Ap and R1 are relatively prime. The resultant R2 of these u· -polynomials with respect to the term up has the form

It is distinct from zero and it does not contain the terms up, Up+1, ... , Um. Continuing in the same way, we obtain a u· -polynomial

R = Tof:h + T1Arl + ... + l1Ar,

(To, ... , Tp E F {Y1, ... , Y.}, r1 = r > r2 = P > ... > rl),

which does not contain U1, ... , Um. Multiplying both sides of equation (6.6.5) by To, we get

Tof:hM = (R - T1Arl - ... -l1Ar,)M

= ToC~ A1 + ... + TOC:"_l' l.e.

RM = D1Al + ... + Dm-IAm- l (6.6.6)

for some u· -polynomials D l , .•. , Dm - 1 , and

degui (RM) = degui (M) < d; for i = 1, ... , m - 1.

Thus, we arrive to the contradiction with the minimality of the length of representation (6.6.2). The Lemma is proved. 0

6.6.8. THEOREM. Let F be an ordinary u·-neld with a basic set u = {a}, R = F{Yl, ... , Y.} the ring of u·-polynomials in u·-indeterminates YI, ... , Y. over F, and P = [A] a prime u· -ideal of this ring, generated by an irreducible u·polynomial A E R \ F[Y1, ... , Yn]. Then any two u·-polynomials Al = ak'A and A2 = ak2 A, whose leaders are not non-trivial transforms of any Ua.A, constitute a characteristic set of the ideal P.

PROOF. It follows from the definition of u· -polynomials A1 and A2 that these two u· -polynomials constitute an autoreducible set of the ideal P. By Lemma 6.6.7, the ideal P does not contain nonzero u· -polynomials that are reduced with respect to {A1' A2}. Therefore, {AI, A2} is an autoreducible set of the lowest rank in the ideal P, i.e. a characteristic set of this ideal. 0


In the following examples we find the dimension polynomials of systems of linear O'*-polynomials with the help of formula (6.4.7): we construct a characteristic set of the corresponding linear 0'* -ideal of the ring of 0'* -polynomials and then apply formula (6.4.7). The advantage of this method (in comparison with the method based on the construction of a free resolvent of the corresponding module of differentials) is that it can be applied to any (not necessarily symmetric) system of linear 0'* -equations.

6.6.9. EXAMPLE. Let F be an inversive difference field of zero characteristic with a basic set 0' = {aI, a2} and let F{y} be the ring of O'*-polynomials of one O'·-variable y over F.

a) Let us consider the 0'* -equation

(6.6.6)

where elements al and a2 lie in the subfield of constants of the field F (dimension polynomial of a similar equation was calculated in Example 6.5.6(a) by constructing a free resolvent of the O'*-F-module of differentials OF(G), where G is the O'*-field of quotients of the factor ring F{y}/[al(al + ail - 2) + a2(a2 + a2'l - 2)]y).

(-1,0)

(-1,0)

Fig. 6.6.2 The basic rectangle ofthe 0'* -polynomial A = [al(al +ail-2)+a2(a2+a2'1-2)]y

has the form shown in Fig. 6.6.2, whence 0'. -ord A = 8. Applying Theorem 6.6.4, we obtain that dimension polynomial of our O'*-equation (i.e. the prime O'·-ideal [A] of the ring F{y}) has the form w(t) = 4t+b, where b E Z. Therefore, O'*-dim[A] = 0, O'*-type[A] = 1, and O'·-t.dim[A] = 2.

Furthermore, Corollary 6.5.4 shows that a characteristic set of the O'·-ideal [A] consists of the 0'. -polynomials A,

and

whose leaders are equal to alY, aila2Y, and aila2-2y, respectively. Thus, the dimension polynomial w(t) of O'·-equation (6.6.6) coincides with the standard dimension polynomial W c(t) of the set

£ = {(1,0); (-1, 1); (-1,-2)} ~ Z2.


We may use Theorem 2.5.5(3) to find this polynomial:

IV(t) = IV e(t) = W (1 0 0 0) (t) = 0101 00 1 2 o 1 0 1 10 1 0

=W(100)(t) 01 2 1 0 1

= W( 1 1 0) (t) - W( 01 0) (t - 2) = 012 011

= W(llO)(t) - W(100)(t - 3) - W(010)(t - 2)

= C ; 3) -C ; 1) - C; 1) - G) = 4t.

(We used statements 2, 5 and 8 of Theorem 2.2.10 for calculation of Kolchin poly-

nomial of the matrix (~ ~ ~ ~).) o 1 0 1 1 0 1 0

b) Now, let us find the dimension polynomial of the CT*-equation

(6.6.7)

(aI, a2 are elements of the subfield of constants of F), which is obtained by the finite difference approximation of the diffusion equation by the Krank and Nicolson scheme (see [RM67, Chap. 8, Sect. 2]). The basic rectangle MNPQ of the CT*polynomial

has the form (by "cross" we mean the points (i, j) such that the terms oi o~y apper in the CT*-polynomial A). Thus, CT*-ordA = 12 and Theorem 6.6.4 implies that dimension polynomial of CT*-equation (6.6.7) has the form IV(t) = 6t + b, where bE Z. Therefore, CT*-dim[A] = 0, CT*-type[A] = 1 and CT*-t.dim[A] = 3.

In order to determine the polynomial IV(t) completely, we determine a characteristic set oflinear CT*-ideal [A]. By Corollary 6.5.4, this characteristic set consists of the following four CT* -polynomials:

A,

011 A = [a1(1- 011) - a2(02 + 02"1 + 01102 + 01-102"1 - 201 1 - 2)]y,

o2" l A = [at{olo2"l - 0-1) - a2(01 + 0102"2 + 02"2 - 20102"1 - 202"1 -1)]y,

0 1 102"1 A = [a1(02"1 - 0 1 102"1)

- a2(02"2 + 011 + 01102"2 - 202"1 - 201 102"1 + l)]Y.

6.6. COMPUTATION OF DIMENSION POLYNOMIALS

E = {(I, 1); (-1, 1); (1, -2); (-1, -2)} ~ 7l..2 .

Applying Theorems 2.5.5(3) and 2.2.10, we obtain that

'li(t) = 'lie(t) = W(ll 0 0) (t) = W(ll 00) (t) - W(l 00 O)(t - 2) 0110 0110 0010 1002 1002 0012 0012 1010 1010 010 1

= W(~ ~ ~~) (t) - W(002)(t - 2) - G) 1002 001 2

=w 0: m (tl - (' ~ ') - [(' ~ ') - (' ~ ')]- G)

341

=W(~~~~)(t)-W(0100)(t-3)- C~I) -[(t~l) - C~I)] -G) [ C : 4) -C: 2)] -C ~ 1) - G) -C ~ 1) -[C~I) -C~I)] -G) =6(t+l)-7=6t-1.

6.6.10. EXAMPLE. Let F be an inversive difference field with a basic set u = {aI, a2} and let F{Y1' Y2} be the ring of u*-polynomials in u*-indeterminates Y1, Y2 over F. Finite difference approximation of the wave equation leads to the system of u* -equations of the form

{ a1(a1 -1)Y1 - a2(a2 - a;I)Y2 = 0,

a2(a2 - a;I)Yl - al(a1 - I)Y2 = 0, (6.6.8)

where aI, a2 are elements of the subfield of constant of the u* -field F. Using the scheme for construction of a characteristic set of a linear u* -ideal described after Corollary 6.5.4, we find that a characteristic set of the u* -ideal P generated by u*-polynomials Al = al(a1 - I)Y1 - a2(a2 - a;1)Y2 and A2 = a2(a2 - a;I)Y1 -a1 (a1 - 1 )Y2 consists of six u* -polynomials:

AI, A2,

A3 = ada1 -1)A1 - a2(a2 - a2"I)A2 = [a~(a1 _1)2 - a~(a2 - a2"1)2] Y1,

A4 = a2"l Al = a1(a1a2"1 - a2"1)Y1 - a2(1 - a2-2)Y2,

A5 = all A2 = a2(a l 1a2 - a l l a2"1)Y1 - adl - a l 1)Y2,

A -I -lA (-1 -1 -1) (-1 -1 -1) 6 = a1 a2 2 = a2 a1 - a 1 a2 Yl - al a2 - a 1 a2 Y2·


The leaders of these IT·-polynomials are 02Y2, 01Y2, 0~Y1' 02"2y2 , 01102Y1, and -1 -2 t' I 0 1 O 2 Y1, respec lve y.

Using Theorem 6.4.S, we obtain that the dimension polynomial ~(t) of system (6.6.S) can be represented as a sum of the standard dimension polynomials ~e, (t) and ~e.(t), where

£1 = {(2, 0); (-1, 1); (-1, -2)} ~ 0:;2

£2 = {(O, 1); (1, 0); (0, -2)} ~ 0:;2.

Applying Theorems 2.5.5(3) and 2.2.10, we obtain

and

~ e, (t) = W (2 0 0 0) (t) = 6t - 1 o 1 1 0 001 2 1 0 1 0 o 1 0 1

~e2(t) = W(O 1 0 0) (t) = 2t + 1. 1000 0002 1 0 1 0 o 1 0 1

(It is a good exercise to perform the correspondent transformations based on the properties of dimension polynomials, see Theorem 2.2.10.) Thus, ~(t) = ~e, (t) + ~e2(t) = St.

6.6.11. EXERCISE. Let F be an inversive difference field with a basic set lT = {Ot, 02, 03, 03, 04} and let F {Y1, Y2, Y3, Y4} be the ring of IT· -polynomials in IT·-variables Yl, Y2, Y3, Y4 over F. Let us consider the system of IT·-equations from Example 6.5.7 and denote by P the linear IT·-ideal of the ring F{Y1, ... , Y4} generated by the IT· -polynomials

Al = a4(04 - 04'1)Y2 - [ad01 - 011) - a2(02 - 02"1)] Y3 + a3(03 - 0;1)Y4'

A2 = a4(04 - 04'1)Y2 - [a1(01 - 011) + a2(02 - 02"1)] Y3 + a3(03 - 0;1)Y4'

A3 = a3(03 - 0;1)Y1 - [a1(01 - 011) + a2(02 - 02"1)] Y2 - a4(04 - 04'1)Y3,

A4 = [a1(a1 - all) - a2(02 - 02"1)] Y1 - [a3(03 - 0;1)] Y2 - a4(04 - 04'1)Y4,

from the left-hand sides of this system.

a) Show, that a characteristic set of the IT·-ideal P consists of IT·-polynomials


AI, . .. , A 4 , A5,"" A IS , where

A5 = [-ai(al - a l l )2 + a~(a2 - a;-I)2 - a~(a3 - a 31)2 + a~(a4 - a4"1)2] Yl,

A6 = [-ai(al - al l)2 + a~(a2 - a;-1)2 - a~(a3 - a3l )2 + a~(a4 - a4"1)2] Y2,

A7 = -a3(a3 - a 31)Al + [al(al - all) + a2(a2 - a;-l)] A2,

As = all AI,

A -I -lA 9 = a l a 2 1,

Ala = a 3l A2 ,

All = a4" l A3 ,

Au = a;l A 4 ,

A13 = all A 5 ,

A -I -lA 14 = a l a 2 1,

A15 = all A6 ,

A -I -lA 16 = a l a 2 6,

A17 = all A7 ,

A lS = a l l a2-2A7.

b) Consider the leaders of the u" -polynomials AI, ... , A lS and show, that the dimension polynomial \II(t) of the linear u" -ideal P has the form \II(t) = I:J=1 \II Ej (t), where

£1 = &2 = {(2,0,0,0); (-1,2,0,0); (-1,-3,0,0)},

&3= {(2,0,0,0); (-1,2,0,0); (-1,2,0,0); (-1,-3,0,0); (0,0,0,1); (O,O,O,-2)},

&4={(1,O,O,O); (-1,1,0,0); (-1,-2,0,0); (0,0,1,0); (0,0,-2,0); (0,0,0,1);

(0,0,0, -2)} ~ /Z4.

Show that

\IIE1 (t) = \IIE2 (t) = 32 C ~ 3) - 80 C ~ 2) + 80(t + 1) - 40,

\II E3(t) = 32C ~ 2) - 80(t + 1) + 80,

\IIE.(t) = 16(t + 1) - 32.

Conclude that

u"-dimP = 0, u'-typeP = 3, and u"-t.dimP = 8.


6.7. Characteristic Polynomials of Finitely Generated Difference-Differential Modules and their Invariants

Let R be a difference-differential ring whose basic set consists of a family of derivation operators ~ = {db ... , dm } and of families (J' = {al, ... , an}, and c = Vh, ... , {3q}, of injective endomorphisms and automorphisms of the ring R, respectively (see Definition 3.5.1). As in Section 3.5, we say that R is a ~-(J'-c·-ring and denote the set of automorphisms Vh, ... . {3q, {31 1 , ... , {3i I} by c·. The ring of ~-(J'-c·-operators over R (see Definition 3.5.41) will be denoted by F and will be considered as a filtered ring with the filtration (Fr )rEZ introduced in the section 3.5 (so that Fr = {w E F I ordw :S r} for r E Nand Fr = 0 for r E Z, r < 0). As above, by a ~-(J'-c· -R-module we mean a left F-module.

6.7.1. DEFINITION. An ascending chain (Mr )rEZ of R-submodules of a ~-(J'-c·R-module M, such that Mr = 0 for all sufficiently small r E Z, UrEzMr = M, and F. Mr ~ Mr+s for all r E Z, sEN, is called a filtration of M. (Thus, by a filtration of a ~~-c· -R-module M we mean an exhaustive and discrete filtration of M as a left F-module.)

6.7.2. DEFINITION. Let (Mr )rEZ be a filtration of a ~-(J'-c· -R-module M. This filtration is called finite if all Mr (r E Z) are finitely generated R-modules. The filtration (Mr )rEZ is said to be good if there exists a number r E Z such that F.Mr = Mr+. for all sEN. A finite and good filtration of a ~-(J'-c·-R-module M is called excellent.

6.7.3. THEOREM. Let R be an Artinian ~-(J'-c·-ring, where ~ = {dl, ... ,dm }

is a set of derivation operators and (J' = {al, ... , an}, c = {{3I, ... , {3q} are sets of injective endomorph isms and automorphisms of R, respectively. Let M be a ~-(J'-c* -R-module provided with an excellent filtration (Mr )rEZ. Then there exists a numerical polynomial €(t) in one variable t with the following properties;

(1) €(t) = IR(Mr ) for all sufficiently large r E Z; (2) deg€(t) :S m + n + q and the polynomial €(t) can be written in the form

I:(t) = 2"a tm+n+q + o(tm+n+q) where a E Z. .. (m+n+q)!

PROOF. Let us consider the graded ring gr F associated with the filtration (Fr )rEZ of the ring of ~-(J'-c* -operators F over R. Let

Xl, ... , X m, YI, ... , Yn, Zl, ... , Zq, Zq+l, ... , Z2q

be canonical images of the elements

dl, ... , dm , aI, ... , an, {31, ... , {3q, {31 1 , ... , {3i 1,

respectively, in the ring gr F (so that

Xi = di + Fa E F1/Fo = grl F (1 :S i:S m), Yj = aj + Fa E grl F (1 :S j :S n),

Zk = {3k + Fa E grl F and Zq+k = (3;;1 + Fa E grl F for k = 1, ... , q).

6.7. CHARACTERISTIC POLYNOMIALS AND THEIR INVARIANTS 345

Then the elements Xl, ... , Xm, Y1, ... , Yn, Zl , ... , Z2q generate the ring gr F over R, these elements commute with each other, and the following conditions hold: Xia = aXi, Yja = aj(a)Yj, Zka = f3k(a)zk' and Zq+ka = f3;l(a)zk for all a E R (1 ::; i ::; m, 1 ::; j ::; n, 1 ::; k ::; q). A homogeneous component gr. F (s E N) is the R-module generated by the set of all monomials x~' ... x;:'m y~' ... y~n <' ... <' (Ui,Vj,Wk EN for i = 1, ... ,m; j = 1, .. . ,n; k = 1, .. . ,q) such that

m n q

LUi + L Vj + L Wk = S ;=1 j=l k=l

and it - ih #- q for alli = 1, ... , q; h = 1, ... , q. In what follows (until the end of the proof) we shall denote the ring gr F by R{ Xl, ... , Xm; Y1, ... , Yn; Zl, ... , Z2q}'

Let grM = EBsEzgrs M be the graded grF-module associated with the given excellent filtration (Mr )rEZ of the b..-u-c· -R-module M (grs M = Ms I M.- 1 for every s E Z). Precisely as in the proof of Theorem 6.2.5, one can show that gr Mis a finitely generated gr F-module. Therefore, to prove the Theorem, it is sufficient to show that there exists a numerical polynomial g(t) in one variable T with the following properties:

(a) g(s) = lR(gr. M) for all sufficiently large s E Z (say, for all s> So, where So is some integer);

(b) the polynomial g(t) can be represented in the form

2q a g(t) = t m+n+q- 1 + o(tm+n+q- 1)

(m+n+q-l)!

where a E Z.

(Indeed, if such a polynomial g(t) exists, then, by Proposition 2.1.5, there exists a polynomial e(t) = (m~~~q)' tm+n+q + o(tm+n+q) which meets the requirements of

the theorem, i.e., e(r) = Ls<r g(s) = Ls<r iR(gr. M) = iR(Mr) for all sufficiently large r E Z.) - -

We shall prove the existence of a polynomial g(t) with the properties (a), (b) by induction on q = Card c. If q = 0, then the existence of such a polynomial is established by Theorem 6.1.3 (if the basic set of the difference ring R in the condition of Theorem 6.1.3 consists of m identical automorphisms and of n injective endomorphisms a1,"" an).

Let q > 0 and t: = {j31,' .. ,f3q}. Let us consider the exact sequence of graded R{X1, ... , Xm; Y1,···, Yn, Zl,"" z2q}-modules

e, 0-+ KerOq -+ M -4 zqM -+ 0 (6.7.1)

where Oq(u) = zqu for all U E M. Precisely as in the proof of Theorem 6.3.3, it can be shown that there exist numerical polynomials

2q- 1 a gl(t) = tm+n+q- 1 + o(tm+n+q- 1),

(m+n+q-l)! 2q- 1a

g2(t) = tm+n+q- 1 + o(tm+n+q-1) (m+n+q-l)!


in one variable t such that gl(8) = IR(Ker(Oq\gr. M))' g2(8) = IR(Oq(gr. M)) for all sufficiently large 8 E Z and a E Z. Now, by the exactness of the sequence (6.7.1) and by the additivity of the function lr (-) in the class of all finitely generated R-modules, we obtain that the polynomial

satisfies the conditions (a), (b). This completes the proof. 0

6.7.4. DEFINITION. Let R be an Artinian A-/T-c*-ring, and let (Mr)rEZ be an excellent filtration of a A-/T-c* -R-module M. Then the polynomial {(t) whose existence is established by Theorem 6.7.3 is called a dimension (or characteristic) polynomial of the A-/T-c*-module M associated with the filtration (Mr)rEZ,

6.7.5. REMARK. Let R be a A-/T-c*-field, and let M be a finitely generated A-/T-c*-R-module (i.e. M is a finitely generated left .1"-module where .1" is the ring of A-/T-c* -operators over R provided with the filtration (.1"r )rEZ mentioned above). Then there always exist excellent filtrations of the module M. Indeed, if Xl, ... , xp is any finite system of generators of M as a A-/T-c* -module (so that M = L:f=l.1"xi), then the chain (L:f=l.1"rX;)rEZ is an excellent filtration of this A-/T-c* -R-module.

6.7.6. EXAMPLE. Let R be a A-/T-c*-field whose basic set consists of a family of derivation operators A = {c51, ... , c5m} and families /T = {a1, ... , an}, c = {/h, ... , {3q} of injective endomorphisms and automorphisms of the field R, respectively. Let .1" be the ring of A-/T-c* -operators over R provided with the filtration (.1"r)rEZ, and let €;1"(t) be the dimension polynomial of the A-/T-c*-R-module .1" associated with this excellent filtration. For any r EN, the basis of the vector R-space .1"r consists of the elements of the form

where U1, ... , Um, VI, ... , Vn EN, WI, ... ,Wq E Z, and

m n q

L U; + L Vj + L \WIe \ :S r. ;=1 j=l Ie=l

Therefore, the number of elements of this basis is equal to the number of points (U1, ... ,Um ,V1, ... ,Vn,Wl, ... wq ) of the set B = UO~h~rBh where

Bh = {(U1, ... , Um, VI,"" Vn ,W1, ... ,Wq) E Zm+n+q \ U;, Vj EN

(1 :S i :S m, 1 :S j :S n), m n q

LUi+ LV; =h, and L\wle\:S r-h} ;=1 j=l 1e=1


for h = 0, ... , r. Since Bhl n Bh~ = 0 for hI 'I h2 (1 :S hI, h2 :S r), we have

r

Card B = L Card Bh

h=O r

= L p(m + n, h) . p(q, r - h) h=O

= t(-1)Q-1c21c(q) t (m+n +: -1) (r- hk+k) Ic=O k h=1 m + n 1

(see Proposition 2.1.8). Now, the combinatorial identity

with M, N, j EN (see [Rio68, § 1.3, formula (2)]) shows that

~ (m + n + h - 1) (r - h + k) = (m + n + k + r) ~ m+n-1 k m+n+k h=O

(one has to put j = n + m, M = r, N = m + n + k + r in the identity mentioned). Thus,

Card B = t( _1)Q- 1c 21c (q) (m + n + k + r) Ic=O k m+ n+ k

whence

(6.7.2)

6.7.7. EXAMPLE. Let R be a ~-O"-c·-field of the previous example (so that Card~ = m,CardO" = n,Cardc = q), and let:F be the ring of ~-O"-c·-operators over R provided with the filtration (:Fr )rEZ. Let Hp be a free left :F-module with free generators hl' ... , hp (in this case we say that Hp is a free ~-O"-c· -R-module of rank p with the free generators hI, ... , hp ), and let H~ (I E Z) denote the free ~-O"-c· -R-module Hp provided with the excellent filtration ((H~)r )rEZ where (H~)r = Ef=l :Fr_IXj for every r E Z. If {p,l(t) is the dimension polynomial of the filtered ~-O"-c·-R-module H~, then, by formula (6.7.2), we have

(6.7.3)

By a free filtered ~-O"-c· -R-module we mean a direct sum of filtered ~-O"-c· -Rmodules of the form H~ where Hp is a free ~-O"-c· -R-module ofrank p and I E Z. If


~L (t) is the dimension polynomial associated with the free filtered !:!.-u-c· -R-module L = H~~ EB··· EB H~~, then formula (6.7.3) shows that

(6.7.4)

Let R be a !:!'-u-c' -ring with a basic set of derivation operators!:!. = {01, ... , Om} and with basic sets U = {Q1, ... , Qn} and £0 = {,81, ... ,,8q} of injective endomorphisms and automorphisms of R, respectively. Let F[x] denote the ring of polynomials in one indeterminate x over the ring of !:!'-u-c' -operators F over R, and let i be the subring EB Fr @R Rxr of the ring F @R R[x]. Furthermore, if (Mr )rEZ

rEN

is a filtration of a finitely generated !:!.-u-c· -R-module M, then the left i-module EB M r @R Rxr, will be denoted by M. The proofs of the following three lemmas rEN and Theorem 6.7.10 are analogous to the proofs of the corresponding statements for differential and difference modules (Lemmas 5.1.13, 5.1.14, 6.3.8, Proposition 5.1.15 and Theorem 6.2.12).

6.7.8. LEMMA. Let R be a !:!.-u-c· -ring, and let (Mr )rEZ be a finite filtration of a !:!.-u-c· -R-module M. Then the following conditions are equivalent:

(1) The filtration (Mr )rEZ is excellent; (2) M is a finitely generated i-module.

In what follows we shall often consider !:!.-u-c· -rings R such that elements of u are automorphisms of R (as well as those of c). To indicate this fact we shall write" !:!.-u a-c' -ring" instead of" !:!.-u-c' -ring". (The other notions connected with the !:!'-ua-c'-ring, will be still marked by the prefix" !:!'-u-c'-" (the prefix" !:!.-ua-£0' - can also be used). For example, one may talk about !:!.-u-c· -R-modules over a !:!.-u a-co -ring R).

6.7.9. LEMMA. Let R be a noetherian !:!.-ua-c'-ring, F the ring of !:!.-u-c'operators over Rand i the ring EB Fr @R Rxr considered above. Then the rings

rEN

F and i are left noetherian.

6.7.10. THEOREM. Let R be a noetherian !:!'-ua-c'-ring, and let 1 : N -+ M be an injective homomorphism of filtered !:!.-ua-c· -R-modules. If the filtration of M is excellent, then the filtration of N is also excellent.

6.7.11. EXERCISE. Show that the ring of !:!'-u-c'-operators over a !:!'-ua-c'-field is a left Ore ring.

Let R be a !:!.-u-c' -field, M a finitely generated !:!.-u-c· -module, and ~(t) a dimension polynomial of M associated with some excellent filtration of this module (by Remark 6.7.5, such filtration of M always exists). As in the case of differential and difference modules, we have the following statement which shows that the degree and the leading coefficient of ~(t) are independent of the choice of an excellent u-filtration which this polynomial is associated with.


6.7.12. PROPOSITION. Let R be a Ll-u-c*-field such that the set Ll consists of m derivation operators and the sets u and c consist of n injective endomorphisms and of q automorphisms of R, respectively. Let M be a finitely generated Ll-u-c* -Rmodule, and let e(t) and el (t) be dimension polynomials of this module associated with excellent filtrations (Mr)rEZ and (M:)rEZ of M, respectively. Then

where d = deg e(t) (here Lli !(t) (i E N) denotes the i-th finite difference of the polynomial !(t)).

Since the leading coefficient of a numerical polynomial !(t) of degree dis AdJ(t),

Proposition 6.7.12 justifies the following definition.

6.7.13. DEFINITION. Let R be a Ll-u-c* -field such that Card Ll = m, Card u = nand Cardc = q. Let M be a finitely generated Ll-u-c*-R-module, and let e(t) be a dimension polynomial associated with an excellent filtration of this module. Then the numbers Llm+n+qe(t)/2Q, d = dege(t) and Llde(t) are called the Ll-u-c*dimension, the Ll-u-c" -type, and the typical Ll-u-c* -dimension of Mover R; they are denoted by Ll-u-c* -dimR M, Ll-u-c* -t(M) and tLl-u-c*dimR M, respectively. (in what follows, while considering these notions, we shall also use the adjective "difference-differential" instead of the prefix "Ll-u-c*").

The proofs of the following Propositions 6.7.14 and 6.7.16 are easy adaptations of the proofs of similar assertions for differential and difference modules (see Proposition 5.2.11, 5.2.12, 6.2.16 and 6.2.17).

6.7.14. PROPOSITION. Let R be a Ll-ua-c*-R-field, and let

be an exact sequence of finitely generated Ll-u-c* -R-modules (imath and jmath are Ll-u-c*-homomorphisms). Then Ll-u-c*-dimR M =Ll-u-c*dimR M +Ll-u-c*dimR P.

6.7.15. DEFINITION. Let R be a Ll-u-c* -field whose basic set consists of a family of derivation operators Ll = {(h, ... , t5m } and of families u = {al, ... , an}, c = {lh, ... , f3q} of injective endomorphisms and of automorphisms of the field R, respectively. Let A be the commutative semigroup of elements of the form A = t5~1 ... t5:; ... 'a~l ... a~f3rl ... f3~q (ki' li EN; u/ E :;z for i = 1, ... ,m; j = 1, ... , nj I = 1, ... , q) considered in Section 3.5, let F be the ring of Ll-u-c*-operators over R, and let M be a Ll-u-c*-R-module. Elements Xl, ... , Xp E M are called Ll-u-c" -linearly dependent over R iff the family {AXi I >. E A, 1 ~ i ~ p} is linearly dependent over R (i.e. the elements Xl, ... , Xp are linearly dependent over the ring F). Otherwise, the elements Xl, ... , xp are said to be Ll-u-c* -linearly independent over the field R.

6.7.16. PROPOSITION. Let R be a Ll-ua-c*-field, and let M be a finitely generated Ll-u-c" -R-module. Then the Ll-u-c" -dimension of Mover R is equal to the maximal number of elements of the module M which are Ll-u-c* -linearly independent over R.


6.7.17. EXERCISE. Let R be a Ll-CT-c·-field such that Ll consists of a single derivation operator d, CT = 0, and c consists of a single automorphism f3 of R. Let M be a Ll-CT-c·-R-module with one generator x and with one defining relationship (d2 + f32 + f3-2)x = 0 (one can treat M as a Ll-CT-c· -R-module H /.1"(d2 + f32 + f3-2)e where .1" is the ring of Ll-CT-c· -operators over R, and H is a free left .1"-module with a single free generator e). Find the dimension polynomial of M associated with the excellent filtration (.1"rX)rEZ.

Starting afresh, let R be any Ll-CT-c· -field whose basic set consists of a family of derivation operators Ll = {d1, ... , dm } and of families CT = {0"1, ... , O"n}, c = {f31, ... , f3q} of injective endomorph isms and automorphisms of R, respectively. Let 0, T and 0T be commutative semigroups generated by the families Ll, CT and Ll U CT, respectively, let f be the free commutative group generated by the elements f31, ... , f3q, and let 0f, Tf, Tf and A be commutative semigroups of the elements O'Y (0 E 0, 'Y E f), T'Y(T E T, 'Y E f) and OT'Y (0 E 0, T E T, 'Y E f), respectively. If Al = d~l ... d~mO"il ... O"~" f3r l ... f3~q and A2 = dfl ... d!:;"O"~1 ... O"~" f3r l ... f3~q are any two elements of A (k;, Pi, lj, rj E N and Ut, Vt E ~ for 1 ::; i ::; m, 1 ::; j ::; m, 1 ::; t ::; q), then Al = A2 iff k; = Pi, lj = rj and Ut = Vt for all i = 1, ... ,mj j = 1, ... nj t = 1, ... ,q. In accordance with the notation of Section 3.4, the subset {f31, ... ,f3q,f311, ... ,f3;1} of f will be denoted by c·. If A = d~l ... d~m O"il ... O"~" f3r I ••• f3~q E A, then the numbers

m n q

ordA = Ek; + Elj + E IUtl, ;=1 m

ord~ A = E k;, ;=1

n

orda A = Elj, j=l

q

ordt A = E IUtl, t=l

j=l

m n

ord~a A = Eki + Elj, ;=1 j=l m q

t=l

ord~t A = E k; + E Iutl and ;=1 t=l

n q

ordat A = E lj + E Iutl j=l t=l

are called the order, the Ll-order, the CT-order, the c-order, the Ll-CT-order, the Ll-c-order and the CT-c-order of A, respectively. As above, we can consider the ring of Ll-CT-c·-operators.1" over R (see Definition 3.5.41) whose elements are of the form E>'EA a>.A where a>. E R (A E A) and almost all elements a>. are equal to zero (i.e. the sum E>'EA a>.A contains a finite number nonzero terms).


If w = E>'EA a>.A E :F then the order, D.-order, u-order, e-order, D.u-order, D.e-order, and ue-order of w, respectively, are defined as follows:

ordw = max{ordA I a>. i= O},

ord6 w = max{ord6 A I a>. i= O}, ... ,

ord.,.£ w = max{ord.,.. A I a>. i= O}.

In accordance with these notions, one may consider the filtrations (:Fr )rEZ, (:F;- )rEZ, (:F:)rEZ, (:F;)rEZ, (:F;-"')rEZ, (:F;-£)rEZ and (:F;r")rEZ of the ring :F such that :Fr = :F;- = ... = :F;r' = 0 for all r E Z, r < 0, and

:Fr = {w E :F lord w :::; r},

:F;- = {w E :F I ord6 w :::; r}, ... ,

:F;£ = {w E :F lord.,.. w :::; r}

for all r E f::J. These filtrations will be called the standard filtration, the standard fl.-filtration, ... , the standard ue-filtration of :F, respectively.

If U = 0 and e = 0, then A = e and the notion of D.-u-e* -operator coincides with that of D.-operator over the differential field R with the basic set D.. Similarly, if D. = 0, e = 0, then A = T and :F is the ring of difference (u-) operators over R, etc. Thus, one may consider the following subrings of the ring of D.-u-e* -operators :F over a D.-u-e*-field R: the subring D of D.-operators over R, the subring B of u-operators over R, the subring £ of inversive difference (e* -) operators over R and the subrings DB, D£ and B£ of D.-u-, D.-e* - and U-e" -operators over R, respectively. These subrings may be thought of as filtered ones with the filtrations (Dr = :Fr n D).Ez, (Br = :Fr n B)rEZ, ... , «B£)r = :Fr n B£)rEZ, respectively (such filtrations will be called standard). The rings DB, D£ and B£ may be also considered with the filtrations which are induced by the standard D.-, u- and e· - filtrations of:F. The corresponiding D.-, ... , e- filtrations «DB)~ = :F;- n DB)rEz, ... , «B£)~ = :F;nB£)rEz will be also called standard. Let M be a D.-u-e*R-module. Then, in accordance with Definition 6.7.1, by a D.- (u-, eo, D.u-, D.e-, ue-) filtration of M we shall mean an ascending chain (Mr )rEN of B£- (respectively, D£-, DB-, £-, B-, D-) submodules of M which is an exhaustive and discrete filtration of M over the ring :F with the standard D.- (respectively, U-, eo, D.u-, D.u-, D.e-, Ue-) filtration. In particular, if M is a finitely generated left :F-module with generators Xl, ... , xP ' then (Ef=l :F;-XdrEZ' ... , (Ef=I:F;r Xi)rEZ, (Ef=l :F;r' Xi)rEZ are the fl.-, U-, ... , ue-filtrations of M, respectively.

6.7.18. DEFINITION. A D.- (u-, ... , Ue-) filtration of a D.-u-e*-R-module M is called excellent if all Mr (r E Z) are finitely generated B£- (respectively, DB-, ... , D-modules) and there exists ro E Z such that :Fi" Mr = Mr+. (respectively, :F': Mr = Mr+., ... , :F':£ Mr = Mr+.) for all r ~ ro, s E f::J.

Note, that the above filtrations of a finitely generated D.-u-e*-R-module are excellent.

REMARK. Remind that if M is a finitely generated D.-u-e*-R- (i.e. left :F-) module, then its D.-u-e* -dimension over R is denoted by D.-u-e* -dimR M. Similarly, if M is a finitely generated U-e" -R-module (i.e. a left 8£-module), then its


(1'-c· -dimension over R (i.e. the maximal number of elements of M which are linearly independent over the ring BE) will be denoted by (1'c· dimR M. In the same sense the ~-(1'-, ~-c·-, ~-, (1'- and c·-dimensions of a finitely generated ~-(1'-R,

... , c·-R-module M, respectively, are treated. These dimensions will be denoted by ~(1' dimR M, ~c· dimR M, ~ dimR M, (1' dimR M and c· dimR M, respectively. (We use this notation in order to unify our considerations. Of course, c· dimR M is inversive difference dimension of M over the inversive difference field R with the basic set c; such dimension was denoted by iOe (M) in Section 6.3. A similar remark can be done about the ~-dimension).

6.7.19. THEOREM. Let R be a ~-(1'a-c·-field whose basic set consists of a family of derivation operators ~ = {O 1 , ... , Om} and families (1' = {(}:1, ... , (}:n}, c = {.lh, ... , {3q} of automorphisms of R. Let (Mr )rEZ be an excellent ~-fi1tration of a finitely generated ~-(1'-c· -R-module M. Then there exists a polynomial p(t) in one variable t with rational coefficients such that the following properties hold:

(1) p,(r) = (1'c· dimR Mr for all sufficiently large r E Z; (2) deg p, ~ m and the polynomial p,(t) can be written as p,(t) = ~! t m + o(tm )

where a E Z.

PROOF. Let us consider the graded ring gr~ F associated with the ring of ~-(1'c· -operators F provided with the standard ~-filtration:

00 00

gr~ F = 61gr~ F = 61(F: /F:- 1 ) ~ BE[Xl' ... ' Xm], p=O p=O

where Xi = Oi +BE E gr~ F (i = 1, ... , m). Here BE[X1 , ... , Xm] denotes the ring of polynomials in pairwise commuting indeterminates X 1, ... , Xm over the ring BE; this ring of polynomials is considered as a graded ring with the natural grading. The graded gr~ F-module associated with the given excellent ~-filtration of the ~-(1'-c·-R-module M has the form

00 00

gr~ M = 61gr~ M = 61(Mp/Mp-d p=O p=O

(without loss of generality, we may assume that Mp = 0 for all p < 0). We can also assert that gr~ M is a finitely generated BE[X 1, ... , Xm]-module (this fact can be established in the same way as the similar result for the filtered differential or difference modules; see, for example, the proof of Theorem 6.2.5). Let M(p) = gr~ M (p E N) and let S = BE[X1 , ... , Xm]. Then the polynomial ring Scan be naturally considered as a graded one: S = Ee;:o Sip) where Sip) is the left B£-module generated by the set of all monomials of degree p.

We are going to show that there exists a polynomial c,o(t) E Q[t] such that c,o(t) = (1'10· dimR(M(p») for all sufficiently large q EN, deg c,o ~ m - 1, and the polynomial c,o(t) can be written as c,o(t) = (m~1)! tm- 1 + o(tm- 1) where a E Z.

We proceed by induction on m = Card~. If m = 0, then F = BE and the definition of an excellent ~-filtration shows that Mr = Mr+1 = ... for some r EN, so that M(p) = 0 for all sufficiently large pEN. Thus, for m = 0 our statement is


valid (in this case we have <,oCt) = 0, deg<,o = -1). Now, let us suppose that our statement is valid for Card.6. = m - 1, and let Card.6. = m, .6. = {81> ... , 8m }.

Let us consider the mapping p of degree 1 from the graded S-module gr~ M = ffi;:o M(p) into itself such that PY = €mY for any element Y E gr M. For every p E ~, we have the exact sequence of finitely generated BE-modules

(6.7.5)

where N(p) = Ker(pIM(p»), L(p) = Coker(pIM(p») (p E ~). Since N = Ker p = ffi;:o N(p) and L = Coker p = ffi;'=o LCP) are graded S-modules which are annihilated by Xm , these modules can be considered as graded BE[Xb .. ·, Xm-dmodules. By the inductive hypothesis, there exist polynomials c,ol(t), c,02(t) E Q[t] such that c,ol(p) = Ug* dimR(N(p»), c,02(p) = Ug* dimR(L(p») for all sufficiently large P E ~ (say, for all P ~ Po where Po E ~), and c,oi(t) = (m~2)!tm-2 + o(tm- 2) where aj E IZ (i = 1,2). Now, the exactness of sequence (6.7.5) implies the equalities

Ug* dimR M(P+1) - Ug* dimR M(p) = Ug* dimR L(p) - Ug* dimR N(p)

= c,02(P) - c,ol(P) = c,03(P)

for all sufficiently large P E ~ (where c,03(t) = (~=~)!tm-2 + o(tm- 2) E Q[t]).

Thus, for all p E ~, P ~ Po, we have Ug* dimR M(p) = Co + Ef::;o c,03(i) where Co = Ug* dimR M(po). By Proposition 2.1.5, there exists a polynomial c,o(t) E Q[t] such that c,o(t) = Ug* dimR M(p) for all P ~ Po, deg c,o = deg c,03 + 1 ::; m - 1, and the polynomial c,o(t) can be written as c,o(t) = (m~1)!tm-1 + o(tm- 1), where a = a2 - a1 E 1Z. Since Ug* dimR Mr = E;=o Ug* dimR M(p) for any r E~, Proposition 2.1.5 shows that there exists a polynomial/-l(t) E Q[t] such that /-I(r) = Ug* dimR Mr

for all sufficiently large r E~, deg/-l::; m, and /-I(t) = ~!tm +o(tm), where a E 1Z. This completes the proof. 0

6.7.20. EXERCISE. Let R be as in the conditions of Theorem 6.7.19. Formulate and prove theorems on dimension polynomials for a finitely generated C1-u a-g* -Rmodule with an excellent u*- (g*-, u*-g*-, C1-u*-, .6.-g*-) filtration.

6.7.21. EXERCISE. Use the scheme of the proof of Theorem 6.4.11 to prove the analogues of statements (1)-(3) of Theorem 6.4.1 for .6.-u-g*-field extensions.

CHAPTER VII

SOME APPLICATION OF DIMENSION POLYNOMIALS

IN DIFFERENCE-DIFFERENTIAL ALGEBRA

7.1. Type and Dimension of Difference-Differential Vector Spaces

Let R be a commutative ring, M an R-module and U a family of R-submodules of M. Furthermore, let Bu denote the set of all pairs (N, N') E U x U such that N 2 N', and let Z be the set obtained by the adjoining of a new symbol 00 to the set of integers ::£:. We shall consider Z as a linearly ordered set whose order < is the extension of the natural order of ::£: such that a < 00 for all a E ::£:.

7.1.1. PROPOSITION. With the above notation, there exists a unique map pu : Bu -+ Z with the following properties:

(1) pu(N,N') ~ -1 for every pair (N,N') E Bu; (2) if dEN, then the inequality pu(N, N') ~ d holds if and only if N t= N'

and there exists an infinite chain N = No ::> Nl ::> ::> N' such that PU(Ni-l, N;) ~ d - 1 for all i = 1,2, ....

PROOF. We shall construct the desired map Pu : Bu -+ Z as follows. If N E U, then we set pu(N, N) = -1. If (N, N') E Bu, N t= N' and the condition (ii) of the proposition holds for all dEN, then we set pu(N, N') = 00. Finally, if a pair (N, N') E Bu satisfies the condition (ii) of the Proposition for some dEN, then we define pu(N, N') as the greatest of such numbers d. Obviously, the map PU is well-defined, it is uniquely determined, and the condition (i), (ii) hold. 0

7.1.2. REMARK. The conditions (i), (ii) of Proposition 7.1.1 show that the map PU has the following properties.

(a) pu(N, N') = -1 if and only if N = N'; (b) pu(N, N') = 0 if and only if N ~ N' and any chain of R-submodules

N = No 2 Nl 2 Nl 2 ···2 N' (Ni E U for all i = 0,1, ... ) becomes stable (that is Nj = NjH = ... for some j EN).

(c) pu(N, N') ~ 1 if and only if there exists an infinite strictly descending chain

where Ni E U (i = 1,2, ... ).

7.1.3. DEFINITION. With the above notation, the least upper bound of the set {p(N, N') I (N, N') E Bu} is called the type of the R-module M over the family of its submodules U. It is denoted by typeu M.

355


356 VII. SOME APPLICATION OF DIMENSION POLYNOMIALS

7.l.4. DEFINITION. Let R be a ring, M an R-module and U a family of Rsubmodules of M. The least upper bound of the lengths of chains

such that J.lu (Ni -1, Ni) = typeu R (i = 1, ... , p) is called the dimension of Mover the family of ideals U and is denoted by dimu M. (As usual, by the length of the chain

N = No :J Nl :J Nl :J ... :J Np

we mean the number of "gaps" p).

7.1.5. EXAMPLE. Let K be a field and V a vector K -space of finite dimension n. Let U be the family of all vector K-subspaces of V. Then typeu V = 0 and dimu V = n. Indeed, the first equality follows from Remark 7.1.2(b). In order to prove the second equality, let us consider the strictly ascending chain of vector K-spaces

such that J.lu([el, ... , ek], h,· .. , ek-l]) = 0 = typeu V

for every k 2, ... , nand J.lu([ed,O) = 0 (here el, ... , en is a basis of V and [el, ... , ek] (k = 1, ... , n) denotes the vector subspace of V with the basis el, ... , ek). This chain shows that dimu V 2: n. Conversely, if Wo :J WI ~ ... ~ Wp is some chain of vector K -subspaces of V such that J.lu (Wi-I, ~) = 0 (i = 1, ... , p), then

n = dim V 2: dim Wo > dim WI > ... > dim Wp 2: 0

hence dimu V ~ n (by dim Wi we denote the dimension of a vector space Wi in the usual sense). Thus, dimu V = n = dim V.

7.1.6. EXAMPLE. Let K be a field and V a vector K -space of infinite dimension which has a countable basis. Let U be the family of all vector K-subspaces of V. Then typeu V = 00 and dimu V = O. Indeed, if B is a countable basis of V, then B can be represented as a countable union U~l Bi of pairwise disjoint countable

sets. For any k = 1,2, ... , let Ck l ) denote the vector subspace of V with the basis

B \ U~=l Bi and let C~l) = V. Then we have the following chain of elements of U:

(7.1.1)

Since every vector K-space Ck~l/C~l) (k = 1,2, ... ) has a countable basis, for every k = 1, 2, ... , there exists an infinite strictly descending chain of vector spaces C(2) = e l ) :J d 2) :J ... :J C(1) such that the vector spaces d 2) /d2) (i = k-l:t:. l:t:. :t:. k 1-1 I

1,2, ... ) have countable bases, etc. By Definition 7.1.3 we obtain that typeu V = 00. Furthermore, for chain (7.1.1) we clearly have J.lU(C~~l' C~l») = 00 = typeu V, so that dimu V = 00.

Obviously, if typeu M < 00, then dimu M 2: 1. The following example shows that if typeu M = 00, then it may be dimu M = o.

7.1. TYPE AND DIMENSION OF VECTOR SPACES 357

7.1. 7 . EXAMPLE. K be a field and V1, V2 ... , Vn, . .. finitely generated vector K-spaces such that dim V1 = 1, dim V2 = 2, ... , dim Vn = n, .... Let V = EB~=1 Vn and let U be the family of all subspaces W of V such that 7I"n(W) =I- 0 for only one number n (7I"n denotes the projection of V onto its n-th component Vn ). It is easy to see (in view of Example 7.1.5) that typeu V = 00 and dimu V = O.

7.1.B. EXERCISE. Let A be the set of functions of one real variable t which are defined and continuous on the whole coordinate line JR. We consider A as an algebra over the field ofreal numbers JR and, for any open interval (a,b) (a,b E JR, a < b), set O(a, b) = {f(t) E A I f(c) = 0 for all c E JR \ (a, b)}. Obviously, O(a, b) is an ideal of A. Let U denote the set of all such ideals. Find typeu A and dimu A.

7.1.9. EXERCISE. Let A be a commutative ring with a finite Krull dimension dimA and let U be the family of all prime ideals of A. Show that typeu A = 0 and dimu A = dimA. In particular, if A is artinian, then typeu A = dimu A = O.

Theorem 7.1.10 stated below shows, in particular, that for every numbers m, n E N, there exist a field K, a vector K -space M, and a family U of vector K -subspaces of M such that typeu M = n, dimu M = m.

Let R be a A-O'-£*-field (see Definition 3.5.1) whose basic set consists of a set of derivation operators A = {81, ... , 8m } of the field R into itself and of two sets of automorphisms 0' = {a1,"" an} and £ = {,81, ... ,,8q} of this field such that any two maps of the set AU 0' U £ commute with each other. Let F denote the corresponding ring of A-O'-£" -operators over R. Recall that each element of F (also called a A-O'-£" -operator over R) is a finite sum EAEA aA.x where A is the commutative semi group of elements of the form

\ - rk, rkm~I, In RU' RU q (k k I lEw E '71) 1\-01 "'Om '""1 ... a n I-'1 '''I-'q 1, .. ·, m, 1,"" n l'lj U1, .. ·,Uq ILJ

(7.1.2) and aA E JR for all .x E A. The number ord.x = E~1 kj + Ej=1lj + E~=1 lulIl is called the order of the element .x and the order of A-O'-£* -operator w = EAEA a>..x E F is the number ordw = max{ord'x I a>. =I- O}. The multiplication of the elements of F is defined according to the relationships 8a = a8+8(a), aa = a(a)a,,8a = ,8(a),8, ,8-1a = ,8-1 (a),8-1 (a E R, 8 E A, a E 0',,8 E £) and to the distributive laws. As in § 3.5, we shall consider F as a filtered ring with the ascending filtration (Fr )rEZ where:Fr = {w E F I ordw ~ r} for r E Nand Fr = 0 for r E ~, r < O.

Let M be a finitely generated vector A-O'-£"-space over the field JR (i.e., M is a A-O'-£* -R-module in the sense of Definition 3.5.42 which is finitely generated as a left F-module). As in the case of differential, difference, and inversive difference modules, by a filtration of M we mean an exhaustive and discrete ascending filtration of M as of a left F-module. A filtration (Mr )rEZ of M is said to be excellent if any vector R-space Mr (r E ~) is finitely generated and there exists a number ro E ~ such that FpMr = Mr+p for any integer r > ro and for any pEN. Such fil-

trations of M always exist: if M = E:=l FZj, then (E:=1 Frzj) rEZ is an excellent

filtration of M.

7.1.10. THEOREM. Let R be a A-O'-£*-field with the basic set AUO'U£ described above, let M be a finitely generated vector A-O'-£" -R-space, and let U be the family


of all vector ~-(J'-E:*-R-subspaces of M. Then:

(1) if ~-(J'-E:*-dimRM > 0, then typeu M = m + n + q, dimu M =~-(J'-E:*

dimRM; (2) if ~-(J'-E:*-dimR M = 0, then typeu M < m + n + q.

PROOF. Let (Mr )rEZ be an excellent filtration of M and let N E U. By Theorem 6.7.10, (NnMr )rEZ is an excellent filtration of N, so we may consider the dimension polynomial'll N(t) of N associated with this filtration. (According to Theorem 6.7.3, 'IIN(r) = dimR(N n Mr) for all sufficiently large rEa: and the polynomial 'II(t) can be written in the form 'IIN(t) = 2 q a tm+n+q + o(tm+n+q) where a =~-(m+n+q)1 (J'-c*-dimRN). If N,L E U and 'IIN(t), 'II£(t) are their dirr.ension polynomials (associated with the excellent filtrations (NnMr )rEZ and (LnMr )rEZ, respectively), then the inclusion L ~ N, obviously, implies the inequality 'II£(t) :S 'IIN(t) (by"2:" we mean the natural order on the set of all numerical polynomials, see Definition 2.4.1). Furthermore, in this case the equality L = N holds if and only if 'IIN(t) = 'II£(t). (Indeed, if'llN(t) = 'II£(t), but L ~ N, then there exists ro E a: such that L n Mr ~ N n Mr for all r > ro. Therefore, we would have 'II£(t) < 'IIN(t), that contradicts our assumption).

We shall prove that if L, N E Bu = {(P, Q) E U x U I P 2 Q} and J-Lu(N, L) 2: d (d E a:, d 2: -1), then deg('IIN(t) - 'II£(t)) 2: d.

We proceed by induction on d. Since deg('IIN(t) - 'II£(t)) 2: -1 for all pairs (N, L) E Bu and deg('IIN(t) - 'II£(t)) 2: ° if N ~ L (in this case we have 'IIN(t) > 'IIL(t)), our statement is valid for d = -1 and for d = 0.

Now, let d> ° and let the statement is valid for all integers which are less than d. Let (N,L) E Bu, J-Lu(N,L) 2: d and let

be an infinite strictly descending chain of vector ~-(J'-E:* -R-subspaces of M such that J-LU(Ni-I,Ni) 2: d-l (i = 1,2, ... ). Ifdeg('IIN;_,(t) - 'IIN;(t)) 2: d for some i E N, i 2: 1, then clearly deg('IIN(t) - 'IIL(t)) 2: d and our statement is valid. Let us suppose that deg('IIN;_,(t) - 'IIN.(t)) = d-l for all i = 1,2, .... Then every polynomial 'IIn-i(t) - 'IIN;(t) (i EN, i 2: 1) can be written in the form

'II (t) - 'II .(t) = Cj td- 1 + O(t d- 1 ) N._, N. (d _ I)!

where Ci E a:, Cj > ° (Ci is the ith finite difference of f(t), see formula (2.1.13)). In this case we have

where bi = Cl +- . +Cj, so that bi < b2 < .... It follows that deg('II N (t)- 'II £(t)) 2: d. By Theorem 6.7.3, we have

deg('IIN(t) - 'II£(t)) :S m + n + q

7.1. TYPE AND DIMENSION OF VECTOR SPACES 359

for any pair (N,L) E Bu. This inequality together with the above statement show that J.Lu(N, L) ~ m + n + q for all N, L E Bu, whence

typeu M ~ m + n + q. (7.1.3)

If ~-u-e·-dimR M = e > 0, then Proposition 6.7.16 shows that there exist I elements Xl, ... ,XI E M which are linearly independent over the ring of ~-(J"-e·operators F over R. Obviously, if U' is the family of all F-submodules of the F-module FX1, then typeu' FX1 ~ typeu M. We shall show that typeu' FX1 ~ m + n + q (by virtue of the inequality (7.1.3), it will imply the validy of the first statement of our theorem). Let U" denote the family of all F-submodules of FX1 which can be represented as finite sums of F-modules of the form

Fhtl ... 8~m(a1 - 1)'1 ... (an - I)'n (.81 -1)u I ... (.8q - l)u· X1

(k1, ... , km ,/1, ... ,In, U1, ... , Uq EN).

We shall show that J.LU,,(FX1,0) ~ m + n + q by induction on m + n + q. For m + n + q = 0 the inequality is trivial. Let m + n + q > 0, that is at least one of the number m, n, q is positive. At first we shall consider the case where m > O. Since Xl is linearly independent over F, we have the strictly decreasing chain

of elements of U". Let us show that J.Lu" (F8:;..-l x1 , F8:;"xt) ~ m+n+q-I for every r = 1,2, .... Let L = F8:;..-1 x1/F8:;"X1 and let y be the image ofthe element 8:;"-1 Xl under the canonical epimorphism F 8:;"-1 Xl ~ L. Then 8m y = y and 8m (}' y = (}' 8m y for any element (}' E 6', where

m-1 n q

6' = {II II II 87;(a1 - 1)'; (.8" -1)u~ I k1,.·., km - 1, it, ... ,In, U1,···, uq EN}. ;=1 j=l,,=l

Therefore, if ~ = {81 , ... ,8m -d, here R is treated as a ~'-u-e·-field, and F' is the ring of ~'-u-e·-operators over R, then L = F'y, and the elemenet y is linearly independent over F', Furthermore, J.Lu,,(FO:;..-l x1 ,T8:;"xt} = J.Lu~'(L,O), where U{' is the family of all F'-submodules of L which can be represented as finite sums of :F'-modules of the form :F'(}'y ((}' E 6'). Using the inductive hypothesis, we get J.Lu~'(L, 0) ~ m + n + q - 1, whence J.LU" (F8:;..-l x1 , F8:;"xt} ~ m + n + +q - 1 (r = 1,2, ... ). Thus, J.LU,,(FX1, 0) ~ m + n + q.

Now, let m + n + q > 0 and n > O. Let us consider the strictly decreasing chain of F-modules FX1 :) F(an - I)X1 :) F(an - 1)2x1 :) ... :) O. Similary to the case m > 0, for every r = 1,2, ... we can consider the ~-(J"'-e·-R-module P = F(an - It- 1xt/F(an -ltx1 (where (J"' = {a1, ... , an-d) and obtain that

J.Lu,,(F(an - It-1xt, F(an - It Xl) ~ m + n + q - 1

whence J.LU,,(FX1, 0) ~ m+ n+ q. The validity of this inequality in the case m ~ 0, n ~ 0, q > 0 can be established just as in the case n > o.


Thus, typeu M 2: typeu' FXl 2: typeull FXl 2: m + n + q.

Applying inequality (7.l.3), we obtain that

typeu M = m + n + q.

Our proof of the last equality shows that typeu' F Xi = m + n + q for every i = 1, ... ,I (here Uf denotes the family of all F-sub~odules of the F-module F Xi).

Therefore, J.lU(L:~=1 Fx;, L:~;;11 FXi) = m+n+q = typeu M for every k = 1, ... , I, and the chain

I 1-1 M = LFxi :J LFx; :J ... :J FXl :J 0

;=1 i=1

leads us to the inequality

dimu M 2: !:!..ac· dimR M (7.1.4)

Let No :J Nl :J ... :J Np be a chain of !:!..-a-c·-R-subspaces of a vector !:!..-a-c*R-space M such that J.lU(Ni-l, Ni ) = typeu M = m + n + q for i = 1, ... ,p. Then the above reasonings show that the degree of each polynomial \IINi_1 (t) - \liN, (t) (1 ~ i ~ p) is equal to m + n + q, hence such polynomial can be written as

2qa· \II N· (t) - \II N (t) = • tm+n+q + O(tm+n+q)

,-1 , (m + n + q)!

where aj = !:!..-a-c * -dimR Ni_l-!:!..-a-c· -dimR Ni 2: 1. Now we can write

P

\IINo(t) - \IINp(t) = L(\IINi-1 (t) - \IIN,(t)) i=1

where a = L:f=1 aj 2: p. On the other hand,

2q · (!:!..-a-c·-dimM) \liN (t) - \liN (t) < \liN (t) < \IIM(t) = tm+n+q + o(tm+n+q)

o p - 0 - (m + n + q)! '

hence !:!..-a-c· -dimR M 2: a 2: p. It follows that !:!..-a-c* -dimR M 2: dimu M, and, in view of inequality (7.1.4), we obtain the desired equality dimu M =!:!"-a-c· -dimR M.

Now, we are going to prove the last statement of the theorem. If tl.-a-c·dimR M = 0, then for any pair (N, L) E Bu we have the equality

!:!..-a-c· -dimR N =!:!"-a-c· -dimR L = 0,

whence deg \IIN(t) < m + n + q, deg \IIL(t) < m + n + q and deg(\IIN(t) - \IIL(t)) < m+n+q. As indicated above, the last inequality implies the inequality J.lu(N, L) < m+n+q. Thus, if !:!..-a-c·-dimR M = 0, then typeu M < m+n+q. This completes the proof. 0

7.2. TYPE AND DIMENSION OF DIFFERENCE-DIFFERENTIAL ALGEBRAS 361

7.2. Type and Dimension of Finitely Generated Difference-Differential Algebras

Let R be a commutative ring, U a family of its ideals and Bu = {(P, Q) E U x UIP ;2 Q}. Then, by Proposition 7.1.1, there exists a unique mapping }lu Bu -+ ~ = Z U {oo}, with the following properties:

(1) }lu(P,Q) ~ -1 for any pair (P,Q) E Bu; (2) if dEW, then the inequality /-Iu(P, Q) ~ d holds if and only if P ~ Q

and there exists an infinite chain P = Po :J PI :J ... :J Q such that }l(Pi-l, Pi) ~ d - 1 for all i = 1,2, ....

In accordance with Definitions 7.1.3, 7.1.4, the number

typeu R = sup{}lu(P, Q)I(P, Q) E Bu}

is said to be the type of the ring R over the family of ideals U and the least upper of the lengths of chains of ideals

(PiEU for i=O,I, ... ,k)

such that }lU(Pi-1> Pi) = typeu R (1 ~ i ~ k) is called the dimension of the ring R over the family U and is denoted by dimu R.

Let K be a Ll-O"-£* -ring whose basic set consists of a family Ll = {6l , ... , Jm } of derivation operators of the ring K into itself and offamilies 0" = {01, ... , On}, 0: = {/h, ... , {iq} of automorphisms of K such that any two elements of LlUO"Uo: commute with each other. As usual, the set of automorphisms {{iI, ... , {iq, {ill, ... , {iiI} of the ring K, will be denoted by £*. Recall, that an algebra R over the !:!..-O"-o:* -ring K is said to be a !:!..-O"-o:* -K -algebra if the elements ofthe set!:!.. UO"Uo:* act on R in such a way that R becomes a !:!..-O"-o:*-ring and J(ka) = J(k)a + kJ(a), /,(ka) = /,(kb(a) for any elements 6 E !:!.., /' E 0" U £* , a E R, k E K. A !:!..-O"-o:* -J{ -algebra R said to be finitely generated if there exists a finite family '1 = {'1l, ... , '1.} of elements of R such that R = K {711, ... , '1.}. Recall, that if A is a semigroup of all elements of the form

then K {711> ... , 71.} coincides with the ring R = K[{ >''1i I>' E A, 1 ~ i ~ s }]. If a !:!..-O"-o:* -algebra R over a !:!..-O"-£* -field K is an integral domain, then the

!:!..-O"-£* -transcendence degree of the quotient field of Rover K will be denoted by !:!..-O"-£* -trdegK R.

7.2.2. THEOREM. Let K be a !:!..-O"-£*-/ield with the basic set Ll U 0" U 0: described above, let R = K{711> ... , 71.} be a !:!..-O"-£*-K-algebra without zero divisors generated by the family '1 = (711, ... ,71.), and let U be the family of all prime !:!..-O"-£* -ideals of R. Then the following statements hold.

(1) typeu R ~ m + n + q; (2) if Ll-O"-£*-trdegK R = 0, then typeu R < m + n + q, (3) iftypeu R = m + n + q, then dimu R ~ Ll-£*-trdegK R


PROOF. Let A(r) (r EN) denote the set of all elements

\ _rlcl rlc m 11 1"(JUl f.lUq 1\ -U1 .. . Um a 1 .• . an 1 .. ·fJq

(k1 , ... ,km ,lt, ... ,lnEN; U1, ... ,Uq EZ),

such that m n q

ordA = Lk; + Llj + L Iuvl ~ r, ;=1 j=l v=l

let Rr = K[A(r)7Jl U ... U A(r)7J.l for r E Nand Rr = K for r E Z r < O. Furthermore, for every prime l1-u-c· -ideal P of the algebra R, let iiI, ... , ii, be the canonical images of the elements 7Jl, ... , 7J" respectively, in the ring R/ P.

It is easy to see that if r E Nand P is a prime l1-u-c· -ideal of R, then P n Rr is a prime ideal of the ring Rr and the quotient fields of the rings Rr / P n Rr and K[A(r)iil U· . ·UA(r)ii,l are isomorphic. By Theorem 6.7.3, there exists a numerical polynomial Xp(t) in one variable t such that

xp(r) = trdegK K[f(r)ii1 U ... U r(r)ii,l = trdegK(Rr / P n Rr)

for all sufficiently large r E Z, deg XP (t) ~ m + n + q and the polynomial XP (t) can be written as

2qap Xp(t) = tm+n+q + o(tm+n+q),

(m+n+q)!

where ap =11-u-c·-trdegK(R/P). Obviously, if P,Q E U and P:J Q, then Xp(t) ~ XQ(t). Furthermore, if P, Q E U, P 2 Q and Xp(t) = XQ(t), then P = Q (see the similar statement at the beginning of the proof of Theorem 7.1.10).

Now the same arguments that were used in the proof of Theorem 7.1.10 show that, for any d E Z, d ~ -1 and for any pair (P, Q) E Bu, the inequality J-Iu (P, Q) ~ d implies the inequality deg(XQ(t) -Xp(t)) ~ d. Since deg Xp(t) ~ m+n+q, for any prime l1-u-c·-ideal P of Rand deg Xp(t) < m+n+q if l1-u-c·-trdegK R = 0 (in this case ap =11-u-c*-trdegK(R/ P) = 0), we have deg(XQ(t) - deg Xp(t)) ::; m + n + q for any pair (P, Q) E Bu and this inequality becomes strict if ~-u-c·-trdegK R = O. Therefore, if (P, Q) E Bu, then J-Iu(P, Q) ::; m + n + q, and J-Iu(P, Q) < m + n + q if l1-u-c· -trdegK R = o. Thus, typeu R ::; m + n + q and typeu R < m + n + q if l1-u-c*-trdegK R = O.

Now, let us prove the last statement of the theorem. Let typeu R = m + n + q and let Po :J PI :J ... :J Pd (d E N) be a descending chain of prime l1-u-c· -ideals of the algebra R such that J-IU(Pi- 1 , Pi) = typeu R = m + n + q (i = 1, ... , d). Clearly, in order to prove the inequality dimu R ~ ~-u-c· -tr degK R, it is sufficient to show that d ~ ~-u-c·-trdegK R.

First of all, we can write

d

xpAt) - XPo (t) = L(XPi (t) - XPi_l (t)) = ;=1

d [ 2q ] = ""' (ap. - ap._ )tm+n+q + o(tm+n+q) = ~ (m + n + q)! • • 1 .=1


Since PU(Pi-1, Pi) = m + n + q, by the above consideration we have

deg(XPi (t) - XPi_, (t)) ?: m + n + q,

where X'IIK (t) is the dimension polynomial of the ~-(T-c" -field extension K (T)1 , ... , T)s) of the field K. Since

(see Theorem 6.7.3), the last chain of inequalities shows that apd - apo :5 ~-(T-c"trdegK R. Thus, d :5 apd - apo :5 ~-(T-c" -trdegK R. This completes the proof of the theorem. 0

7.2.3. THEOREM. Let K be a~-(T-c"-field with the basic set ~U(TUc" described above, let R = K {Y1, ... , y.} be a ~-(T-c* -polynomial algebra in a family of ~-(T-c" -indeterminates Y1, ... ,Ys over the field K, and let U be the family of all prime ideals of R. Then typeu R = m + n + q, dimu R = s.

PROOF. Let U1 denote the family of all homogeneous linear ~-(T-c' -ideals of R. Since every ideal of the family Ut is prime (see ... ), typeu, R ?: typeu R. By the first statement of Theorem 7.2.2 we have typeu, R:5 m+n+q, so in order to prove the equality typeu, R = m+n+q it is sufficient to show that typeu, R?: m+n+q. This inequality, in turn, is the consequence of the inequality

pu, ([Y1, ... , YP], [Y1, ... , Yp-1]) ?: m + n + q, (7.2.1)

(1:5 P :5 s) which we are going to prove. Let U~p) denote the set of all homogeneous linear ~-(T-c· -ideals P E U1 such that

[Y1, ... , Yp] 2 P 2 [Y1, ... , Yp-d}· Then the inequality (7.2.1) is equivalent to the inequality

(7.2.2)

Furthermore, the canonical ~-(T-c' -isomorphism

induces the bijective mapping of the family uip) onto the family U~p) of all homogeneous linear ~-(T-c' -ideals of K {YP' ... ,Y.} whose generators are of the form

2:;=1 ajAj(Yp), where aj E K, Aj E A (k is a positive integer). Of course, this mapping preserves inclusions of the ideals.

Let U~p) (1:5 P :5 s) be the family of all homogeneous linear ~-(T-c"-ideals of the

~-(T-c' -algebra K {Yp}. It is easy to see that if J E U~p), then J . K {YP' ... , Ys} E


u~p) and J. K {YP' ... , Y.} n K {Yp} = J. Thus, there exists a one-to-one correspon

dence between the families U~p) and U~p) that preserves inclusions. By the above reasoning, in order to prove inequality (7.2.2) (and, threfore, in

equality (7.2.1)) it is sufficient to show that if /( {y} is a Ll-u-c· -polynomial algebra in one determinate Y over a Ll-u-c· -field /( and B is the family of all homogeneous linear Ll-u-c·-ideals of /( {y}, then typeB{Y} ~ m + n + q. We shall prove the last inequality by induction on m+n+q. If m+n+q > 0, then this inequality is trivial. Let Ll' = {d'l' ... , d'm-l} and let us consider the descending chain of Ll-u-c" -ideals

[y) :) [d'm y) :) ... :) [d';;' y) :) ... :) (0). (7.2.3)

For every r EN, let Br denote the set of all ideals I E B such that [d':;' y) :) I :) [d';;..+ly). If (r is the canonical image of the element d':;'y in the Ll-u-c·-ring /({d':;'y}j[d':;,+ly], then this ring coincides with the ring /({(r} and can be treated as a Ll'-u-c· -polynomials ring in one Ll'-u-c· -indeterminate (r over /( (the structure of the Ll-u-c· -ring /( {(r} is determined by the condition d'm(r = 0). Furthermore, there exists a natural one-to-one correspondence between Br and the family of Ll'u-c" -ideals of the ring /( {(r } generated by various sets of elements of the form>..' (r where>..' is an element ofthe semigroup A whose expression (7.1.2) does not contain powers of d'm. This correspondence preserves the inclusions of ideals, so that

JlB. ([d';;'y) , [d';;'+l y]) ~ m + n + q - 1

(by the inductive hypothesis). Therefore, JlB.([y], (0)) ~ m + n + q - 1 and typeB/({Y} ~ m+n+q.

Now, let m + n + q > 0 and n > 0 (the case m + n + q > 0, q > 0 can be considered similarly). Then the descending chain of Ll-u-c· -ideals

[y) :) [(an -l)y) :) ... :) [(an - 1ty) :) '" :) (0)

can be used for the step of induction (as the chain (7.2.3) was used in the case m > 0). More precisely, if 0" = {al, ... ,an-l} and "Ir (r E N) is a canonical image of the element (an - 1 r y in the Ll-u-c· -ring /( {( an - 1 Y y} j[( an - 1 y+l y], then the last ring can be treated as a Ll-u'-c· -polynomial ring K {"Ir} in one Llu'-c· -indeterminate "Ir over the field /( (the structure of Ll-u-c· -ring on /( {"Ir} is determined by the condition an"lr = "Ir). Let Ar be the family of all Ll-u'-c*ideals of /( {"Ir} generated by various sets of elements of the form ).."Ir where ).. is an element of the semigroup A whose expression (7.1.2) does not contain powers of an. Then

so the inductive hypothesis leads to the inequality

JlB)[(an -lty], [(an - W+1y]) ~ m + n + q - 1

for any r E N, hence JlB.([y), (0)) ~ m + n + q - 1, so that typeB{Y} ~ m + n + q. Thus, inequality (7.2.1) is proved, so the equality typeB{y} = m + n + q is also proved. Moreover, the above reasonings show that

JlU([YI, ... , yp], [YI, ... , Yp-I]) = typeu R = m + n + q


for all p = 1, ... , s hence dimu R ~ m + n + q. On the other hand, by the last statement of Theorem 7.2.2 we have dimu R::; m+n+q. Thus, dimu R = m+n+q, so the theorem is proved. 0

In the case of finitely generated differential algebras without zero divisors the result of Theorem 7.2.2 can be essentially strengthened. Namely, the following theorem of J. Johnson (see [J069c]) is valid.

7.2.4. THEOREM. Let F be a differential field of zero characteristic with a basic set of derivation operators .1. = {51, ... ,5m } and let R = F {7]1, ... , 7].} be a finitely generated .1.-F -algebra without zero divisors (7] = {7]1, ... , 7].} is the set of .1.-generators of Rover F). Let U be the family of all prime .1.-ideals of the algebra R. Then the following statements are valid:

(i) if .1.-trdegF R > 0, then typeu R = m and dimu R = .1.-trdegF R; (ii) if .1.-trdegF R = 0, then typeu R < m.

PROOF. First of all, it should be noted that by Theorem 7.2.2, the following conditions hold:

typeu R ::; mj

if .1.- trdegF R = 0, then typeu R < mj

if typeu R = m, then dimu R ::; .1.- trdegF R

(7.2.4)

(7.2.5)

(7.2.6)

The second of these statements shows that we need to prove only the statement (i) of the theorem.

The proof will be based on the following well-known results of Commutative Algebra (see, for example, [Bou61, Chapter V, Sect. 3, Corollary 1]).

7.2.5. LEMMA. Let A be a finitely generated commutative algebra without zero divisors over a field K (Char K = 0), and let Xl, ... , Xd be a transcendence basic of A over K. Then there exists an element T E K[XI, ... , Xd] such that A [~] is an integral extension of the ring K[XI, ... , Xd,,f.-] (the element ,f.- is considered here as an element of the quotient field of A).

7.2.6. LEMMA. Let A be an integrally closed commutative ring, let a commutative ring B be an integral ring extension of A, and let 0 be the only element of A that is a zero divisor in B. Furthermore, let PI and P2 be prime ideals of A such that PI ::> P2, and let Q2 be a prime ideal of B such that Q2 n A = P2. Then there exists a prime ideal QI of B such that QI ::> Q2 and QI n A = Pl.

The last lemma implies the following result.

7.2.7. COROLLARY. Let A be a subring of a commutative ring B such that B is integral over A. Let Po, PI, P2 be prime ideals of A such that Po ::> PI ::> P2, let Qo, Q2 be prime ideals of B such that Qo ::> Q2, Qo n A = Po and Q2 n A = P2, and let the ring AI Po is integrally closed. Then there exists a prime ideal QI of B such that Qo ::> QI ::> Q2, QI n A = PI and Ql is a prime component of the ideal PI B in the ring B.

The proof of Theorem 7.2.4 for m = o.


Let R be a finitely generated commutative algebra without zero divisors over a field F (Char F = 0). Let d = trdegF R > 0 and let Xl, ... , Xd be a transcendence basis of Rover F. By Lemma 7.2.5, there exists an element T E F[Xl, ... , xdl such that R[ +-] is an integral extension of the ring F[ Xl, ... , Xd, +-]. Let (1, ... , (d

be elements of R such that T((l, ... ,(d) =1= 0 (since the field F is infinite, one can choose such elements) and let Qi (i = 1, ... , d) denote the prime ideal of the ring F[xl, ... , Xd, +-1 generated by the elements Xl - (1, ... , Xi - (i. By the "going up" theorem (see Theorem 1.3.58), the chain of prime ideals

of the ring F[Xl, ... , Xd, +-1 can be lifted to a chain

Qd ~ Qd-l ~ ... ~ Q~ ~ Q~ = (0)

of prime ideals of the ring R[+-l such that Q: n F[Xl, ... , Xd, tl = Qi (0 :s i :s d). Setting Q~' = Qi n R (0 :s i :s d) we obtain the strictly descending chain of prime ideals of the algebra R:

Q" :J Q" :J ... :J Q" :J Q" = (0) d ~ d-l ~ ~ 1 ~ 0 .

Therefore, dimu R ~ d = trdegF R (as above, U denotes the set of all prime ideals of R). Combining the last inequality with inequality (7.2.6) we obtain the desired equality dimu R = trdegF R.

The proof of Theorem 7.2.4 for m > O. First of all we shall prove a number of auxiliary statements (see Lemmas 7.2.8-

7.2.13 below and their corollaries).

7.2.8. LEMMA. Let A be a subring of a commutative ring B and let UA, UB be families of ideals of the rings A and B, respectively, such that the following conditions hold:

(i) if I, l' E UA and I ~ I', then there exist ideals J, J' E UB such that J ~ JI, J n A = I and J' n A = I';

(ii) let Jo, J2 E UB and Jo ~ h. If It E UA and Jo n A ~ It ~ J2 n A, then there exists an ideal h E UB such that Jo ~ Jl ~ J2 and Jl n A = It.

Then typeuA A:S typeuB B. Furthermore, iftypeuA A = typeuB B, then dimuA A :s dimuB B.

PROOF. Let I, l' E UA; J, J' E UB and I ~ I', J ~ JI, 1= J n A, I' = J' n A. Let pEN and J.l.uA (1', I) ~ p. We shall show that J.l.UB (J/, J) ~ p by induction on p. If p = 0, then this inequality is trivial. Let p > o. Then there exists an infinite descending chain of ideals l' = Io :J It :J ... :J I of the ring A such that J,. E UA and J.l.UA (Ik' Ik+1) ~ p - 1 (k = 0,1, ... ). By condition (ii) there exists an infinite descending chain of ideals J' = Jo :J Jl :J ... :J J of the ring B such that Jk E UB and Jk n A = J,. (k = 0,1, ... ). By the inductive hypothesis we have J.l.UB(Jk,Jk+l) ~ p-I for k = 0,1, ... , hence J.l.UB(J/,J) ~ p. Therefore, typeuA A :s typeUB·


Now, let typeuA A = typeuB B. It follows from the conditions (i) and (ii) that, for any chain 10 2ft 2ft 2···2 IdofidealsofthefamilyUA with/-luA (Ik,h+1) = typeuA A (k = 0,1, ... , d - 1), there exists a chain of ideals Jo 2 J 1 2 ... 2 Jd of the family UB such that

(k = 0, 1, ... , d - 1)

and Jk n A = Ik (0 ~ k ~ d). Thus, dimuA A ~ dimuB B, so the lemma is proved. 0

7.2.9. DEFINITION. Let R be a differential integral domain with a basic set ~ = {db ... , dm} and let F be a ~-subring of R. We shall introduce the notion of parametric subring of Rover F by induction on m = Card~.

1. Let m = O. Then a subring Ro of R is called a parametric subring of Rover F, if the following conditions hold:

(1) the quotient field of R is a finite algebraic extension of the quotient field of Ro;

(2) there exists a finite set {1/1, ... , 1/n} of elements of R such that Ro = F[ 1/1, ... , 1/n] and elements 1/1, ... , 1/n are algebraically independent over F.

2. Let m > O. In this case a subring Ro of R is called a parametric subring of the ~-ring Rover F, if there exist ~-algebraically independent over F elements 1/1, ... , 1/n E Ro such that Ro is a parametric subring of Rover F {1/1, ... , 1/n}, when the last two rings are considered as differential rings with the basic set ~(m-1) = {d1, ... ,dm -il·

The following result is the immediate consequence of the definition of parametric subring.

7.2.10. LEMMA. Let R be a differential integral domain with a basic set ~ and let F be a ~-subring of R. Then the following statements are valid.

(1) Let Ro be a subring of R, such that F ~ Ro. Then Ro is a parametric subring of Rover F if and only if Ro is a parametric subring of the quotient ~-field of Rover F.

(2) If Ro is a parametric subring of Rover F, then Ro is a polynomial ring extension of the ring F, and the quotient field of R is a finite algebraic extension of the quotient field of Ro.

7.2.11. LEMMA. Let R be a differential integral domain with a basic set ~ = {d1, ... , dm }, F a ~-subring of Rand Ro a parametric subring of Rover F. Let [( and [(0 be the quotient fields of the rings Rand Ro, respectively, and let a1, ... , an be the basis of the field [( over [(0. Then there exists an element {j E Ro \ (0) such that S = E~=l Ro[;~·]aj is a finitely generated ~-algebra over the ring F.

PROOF. Let M = E~=l Roaj. We are going to prove the lemma by induction on m = Card~. If m = 0, then we can choose an element {j E Ro \ (0) such that {jajaj E M for all i,j = 1, ... ,n. If Ro = F[1/1, ... ,1/d] for some elements 1/1, ... ,1/d E Ro that are algebraically independent over F (see Definition 7.2.9), then S = Ro[;klM = F[1/1, ... , 1/d, ;k, a!, ... , an] is a finitely generated F-algebra.


Now, let m > 0 and let 1]1, ... , 1]d be elements of Ro, that are algebraically independent over F and such that Ro is a parametric subring of Rover F {1]1, ... , 1]d} when the last two rings are considered as differential rings with the basic set Ll(m-1) = {o, ... , om-d (i.e., as Ll(m- 1l-rings). By the inductive hypothesis, there exists an element f3 E Ro \ (0) such that Ro[~]M is a Ll(m- 1l-algebra over the

Ll(m- 1l-ring F{1]l, ... , 1]d} with some finite set of Ll(m-1l-generators (1,.··, (q. Let us choose an element, E Ro \ (0) such that ,om((d E Ro[~]M for every

i = 1, ... , q. It is evident that the F-algebra RO[/-y]M is closed under the deriva

tion operators 01, ... , Om-1 and Om(i E Ro[p\]M for every i = 1, ... , q. Therefore,

OmB(i E RO[/-y]M for any B E e(m-1) = {0~1 ... 0:':..-11 E fiI} (1 ~ i ~ q), hence

Ro[p\]M is a finitely generated Ll-algebra over F. This completes the proof. 0

7.2.12. COROLLARY. Let Rand F be differential integral domains with a basic set Ll such that R is a finitely generated Ll-extension of F. Let Ro be a parametric subring of the Ll-ring Rover F. Then there exists an element T E Ro \ (0) such that R[~] is a finitely generated module over the ring Ro[~].

PROOF. Let a1, ... , an be a basic of the quotient field J{ of the ring R over the quotient field J{o ofthe parametric subring Ro. By Lemma 7.2.11, one can choose an element f3 E Ro \ (0) such that Ro-module S = 2:::7=1 Ro[~]ai is a finitely generated Ll-F-algebra. Let elements (1, ... ,(p generate R as a Ll-F-algebra. Then there exists an element, E Ro \ (0) such that (k E 2:::~1 Ro[p\]ai for every k = 1, ... ,po

Since R[p\] = 2:::7=1 Ro[p\]ai, one can set T = f3, and obtain the desired result:

R[~] is a finitely generated Ro[~]-module. 0

7.2.13. LEMMA. Let R be a differential integral domain with a basic set Ll = {Ol, ... ,Om}, let CharR = 0, and let F be a Ll-subring of R. Furthermore, let the quotient field J{ of the ring R be a finitely generated Ll-field extension of the quotient field Ko of the ring F, and let 1]1, ... , 1]d be a Ll-transcendence basis of Rover F. Then there exists a nonsingular m x m-matrix G = (cijh$:i,j$:m over the field of constants G(Ko) of the Ll-field Ko with the following property: if all differential algebraic structures considered above are treated relative to the basic set of derivation operators Ll' = {o~, ... , O:,,} with 0: = 2:::';=1 CijOj (1 ~ i ~ m), then the Ll'-ring R contains a parametric subring over F {1]1, ... , 1]d}.

PROOF. We proceed by induction on m = Card Ll. 1. If m = 0, then K is a finite algebraic extension of the quotient field of

the ring F[1]1, ... , 1]d], hence F[1]l> ... , 1]d] itself is a parametric subring of Rover F[1], .. . , 1]d].

2. Let m > O. By Theorem 5.6.3, there exists a nonsingular m X m-matrix G' = (C:jh$:i,j$:m over the field G(Ko) with following properties: if 0: = 2:::~1 cijoj (1 ~ i ~ m}, and Ll:"_l = {o~, ... ,0:"_1}' then J{ is a finitely generated Ll:"_1-extension of the quotient field of the Ll:"_cring F {1]1, ... , 1]dh;"_I. By the inductive hypothesis, there exists a nonsingular (m - 1) x (m - I)-matrix Gil = (cijh$:i,j$:m-1

over the field G(Ko) such that the following condition holds: if 0:' = 2:::,;=-;.1 cijoj (1 ~ i ~ m - 1) and Ll~_1 = {o~, ... , 0::'_ tl, then there exists a parametric sub-ring Ro of the ring Rover F { 1]1, ... , 1]d}, when the last two rings are considered


as A~_1-rings. Now, the matrix C which describes the transition from the basic set A to the basic set A" = {O~, ... , 0::"_1'0:"} is the matrix whose existence is asserted by the Lemma. This completes the proof. 0

Now we are going to complete the proof of Theorem 7.2.4. Let R be a finitely generated differential algebra without zero divisors over a differential field F (Char F = 0) with a basic set A = {Ol, ... ,Om} (m > 0). Let d = A-trdegpR > 0 and let U be the family of all prime A-ideals of the algebra R. It follows from the conditions (7.2.4) and (7.2.6) that Theorem 7.2.4 will be proved if we show that (m, d) -< (typeu R, dimu R), where -< is the lexicographic order on the set Z2.

Let 1/1, ... , 1/d be a A-transcendence basic of the algebra Rover F. By Lemma 7.2.10 and Corollary 7.2.12 there exist a subring Ro of R and an element T E Ro \(0) such that the following conditions hold:

1) Ro is a polynomial ring extension of the ring F{1/l, ... , 1/d}; 2) the algebra R[,j-] is a finitely generated Ro[,j-]-module. Let 6 be the free commutative semigroup, generated by the elements 01, ... , Om

and let (i = ()1/i (1 ~ i ~ d) for some element () E 6. Let Ul denote the family of all A-ideals of the ring F{(l, ... , (d}, which are generated (as A-ideals) by finite subsets of the set (6(1 U ... u 6(d). We also suppose that the order of () is so large that the element T does not belong to any ideal of the family Ul .

Let B be the family ofall prime A-ideals of the ring R[,j-], whose intersection with F{(l, ... , (d} belong to Ul . It is easy to see, that the family B can be identified with the family of all prime A-ideals J of R such that T f/. J and J n F {(I, ... ,(d} E Ul. Now, in order to complete the proof of Theorem 7.2.4, it is sufficient to show that (m, d) -< (typeB R, dimB R). Moreover, it follows from Lemma 7.2.8 and Theorem 7.2.2 that we only need to check the validity of the conditions (i) and (ii) of Lemma 7.2.8 (for A = F{(lJ ... ,(d}, B = R, UA = Ul, UB = B).

7.2.14. REMARK. With the above assumption, the ring Ro, being a polynomial ring over F { 1/1, ... , 1/d}, is also a polynomial ring over F { (1, ... , (d}. Furthermore, if I E Ul, then the ideal I Ro of the ring Ro is generated by linear homogeneous elements of the ring, so that Roll Ro is a polynomial ring over F. Therefore, for any ideal I E Ul , the ring Ro[,j-]IIRo[,j-]' is integrally closed. Finally, let IE Ul and let P be a prime ideal of the ring R[,j-], containing I Ro [,j,l Then P is a minimal prime component of the ideal I R[,j-] hence P is a prime A-ideal (see, for example [KoI73, p. 64]).

Let I, I' E Ul and I ~ I'. Since R[,j-] is an integral extension of the ring Ro[,j-], there exist prime ideals J and J' of R[ ~] such that J ~ J' and J n Ro [,j-] = I Ro [~], J' n Ro[,j-] = I' Ro[~]. Then J, J' E B and these ideals satisfy the condition (i) of Lemma 7.2.8, so that this condition holds.

Let us show that the condition (ii) of Lemma 7.2.8 holds too. Let J o, J 2 E B, h EU1 and let Jo ~ J2, JonF{(lJ···,(d} ~ h ~ J2 nF{(t, ... ,(d}. Furthermore, let 10 = Jo n F{(lJ ... , Cd}. By Corollary 7.2.7, there exists a prime ideal J1 of the ring R[,j-] such that Jo ~ J l ~ hand J l n Ro[~] = hRo[,j-].

Thus, both conditions of Lemma 7.2.8 hold, whence (m, d) -< (typeB R, dims R). This completes the proof of Theorem 7.2.4. 0


7.3. Difference-Differential Local Algebras

Let K be a t::..-u-c· -field of zero characteristic, whose basic set consists of of a finite family of derivation operators t::.. = {dl' ... , dm } of the field K into itself and two finite families u = {aI, ... , an}, c = {.lh, ... ,.Bq } of automorphisms of K such that any two elements of t::.. U u U c commute which each other. As above, we set c· = {.Bl, ... , .Bq , .Bll , ... , .B; I}. Furthermore, if t::.. = 0 and u = 0 (respectively, t::.. = 0 and c = 0 or u = 0 and c = 0), then we treat K as an inversive difference field with the basic set u or as a differential field with the basic set t::...

Let A be the commutative semigroup of elements of the form

(7.3.1)

and let D (or DK) denote the ring of t::..-u-c·-operators over the field K (see Definition 3.5.41), provided with the filtration introduced in Section 3.5. (Recall that the order of the element). of the form (7.3.1) is the number

m n q

ord)' = Ek; + Eli + E luvL ;=1 i=l v=l

the order of a t::..-u-c·-operator w = L.\EA a.\). E D (a.\ E K for all ). E A and a.\ = 0 for almost all ). E A) is the number ord w = maxi ord)' I a.\ i- O}, and the filtration (Dr )rEZ of the ring D is defined as follows: Dr = {w E D lord w ~ r}, for r E N, and Dr = 0, for r < 0).

7.3.1. DEFINITION. A t::..-u-c· -algebra A over the t::..-u-c· -field K is called a local t::..-u-c· -K -algebra, if A is a local ring whose maximal ideal m is a t::..-u-c· -ideal.

If A is a local t::..-u-c· -algebra with a maximal ideal m over a t::..-u-c· -field K, then the field of residue classes k = Aim, obviously, is a t::..-u-c* -extension of K, and m/m2 can be naturally consider as a vector t::..-u-c*-space over the t::..-u-c*-field k.

7.3.2. EXERCISE. Let the maximal ideal m of a local t::..-u-c*-K-algebra A be a finitely generated t::..-u-c· -A-module and let the canonical images of elements It, ... ,/v Em in m/m2 generate m/m2 as a vector t::..-u-c*-k-space (k = Aim). Show that elements It, ... ,/v E m generate m as a t::..-u-c· -ideal of A (in this case we write m = [It,· .. , tv])·

Let A be a commutative ring and B an A-algebra. As in Section 1.4, by n B / A, we shall denote the module of Kahler differentials of B over A, and the corresponding derivation of B into in nB / A will be denoted by dB/A.

7.3.3. PROPOSITION. Let A be a t::..-u-c* -ring, whose basic set consists of derivation operators t::.. = {dl, ... , dm } and two families of automorphisms u = {aI, ... , an} and c = {.Bl, ... , .Bq } of this ring. Let c* = {.Bl, ... , .Bq , .Bl l , ... , .B;l}, A the commutative semigroup of the elements of the form (7.3.1), D the ring of t::..-u-c*-operators over A, and B a t::..-u-c*-algebra. Then the B-module nB/Acan

7.3. DIFFERENCE-DIFFERENTIAL LOCAL ALGEBRAS 371

be uniquely provided with a structure of L::i-u-c*-B-module (i.e., with a structure of a left D-module) such that

(7.3.2)

for any elements "{ E L::i U u U c·, b E B (it follows that the same equality holds for any bE B, "( E A).

PROOF. Let us define the action of elements L::iUuUc· on the A-algebra B®A B as follows:

"((b ® b') = "((b) ® "((b'), l5(b ® b') = l5(b) x b' + b ® l5(b')

for all b, b' E B, "{ E u U c·, 15 E L::i. Then B ®A B becomes a L::i-u-c· -A-algebra and the canonical homomorphism p: B ®A B ~ B becomes a L::i-u-c·-homomorphism (indeed, if b, b' E Band "( E u U c·, then

p("{(b ® b')) = p("{(b) ® "((b')) = ,,((bh(b') = "((p(b ® b'))j

similarly pl5 = 15 p for any 15 E L::i). Therefore, if / = Ker p, then /1/2 is a L::i-u-c·B-module such that the action of arbitrary element "{ E L::i u u U c· satisfies the condition

"(((b ® 1 - 1 ® b) + /2) = ("{(b) ® 1 - 1 ® "((b)) + /2,

i.e. "((dB/Ab) = dB/A"{(b) for any element bE B. The uniqueness of the structure of L::i U u U c·-B-module on 0B/A follows from

the fact that elements dBIAb bE B) generate OB/A as a B-module and the images of these elements under the mappings "{ E L::i U u U c* are uniquely determined by relationship (7.3.2). 0

7.3.4. PROPOSITION. Let K be a L::i-u-c·-field and let A be a local L::i-u-c·-Kalgebra with the maximal ideal m and with the field of residue classes k = Aim. Then there exists an exact sequence of vector L::i-u-c· -k-spaces

(7.3.3)

where L::i-u-c· -homomorphisms p and v act as follows:

(7.3.4)

for all x E m, yEA (jj is the image of y under the canonical epimorphism A ~ Aim).

PROOF. Since K is a field, any short exact sequence of K-modules splits. It follows from Theorem 1.5.17 that there exists a short sequence of vector k-space

I 2 P 1/ n O~mm ~OA/K®Ak-+Hk/K~O,

where the actions of k-homomorphisms p and v satisfy conditions (7.3.4). Since m and m2 are L::i-u-c· -ideals of A, m/m2 can be naturally considered as a vector


Ll-O"-g* -k-space, i.e. as a left -rk-module (as above, the ring of Ll-O"-g* -operators over a Ll-O"-g*-ring B is denoted by -rB). Furthermore, by Proposition 7.3.3 we can consider the module of Kahler differentials Ok/K as a left -rk-module, and the Amodule Ok/K can be considered as a -rA-module. Now, the same actions of elements of the set Ll U 0" U g* on the vector k-space 0A/k ~A k as were considered Remark 3.5.44 turn 0A/K ~A k into a -rk-module. (Recall that the actions mentioned are defined on the generators of the k-module OA/K ~A k as follows:

-y(dA/KX ~ a) = -y(dA/KX) ~ -y(a),

8(dA/KX ® a) = 8(dA/K X) ~ a + dA/KX ~ 8(a)

for any -y E 0" U g*, 8 Ell, a E k, x E A.) It remains to show that the k-homomorphisms p and v are Ll-O"-g* -homomor

phism, i.e. that

(7.3.5)

for any elements

-y E Ll U 0" U g*, x = x + m2 E mjm2 , e E A.

If -y E 0" U g* , then we have

and

where e denotes the canonical image of e in the field k. The appropriate relationships for -y E Ll can be checked by the same way.

Thus, p and v are Ll-O"-g* -homomorphisms of vector k-spaces (7.3.3) is an exact sequence of -rk-modules. This completes the proof. 0

7.3.5. EXERCISE. Show that, with the above notations, Bm is a local ring with the maximal ideal m n Bm.

7.3.6. DEFINITION. Let K be a Ll-O"-g*-field and let A be a local Ll-O"-g*K -algebra without zero divisors. The algebra A is said to be a local Ll-O"-g* -K -algebra of finitely generated type if there exist elements 111, ... , 11. E A such that A = K {111, ... , 11.}m, where m is the maximal ideal of A.

7.3.7. THEOREM. Let K be a Ll-O"-g* -field with the basic set Ll U 0" U g* considered above, and let an integral domain A be a local Ll-O"-g*-K-algebra offinitely generated type with a maximal ideal m and with the field of residue classes k = Ajm. Then:

(i) mjm2 is a finitely generated vector Ll-O"-g* -k-space; (ii) dimk mjm2 ~ Ll-O"-g*-trdegK A - Ll-O"-g*-trdegK k.


PROOF. Let Fk be the ring of t!.-u-c* -operators over k and let

be the exact sequence of Fk-modules described in Proposition 7.3.4. We are going to show that f2A/k ~A k is a finitely generated Fk-module. (Since the ring Fk is left Noetherian (see Lemma 6.7.9), we shall obtain that m/m2 is also a finitely generated Fk-module).

By the condition of the Theorem, there exist elements 711, ... ,71. E A such that A = K {711, ... , 71. }m. Let us show, that every element of the form dAI Ke (e E A) can be written as a linear combination of elements dA/K711, .. . , dA/K71. with coefficients from FA. Indeed, any element e E A can be expressed in the form e = 9 ~" ... '~' where 1 and 9 are t!.-u-c*-polinomials in s t!.-u-c*-indeterminates

9 1)· ." •

and 9(711, .. ·,71.) f/. m. Therefore,

• - 1(1}1 , ... , 71.)dA/K9(711 , ···,71.)1 E E DAdA/K1}i

i=1

(it is easily seen that dA/K 1(711, .. . ,71.) and dA/K9(71b ... ,71.) can be written as linear combinations of the elements dA/K ()..71i) = )..dA/K1}i ().. E A, 1:S i:S s) with coefficients from A).

It follows that the elements dA/K711 ® 1, ... , dA/K71. ~ 1 generate the Fk-module f2A/K ~A k. Indeed, any generator dA/K71~ 1 (71 E A) of the Fk-module f2AIK ~A k can be written as

where aij E A (1 :S i :S s, 1 :S j :S ri), and aij is the image of the element aij under the canonical homomorphism A ~ k = A/m. Since Ej~1 aij'"Yj E Dk (1 :S i :S s), we obtain that the elements dA / K 71i ® 1 (1 :S i :S s) generate the :Fk-module f2A/K ~A k. This completes the proof of the first statement of the theorem.

Let us prove statement (ii). For any r E N, let A(r) denote the set of all elements ).. ofthe form (7.3.1) such that ord)" :S r, and let A(r)71 = {)..(71} I ).. E A(r)} for any element 71 E A. Furthermore, let Ar denote the localsubring K[A(r)711U .. . A(r)71.1m of Ar (see Exercise 7.3.5), and let kr denote the field ofresidue classes Ar/mn Ar of this subring. Then, denoting M n Ar by mr (so that kr = Ar/mr), we obtain (see, for example, [AM69, Chapter 11, Corollary 11.15]) that

(7.3.6)

for every r EN. Let K {Y1, ... , Y.} be a ring of t!.-u-c* -polynomials in t!.-u-c* -indeterminates

Y1, ... , Y. over the field K (recall that as a ring, K {Y1, ... , Y.} coincides with the


polynomialringK[AYlU ... Ay,] infamilyofindeterminatesU:=l AYi = {A(Yi) IA E A, 1 ~ i ~ s}). Let cp: K {Yl, ... , Y.} -t A be the natural fl.-u-c* -homomorphism of fl.-u-c* -K -algebras (such that cp(Yi) = 1Ji for any i = 1, ... , sand cp( a) = a for any a E K) and p = Ker cpo Furthermore, let B denote the local fl.-u-c* -K-algebra K{Yl, ... , Y.},p-l(m) and n denote the maximal ideal of the algebra B. It is easy to see that the homomorphism cp can be naturally extended to a homomorphism of local fl.-u-c*-K-algebras q;: B -t A such that q;(n) = m. Let q = pB, Br (r E N) denote the local subring K[AYI U ... AY.]n of B, and if M ~ B, r EN, then Mr denote the set M n Br. In particular, nr = n n Br and we can identify the field Br/nr with the field kr (with respect to the isomorphism that is naturally induced by the homomorphism q;).

By Theorem 1.5.17, for every r E N there exists an exact sequence

(7.3.7)

and a commutative diagram with exact rows

(7.3.8)

-----+ OB / K ® B k -----+

(the first row of this diagram is obtained by applying of the functor· ®k r k to the exact sequence (7.3.7).)

We are going to show, that the mapping jr in diagram (7.3.7) is injective. Obviously, this statement is the consequence of the fact that for every vector k-space V the induced k-homomorphism of vector k-spaces

is surjective, i.e. the canonical mapping DerK(B, V) -t DerK(Br , V) (see corresponding mapping in the proof of Theorem 1.5.15) is surjective. Since B is the ring of quotients of a polynomial extension of the ring Br , every derivation from Br to a vector K -space V can be extended to a derivation from B to V, the mapping j; is surjective hence jr is injective. Furthermore, this implies that the mapping ir in diagram (7.3.8) is also injective.

Since elements dB/K(AYi) ® 1 (A E A(r), 1 ~ i ~ s) generate the vector kspace Imjr, (Imjr)rEN is an excellent filtration ofthe vector fl.-u-e*-k-space n/n2. Let Nr = Imir (r E N), N = n/n2, and let P denote the vector fl.-u-e*-k-space q + n2 /n2 ~ N. Furthermore, let Pr (r E N) denote the image of the vector fl.-u-c*-k-space (qr + n~/n~) ®k r k under the mapping ir. Then

Nr/Pr = ir((nr/n~) ®kr k)/ir((qr + n~/n~) ®kr k)

= (nr/qr + n~) ®k r k

= (mr/m;) ®k r k,

(Pr )rEN is an excellent filtration ofthe vector fl.-u-e* -k-space P, and m/m2 = N / P.


For every r EN, let (m/m2)r denote the vector k-subspace ofm/m2, generated by the canonical image of ffir in m/m2. Then ((m/m2)r )rEZ is an excellent filtration of the vector D.-u-c* -k-space m/m2 (note, that it is the image of the excellent filtration (Nr )rEN under the canonical mapping N -+ m/m2). Setting P: = P n Nr (r E N), we obtain (see Theorem 6.7.10), that (P:)rEN (r E N) is an excellent filtration of P and, obviously, Nr / P: = (m/m2)r for any r EN.

By Theorem 6.7.3 and by the analogue of Theorem 6.4.1 for D.-u-c*-field extensions (see Exercise 6.7.21), there exists numerical polynomials "IIi(t), XI(t), X2(t), and X3(t) in one variable t such that "IIi(r) = dim/c Nr , XI(t) = diffilc Pr , X2(t) = dim/c P: and X3(t) = trdegK Ar - trdegK kr. Since (Pr )rEZ and (P:)rEZ are two excellent filtrations of the D.-u-c*-k-vector space P, D.m+n+qXI(t) = D.m+n+qX2(t) (here D.m+n+q /(t) denotes the (m + n + q)-th difference of a polynomial /(t)), and D.m+n+qX3(t) = D.-u-c*-trdegK A - D.-u-c*-trdegK k. Since "IIi(r) - xdr) ~ X3(r) for all sufficiently large r E N (see formula (7.3.6)),

D.-u-c*-trdeg/c(m/m2) = D.m+n+q("IIi(t) - X2(t)) = D.m+n+q("IIi(t) - XI(t))

~ Llm+n+qX3(t) = D.-u-c* - trdegK A - D.-u-c* - trdegK k.


CHAPTER VIII

DIMENSION POLYNOMIALS OF FILTERED G-MODULES

AND FINITELY GENERATED G-FIELDS EXTENSIONS

In this chapter we deal with rings, fields and modules on which a finitely generated commutative group G acts. For such structures generalizations of the theorems on difference dimension polynomials and their invariants are derived. The main results of the chapter are Theorem 8.2.1 (which establishes the existence of dimension polynomial of an excellently filtered A-G-module over an artinian G-ring), Theorem 8.2.5 (this theorem describes the invariants of a dimension polynomial of a G-module over a G-field) and Theorem 8.3.16 (an analogue of Kolchin Theorem on differential dimension polynomial (see Theorem 5.4.1) that also generalizes Theorem 6.4.1 on difference dimension polynomial). The last result is helpful for the study of systems of "G-equations", i.e., algebraic equations with respect to indeterminates and their images under the actions of elements of a group G. Using Theorem 8.3.16, for any such system, one can consider the appropriate dimension polynomial and use it to determine the "strength" of the system in the sense of A.Einstein [Ei53].

8.1. Rings with a Group of Operators. G-modules

Let A be a commutative ring with 1 and let elements of a finitely generated commutative group G act on A as automorphisms of this ring. In this case A is called a G-ring. If J is any subring (ideal) of A such that g(J) ~ J for all 9 E G, then J is called a G-subring (respectively, a G-ideal) of the G-ring A. By a prime G-ideal we shall mean a G-ideal that is prime in the usual sense. If E is any subset of a G-ring A and Ao is a G-subring of A, then the intersection of all G-subrings of A containing Ao and E is the unique G-subring of A containing Ao and E and contained in every G-subring containing Ao and E. This intersection is denoted by Ao{E}a (or simply by Ao{E} if the group G is fixed) and is called the G-ring extension of Ao generated by the set E. The set E itself is said to be the set of G-generators of Ao{E} over Ao. (Obviously, Ao{E} coincides with the ring Ao[{g(a) I 9 E G, a E E}] generated by the set {g(a) I 9 E G, a E E} over the ring Ao.) If E consists of a finite number of elements T/1, ... ,TJp (p E N), then the G-extension Ao{E}a is denoted by AO{TJ1,.'" TJp}a or simply by AO{TJ1, ... , TJp} (if the group G is fixed). This ring is called the finitely generated G-ring extension of Ao with the set of G-generators {TJ1, ... , TJp}.

Let A and B be G-rings. A ring homomorphism <p : A ~ B is said to be a G-homomorphism if <p(g(a)) = g(<p(a)) for all a E A, 9 E G. The notions of G-epimorphism, G-monomorphism, G-isomorphism of G-rings are introduced in the usual fashion. It is easy to see that if <p : A ~ B is a G-homomorphism

377


378 VIII. DIMENSION POLYNOMIALS OF G-MODULES AND G-EXTENSIONS

of G-rings, then Ker cp is a G-ideal of the ring A. In this case the natural ring isomorphism between AI Ker cp and 1m cp is an G-isomorphism. Furthermore, if J is a G-ideal of a G-ring A, then AI J may be naturally considered as a G-ring (such that g(a + J) = g(a) + J for all 9 E G, a E A) and the canonical ring epimorphism A -t AI J is a G-epimorphism.

By a G-field we mean a G-ring which is a field (so that elements of the group G act as pairwise commuting automorphisms of this field). It is evident that if a G-ring A is an integral domain, then its quotient field Q(A) can be considered as a G-field such that g(%) = ~ for all a E A, 0 =F b E A, 9 E G. In this case Q(A) is said to be the G-quotient field of the G-ring A.

If I< is a subfield of a G-field L such that g(I<) ~ I< for all 9 E G, then I< is called a G-subfield of Land G-field L is said to be a G-overfield or a Gextension of I<. If E ~ L, then the intersection of all G-subfields of L containing I< and E is the smallest G-subfield of L containing I< and E. This intersection is denoted by I«E}G (or simply by I«E) if the group G is fixed) and is called the G-extension of the G-field I< genemted by the set E. The set E is said to be a set of G-genemtors of I< (E) over I<. (Clearly, I< (E) coincides with the field I« {g(a) I 9 E G, a E E}) generated by the set {g(a) I 9 E G, a E E} over the field f{). If Card E = p < 00, E = {'11, ... , '1p}, then f{(E} is said to be the finitely generated G-extension of f{ and is denoted by f{ ('11 , ... , '1p) (or by f{ ('11 , ... , '1p) G ) .

An algebra R over a G-ring A is called a G-A-algebm (or a G-algebm over A) if R is a G-ring and g(ax) = g(a)g(x) for all a E A, x E R, 9 E G. If R = A{ '11, ... , '1p}G for some elements '11, ... , '1p E R, then R is called a finitely genemted G-A-algebra.

Let A be a G-ring and R a G-A-algebra. A family ('1;)iEI of elements of R is said to be G-algebmically dependent (independent) over A if the set of all pairwise distinct elements of the form g('1i) (g E G, i E I) is algebraically dependent (respectively, independent) over A. If R = A{(Yi)iEdG and the family (Yi)iEI is G-algebraically independent over A, then R is called a G-polynomial algebm in the family of G-indeterminates (Yi)iEI over A. In this case elements of A are said to be G-polynomials in the G-indeterminates (Yi)iEl over the G-ring A. If I = {I, ... , s} (s EN, s ~ 1), then R = A {Y1, ... , Y.} is said to be a G-polynomial algebm in the G-indeterminates Y1, .... , Y. over A.

8.1.1. PROPOSITION. Let A be a G-ring and let I be an arbitrary set. Then there exists a G-polynomial algebra over A in the family of indeterminates (Yi)iEl with the set of indices I. If Rand R' are two such G-polynomial algebras, then there exists a G-isomorphism cp which maps R onto R' and leaves fixed every element ofA.

PROOF. Let R = A[(Yig)iEI,9EG] be the polynomial algebra over A in a family of indeterminates {Yig liE I, 9 E G} with the set of indices I x G. As in the inversive difference case (see Proposition 3.4.4), we obtain that there exists a unique extension of any automorphism 9 E G of the ring A to an automorphism of R such that g(Yih) = Yi,gh for every indeterminate Yih (i E I, h E G). (the extension of the automorphism 9 is denoted by the same letter g).

Let Yi = Yi,l for any i E I (by 1 we mean here the identity of the group G). Then R = A{(Yi)iEI}G and the family of all pairwise distinct elements of the form 9(Yi) = Yi,g (i E I,g E G) is algebraically independent over A, so that R is a

8.1. RINGS WITH A GROUP OF OPERATORS. G-MODULES 379

G-polynomial algebra over A in the family of G-indeterminates (Yi)iEI. The last part of the Proposition can be proved precisely as the corresponding statement of Proposition 3.4.4. 0

Let A be a G-ring and let S be a multiplicative subset of A such that g(S) ~ S for all 9 E G (hence, g(S) = S for all 9 E G). Then the ring of quotients S-1 A can be considered as a G-ring relative to the following actions of elements of G: g(~) = ~ for all f E G, a E A, s E S. This G-ring is said to be the G-ring of quotients of A over S.

B.1.2. EXERCISE. Prove that the actions of elements of G on the ring S-1 A are well-defined and S-1 A is a G-ring with respect to these actions.

Let A be a G-ring and let a primary decomposition of the group G is fixed

(8.1.1)

where {a;}oo is an infinite cyclic subgroup of G with a generator ai (1 ~ i ~ n) and {~j}qj (1 ~ j ~ k) is a cyclic subgroup whose order qj is a degree of a prime integer. Then each element 9 E G has a unique representation of the form

9 _ ~kl ~kn 011 oI m - '""I .. • ... n PI .. ·Pm (8.1.2)

where kl , ... , kn E Zj Ij E Z, 0 ~ Ii < qj (j = 1, ... , m). The number ordg = L7=1 Ikil+ LJ=l li is called the order of the element g. Clearly, ordg ~ 0, ordg = 0 if and only if 9 = 1 (the identity of the group G), and ord(glg2) ~ ordg l + ordg2 for any elements gl, g2 E G.

B.1.3. DEFINITION. Let A be a G-ring. An expression LgEG agg, where ag E A (g E G) and ag = 0 for almost all 9 E G, is called a G-operator over A. Two G-operators LgEG agg and L9EG bgg are equal if and only if ag = bg for all 9 E G.

The set of all G-operators over A has a natural structure of a left A-module. This set becomes a ring if we define the product g1g2 (g1, g2 E G) as in the group G, set ga = g(a)g for all 9 E G, a E A, and demand the validity of the distributive laws. This ring is said to be a ring of G-operators over A and is denoted by :F or by :FA. The ring A can be naturally considered as a subring of :FA.

If P = L9EG agg E :FA, then the number ordP = max{ordg lag =f. O} is called the order o/the G-opemtor P. It is easy to see that ord(PQ) ~ ordP+ordQ and ord( P + Q) ~ maxi ord P, ord Q} for any G-operators P, Q E :FA. Note that the order of a G-operator depends on decomposition (B.1.1) of the group G.

8.1.4. REMARK. Since the study of G-rings and modules over rings of Goperators is closely connected with the study of inversive difference rings and modules, the terminology used here is closely related to that of difference algebra. (In the terms of Ring Theory, :F is a twisted group ring of the group G over the ring A.)

B.1.5. DEFINITION. A left :FA-module is said to be a G-A-module (or a Gmodule over a G-ring A). Thus, an A-module M is said to be a G-A-module if the elements of Gact on M in such a way that the following conditions are satisfied:

(i) each element 9 E G acts as an endomorphism of the additive group of M j


(ii) the identity of G acts as an identity automorphism of M; (iii) gl(g2(X)) = g2(gl(X)) for all gl,g2 E G, x E M; (iv) g(ax) = g(a)g(x) for all g E G, a E A, x E M.

Let M and N be G-A-modules. A homomorphism of A-modules II' : M -+ N is said to be a G-homomorphism (or a homomorphism of G-A-modules) if ip(gx) = gip(x) for all g E G, x E M. The notions of epimorphism, monomorphism and isomorpism of G-A-modules are defined as usual.

Let A be a G-ring and let :F be the ring of G-operators over A. Let :Fr = {p E :F I ordP ~ r} for any r E Nand:Fr = 0 for all r E IZ, r < O. Then (:Fr)rEZ is an ascending filtration of the ring :F. In what follows, if the ring :F is considered as a filtered one, we suppose that its filtration is (:Fr )rEZ. If M is a G-A-module, then by a filtration of M we shall always mean its discrete and exhaustive ascending filtration over the filtered ring :F. In this case M is called a filtered G-A-module.

8.1.6. DEFINITION. A filtration (Mr)rEZ of a G-A-module M is called finite if any A-module Mr (r E IZ) is finitely generated. A filtration (Mr )rEZ of M is said to be good if there exists ro E IZ such that :F.Mr = Mr+. for all r 2: ro, sEN. A finite and good filtration of a G-A-module M is called excellent.

8.1.7. EXERCISE. Let A be a G-field,:F the ring ofG-operators over A and M a finitely generated G-A-module, i.e., M = 2::=1 :FXj for some elements Xl, ... , X. E M (s EN). Show that (2::=1 :FrXj)rEZ is an excellent filtration of the G-A-module M.

8.1.8. EXERCISE. Formulate and prove for G-A-modules the statements that are similar to the assertions of Proposition 3.4.44 and Exercises 3.4.45.

8.2. Dimension Polynomials of Excellently Filtered G-modules

In what follows we keep the notation and conventions of the previous section. While considering G-rings we shall always mean that the primary decomposition (8.1.1) of the group G is fixed.

8.2.1. THEOREM. Let A be an artinian G-ring and let (Mr)rEZ be an excellent filtration of a G-A-module M (associated with decomposition (8.1.1) of the group G). Then there exists a numerical polynomial g(t) in one variable t with the following properties:

(1) g(r) = lR(Mr ) for all sufficiently large r E IZ; (2) degg(t) ~ n (= rankG) and the polynomial g(t) can be written as g(t) =

2;!atn + o(tn) where a E IZ.

PROOF. Let gr:F be the graded ring associated with the filtered ring of Goperators :F over the ring A. It is easily seen that the ring gr:F is generated over A by the pairwise commuting elements Xl, ... , X2n, Y1, ... , Ym which are the canonical images in gr:F of the elements a1, ... , an, all, ... , a;; 1 ,.81, ... , .8m, respectively. Furthermore, Xja = ai(a)xi, xn+ia = a;l(a)xn+i (1 ~ i ~ n), Yi a = .8i(a)Yi (1 ~ j ~ m) for any a E A. The homogeneous components gr,:F (s E :F) of the

8.2. DIMENSION POLYNOMIALS OF EXCELLENTLY FILTERED G-MODULES 381

graded ring gr F are the A-modules generated by the set of all monomials of the form xtl ... x~:'!k ... 'II,.; where ki , Ij EN (1 ::; i::; n, 1::; j ::; m),

n m

Eki+Elj=s, i=l j=l

o ::; Ij ::; qj for all j = 1, ... , m, 1 ::; i 1 , .•. , in ::; 2n, and il' - ill oF n for all J.l = 1, ... , n; 1/ = 1, ... , n. In what follows we denote the ring gr F by Rn, while R~ and R~ denote the graded subrings of Rn whose elements can be written as linear combinations over A of the monomials free of Xn and X2n, respectively.

Let gr M = EB'EZ gr, M be the graded gr F-module associated with the excellently filtered G-A-module M (gr, M = M./M._ 1 for every s E Z). We assert that gr M is a finitely generated gr F-module. To prove this, let us consider F ®A Mr (r E Z) as a filtered left F-module with the filtration ((F®A Mr)')'EZ such that (F ®A Mr). = 2:i+j=. F; ®A (Mr n Mj) for every s E Z. Since the filtration (Mr )rEZ is excellent, there exists r E Z such that Mu = Fu-rMr for all U E Z, u ~ r. Therefore, the F-homomorphism 1r : F ®A Mr ~ M (1r(w ® x) = wx for all w E F, x E Mr) is an epimorphism of filtered F-modules (obviously, 1r((F ®A Mr).) = M. for any s ~ r). Now, the exactness of the sequence of filtered F-modules F ®A Mr ~ Mr ~ 0 implies the exactness of the sequence of graded gr F-modules

gr(F ®A Mr) gf'1") gr M ~ 0

(see Proposition 1.3.41). Using the natural epimorphism

we obtain the exact sequence of graded gr F-modules

(grl1")®p gr F ®A gr Mr ) gr M ~ o.

Since Mr is a finitely generated A-module, gr F ®A gr Mr is a finitely generated gr F-module, whence the graded gr F-module gr M is also finitely generated.

Let M(') = gr. M (s E Z). Since lA(Mr) = 2:.<r lA(M('») for every r E Z, in order to prove the theorem, we only have to establish the existence of a numerical polynomial h(t) in one variable t with the following properties:

(a) there exists So E Z such that h(s) = IA(M('») for all s E Z, s ~ So;

(b) deg h ~ n-l and the polynomial h(t) can be written as h(t) = (;:~)!tn-1 +

o(tn - 1) where a E Z.

(Indeed, if such a polynomial h(t) exists, then, by Proposition 2.1.6, there exists a numerical polynomial g(t) such that g(r) = lA(M('o») + 2::='0+1 h(s) = lA(Mr) for all sufficiently large r E Z, deg g(t) = deg h(t) + 1 ::; n and the leading coefficient of g(t) is equal to l. 2n a = 2n a.)

n (n-l)! n! We shall prove the existence of the polynomial h(t) with the properties (a) and

(b) by induction on n. If n = 0, then gr M is a finitely generated A-module, hence


M(S) = 0 for all sufficiently large 8 E IZ, so we can take h(t) == O. Now, let n > O. Let us consider the exact sequence of graded Rn-modules

o --+ Ker On --+ M ~ XnM --+ 0 (8.2.1)

where On(Z) = XnZ for all Z E M. Clearly, On is an additive mapping and On(az) = an(a)On(z) for all Z E M, a E A. By Lemma 6.1.2, we have

for every 8 E IZ (if N is any graded grF-module, then N(s) (8 E IZ) denotes its 8th homogeneous component). Since the modules Ker On and xnM are annihilated by the elements Xn and X2n, respectively, we can naturally consider Ker On as a graded R~-module and xnM as a graded R~-module.

By the inductive hypothesis, for any finitely generated graded Rn_1-module N = EBpEz N(p), there exists a numerical polynomial 'PN(t) such that 'PN(P) = lA(N(p)) for all sufficiently large P E IZ, deg 'PN :s: n - 2 and the polynomial 'PN(t) can be

written as 'PN(t) = 2(:--":;)rtn-2 + o(tn- 2) where bN E IZ. If L = EBqEz L(q) is a finitely generated graded R~-module, then the first and the last terms of the exact sequence of A-modules

(02n is a multiplication by X2n) are homogeneous components of the finitely generated graded Rn_1-modules EBqEz(Ker 02n)(q) and EBqEz L(q+l) /X2nL(q), respectively. By the inductive hypothesis, there exists a numerical polynomial

such that 'h(q) = lA(L(q+l)) -LA(L(q)) for all sufficiently large q E IZ, say, for all q ~ qo (qo is some integer). Since

q

LA(L(q)) = lA(L(qo)) + L [LA(L(')) -LA(L(s-l))] s=qo+l

for every q > qo, Proposition 2.1.6 shows that there exists a numerical polynomial XL = (:~':)~ tn-1+o(tn- 1) such that XL( q) = LA (L(q)) for all sufficiently large q E IZ. Obviously, the similar assertion is valid for any finitely generated graded R~-module

as well. Thus, there exist numerical polynomials ht{t) = (:~'1)1 tn- 1 + o(tn- 1)

and h2(t) = (:~'l)~tn-l + o(tn- 1) (al,a2 E IZ) such that hl(8) = LA((KerOn)(')),

h2(8) = LA((xnM)(s)) for all sufficiently large 8 E IZ. Furthermore, the exactness of the sequence (8.2.1) implies the equality lA((Ker On)') + LA((xnM)(')) = LA(M(s)) for all 8 E IZ.


Now, it should be noted that lA((xiM)(')) = lA((xn+iM)(')) for all sEll, i = 1, ... , n. Indeed,

Thus,

IA((XiM)(')) = lA(aiM. + M./M.) = lA(aiM. + M.) -IA(M.)

= lA(ai1(aiM. + M.)) -IA(M.)

= lA(ai 1 M. + M./M.) = lA((xn+iM)(·)).

lA(M(')) = lA((KerO)(')) + lA((xnM)('))

= lA((Ker On)(s) + lA((x2nM)(s)) = hds) + h2(S)

for all sufficiently large sEll. The last term of the sequence of graded R~-modules

is a finitely generated graded Rn _ 1-module. By the inductive hypothesis there exists a numerical polynomial h3(t) such that h3(t) = IA((KerOn/x2nM)(')) for all sufficiently large sEll and degh3 ~ n - 2, whence a1 = a2. Now, for the polynomial

we have h(s) = IA(M(s)) for all sufficiently large sEll. This completes the proof. 0

8.2.2. DEFINITION. The numerical polynomial g(t) whose existence is established by Theorem 8.2.1, is called a dimension (or chamcteristic) polynomial of the G-A-module M associated with the excellent filtration (Mr )rEZ of M (and with decomposition (8.1.1) of the group G).

8.2.3. EXAMPLE. Let A be a G-field and let decomposition (8.1.1) of the group G be fixed. Let us consider the G-A-module M = :F with the excellent filtration (:Fr )rEZ and find the corresponding dimension polynomial g:F(t). It follows from the definition of the dimension polynomial that

9:F(r) = dimA:Fr = Card{ a~l ... a~".Bil ... /3:;; I ki,lj Ell, 0 ~ Ij < qj

n m

(1 ~ i ~ n, 1 ~ j ~ m), E Ik;\ + E Ij ~ r} i=1 j=1

for all sufficiently large r E Il. Let Q = Ej=1 (qj - 1) and, for any s E ~, let

<P1(S) denote the number of distinct elements .Bil .. . /3:r:- such that 0 ~ Ij < qj


(j = 1, ... , m), and L:7=llj = s. Furthermore, let c,02(S) (s EN) be the number of

distinct elements a~' ... o:~" such that kl' ... ,kn E ~ and L:7=1 Ikd ~ s. Then

Q

gF(S) = L: c,ol(i)c,o2(S - i) (8.2.2) i=O

for all sufficiently large s E ~, hence our problem will be solved if we express c,ol (s) and c,02(S) in terms of s.

It is easy to see that c,ol (s) = 0 for all s > Q and if 0 ~ s ~ Q, then c,ol (s) coincides with the number of solutions (Xl, ... , xm) of the equation Xl + ... + Xm = S such that Xj E ~, 0 ~ Xj ~ qj - 1 (j = 1, ... , m). Applying Proposition 2.1.8, we obtain that

(m + s - 1) ~ Ie '" (m + s - q. _ ... - q. - 1) c,ol(S) = m -1 + L...-(-1) L...- J:n -1 Jk

Ie=l 1 <iI <···<ik<m. 'iiI +···+qjk"3;-

Since c,02(r) (r EN) is the number of solutions (Xl, ... , xn) E ~n ofthe inequality IxIi + ... + IXnl ~ r, we have c,02(r) = L:7=0(-1)n- i2i (7)(rt i ) (see Proposition 2.1.9). Using formula (8.2.2), we can now get the expression of the polynomial gF(t):

gF(t) = t, [(m: ~ ~ 1) +f)-l)1e L: (m+i-qj~-_·~·-qjk-1)]

1e=1 l<iI< ... <h<m ii, +···+qjk ~i

(8.2.3)

The following result can be proved just as the similar statements for difference and inversive difference modules (see Theorems 6.2.12 and 6.3.10).

8.2.4. THEOREM. Let A be a Noetherian G-ring and let i : N -+ M be an injective homomorphism of filtered G-A-modules. If the filtration of M is excellent, then the filtration of N is also excellent.

In what follows we shall concentrate our attention on finitely generated Gmodules over a G-field.

Let K be a G-field and let the decomposition (8.1.1) of the group G be fixed:

Let :F be the filtered ring of G-operators over K with the filtration (:Fr )rEZ introduced in Section 8.1. In this situation every finitely generated G-K-module M can


be provided with an excellent filtration: if M = 2:~=1 FZj, then (2:~=1 FrZj)rE7I.. is such a filtration (see Exercise 8.1.7). Furthermore, any finitely generated G-K-module M = 2:~=1 FZi can be also considered as an inversive difference module over the inversive difference field K with the basic set u = {a1, ... ,an } (i.e., as a u*-K-module when K is treated as a u*-field). Indeed, if H is the periodical part of the group G (i.e., H = {g E G I gk = 1 for some kEN}), then M is generated over the appropriate ring of u* -operators £ by the finite family of elements {hZi I h E H, 1:S i :S l}. (Note that £ is actually the ring of r -operators over K where r is the free commutative group generated by the elements a1, ... , an). Let

(Mr = 2:~=1 FrZi)rE7I.. be the excellent filtration of the G-K-module M = 2:~=1 FZj considered above, and let (M:)rE7I.. be the excellent filtration of M treated as a u* -K-

I 2 n

module such that M~ = 2:j=l 2:hEH Frhzj for every r E IZ. Let get) = n!atn+o(tn) be the dimension polynomial of the G-K-module M associated with the filtration (Mr )rE7I.., and wet) the characteristic polynomial of the u* -K -module M associated with the filtration (M:)rE7I.. (see Definition 6.3.4). Since ord h :S Q = 2:7=1 (qj - 1) for any h E H, we have the inclusions Mr ~ M: ~ Mr+Q for all r E IZ, hence g(r) :S w(r) :S g(r + Q) for all sufficiently large r E IZ. It follows that a = io,,(M), the degree d of the polynomial get) equals it,,(M), and [).d2il}t) = tio,,(M). Thus, the degree and the leading coefficient of a dimension polynomial of a finitely generated G-K-module M associated with any excellent filtration of M are independent of the choice of such filtration. Let us show that these numbers are independent of the choice of the decomposition (8.1.1) of the group G, as well. Let

(8.2.4)

be another decomposition of the group G into the direct product of infinite cyclic and of finite primary cyclic subgroups. Suppose that the generators of the cyclic subgroups of decompositions (8.1.1) and (8.2.4) are related by the equalities

a· = ",kil ",kin 01 it olim (1 <_ i <_ n), • 11 ···In 1 ···m ,j = a~jl ... a~jn !3~jl ... !3':J m (1:S j :S n)

where ki/J,ijv,uj)..,Vjp E IZ, 0 :S iiv :S qjv, 0 :S Vjp < qp (1 :S Il,).. :S n, 1 :S l.I,p :S m). Then, putting q = q1 ... qm, we obtain that ai = bi)kil ... b~)kin and ,J = (ankjl ... (a~)Ujn (1:S i,j:S n). Now, if we set U1 = {ai, ... ,a~} and U2 = {,f, ... , ,~}, then the ring of ui -operators over the field K coincides with the ring of u2-operators over K (in the former case K is considered as a ui -field and in the latter one K is treated as a u::;-field). We shall denote this ring by t and consider it as a filtered ring with the filtration (t)rE7I.. such that tr = {w E t I ord"l w :S r} for all r EN and t = 0 for all r < O.

As above, let M be a finitely generated G-K-module with generators Zl, ... , Zt

(that is M = 2::=1 Fz;) and let H be the periodical part of the group G. Then elements ha~l ... a~n Zj (h E H), 0 :S kj < q for i = 1, ... , n; (j = 1, ... ) I) generate

- ",I the module M over the ring £. Let Mr = ~j=l Frzj and

1

Mr=LL


for every r E IZ ((Fr )rEZ is our usual filtration of the ring of G-operators :F over K corresponding to the decomposition (B.1.1) of the group G). It is easily seen that (Mr)rEZ and (Mr)rEZ are excellent filtrations of M as of G-K-module and u·-Kmodule, respectively. Let g(t) = 2",atn+o(tn) and qi'(t) = 2",b tn+o(tn) (a, bE IZ) be n. n. the appropriate dimension polynomials whose existence is established by Theorems B.2.1 and 6.3.3, respectively. By Proposition 6.3.15, we have b = iOU1 (M) = ioU2 (M) (this number coincides with the maximal number of elements of M that are linearly independent over the ring e). Since for every element () = a1U1 +111 •• • a~u"+II .. ( Ui ,Vi E IZ, 0 ~ Vi < q for all i = 1, ... , n) we have

the inequality ordu () ~ qr implies the inequality L~l IUil ~ r + n. It follows that Mqr ~ Mr+n for all sufficiently large r E IZ. Since we also have Mr ~ Mqr+nq+Q, the number a = qb .. = ioU1 (M)/qn = ioU2 (M)/qn and the numbers

d = deg g(t), t:. dg (t)/2d are independent of the choice of an excellent filtration (Mr )rEZ connected with decomposition (B.1.1) of the group G as well as of the choice of such decomposition itself. Thus, we obtain the following result.

8.2.5. THEOREM. Let K be a G-field and M a finitely generated G-K-module. Let the rank of the group G equal n and let g(t) be the dimension polynomial of the module M associated with some excellent filtration (Mr )rEZ of M (that corresponds to some decomposition of the group G into a direct product of infinite cyclic and finite primary cyclic subgroups). Then the numbers a = ~ ~~(t), d = deg g(t)

and ad = ~:9Jt) are independent of the choice of excellent filtration (Mr )rEZ as well as of the decomposition of the group G. If u = {al,"" an} is a set of free generators of some free commutative subgroup H of G, then a = iou(M), d = iou(M), d = itu(M), and ad = tiou(M).

8.2.6. DEFINITION. Under the conditions of Theorem B.2.5, the characteristics a, d and ad of a finitely generated G-K-module M are called the G-dimension, Gtype, and typical G-dimension of M, respectively. They are denoted, respectively, by oG(M), tG(M), and toG(M).

The following proposition shows that G-dimension is an additive function in the class of all finitely generated G-K-modules. It can be proved just as the similar statement for difference vector spaces (see Proposition 6.2.1), using Theorem B.2.4 instead of the appropriate assertion for difference modules.

B.2.7. PROPOSITION. Let 0 ~ M ~ N -4 P ~ 0 be an exact sequence of finitely generated G-modules over a G-field K (I and J are G-homomorphisms). Then oG(M) + oG(P) = oG(N).

Let M be a finitely generated G-module over a G-field K. Suppose that decomposition (B.1.1) of the group G is fixed, and let H be the periodical part of G. Furthermore,let q = CardH (so that q = ql .. . qm), let u be the set {al," .,an} of automorphism of the field K, and UI = {a1, ... , a~}. Then K can be considered as a u·-field as well as a ur-field. The proof of Theorem B.2.5 shows that

8.3. SOME GENERALIZATIONS FOR DIFFERENTIAL G-STRUCTURES 387

dG(M) = idu1(M)/qn. Let Go = {odoo x ... x {On}oo be the free commutative subgroup of G with the free generators 01, ... , On. Then, as in the proof of The-orem 8.2.5, we obtain that dGo(M) = idu(M) = idu1 (M)/qn = dG(M). Now, applying Proposition 6.3.15, we get the following result.

8.2.8. PROPOSITION. Let K be a G-field and M a finitely generated G-Kmodule. Let rank G = n, and let 11' = {01, ... , On} be the set of free generators of a free commutative subgroup Go ~ G. Then dG(M) = dGo(M) = idu(M) and this number is equal to the maximal number of elements of M which are linearly independent over the ring of 11'* -operators £u over the field K.

8.2.9. EXERCISE. Let K be a G-field, M a finitely generated G-K-module and U a family of all G-K-modules of M. Show that for the numbers typeu M and dimu M (see Definitions 7.1.3 and 7.1.4) the following statements are valid:

(1) ifdG(M) > 0, then typeuM =rankG and dimuM = dG(M); (2) if dG(M) = 0, then typeU M ~ tG(M).

[Hint: apply the scheme of the proof of Theorem 7.1.10].

THM8.2.10. Let L be a G-field of zero characteristic and let R = (1]1, ... ,1].)G be a G-field extension of L generated by a family 1] = (1]1, ... ,1].). Then there exists a numerical polynomial X(t) with the following properties:

(i) X(t) = trdegL L(G(r)1J1 U ... U G(r)1].) for all sufficiently large r E ~ (we suppose that decomposition (8.3.1) of the group G is fixed);

(ii) deg X ~ n, where n = rank G, and the polynomila X(t) can be written as X(t) = 2~~tn + o(t), where a E ~;

(iii) let H be any free commutative subgroup ofG such that rankH = rankG. Then a is equal to the maximal number of elements of R that are Halgebraically independent over L.

(Such number a is called a G-transcendence degree of Rover L and is denoted by G-trdegL R.)

8.3. Some Generalizations for Differential G-structures

Let R be a differential field with a basic set of derivation operators ~ = {d1 , ... , 15m }. Suppose that elements of a finitely generated commutative group G act on the filed R as differential automorphisms (that is automorphisms which commute with the derivatives 151 , ... ,15m ). In this case the field R is said to be ~-G-field.

Let e denote a free commutative semigroup generated by the elements 151 , ... , dm

and let 0 be a commutative semigroup of the elements 89 (8 E e, 9 E G and (819t}(8292) = (8182)(9192) for all 8191, 8292 EO). Furthermore, we assume that a primary decomposition of the group G is fixed.

(8.3.1)

where {o;}oo is an infinite cyclic subgroup with the generator 0i (i = 1, ... , n) and {,8j }qj (1 ~ j ~ k) is an infinite cyclic subgroup whose order qj is a degree of a prime integer.


Each element 9 E G has a unique representation of the form

where Ul, ... , Un E ~; vi E ~, 0 ~ Vi < qj (j = 1, ... , k), so that each element E 1"\ h' t t' f th r - rl, rl~ u, Un (.IV, (.IV. W H as a umque represen a lOn 0 e 10rm W - Ul .. 'Um a l .. . a n 1-'1 .. 'I-'k

(11, ... ,/m EN; Ul,·· .,Un,Vl,·· .,Vk E IZj 0 ~ Vi < qi where j = 1, .. .. k). In this

case the number ordw = E~l/i + E~=l Iuvl + E~=1 Vj is called the order of w.

The numbers ordA W = E~lli and orda w = E~=1 Iuvl + E~=1 Vj are called the /l-order and the G-order of w, respectively.

8.3.1. DEFINITION. An expression Eweo aww where aw E R (w E 0) and aw = 0 for almost all W E 0 is called a /l-G-operator over R. Two /l-G-operators :Eweoaww and :Eweobww are called equal if and only if aw = bw for all w EO.

The set of all /l-G-operators over a /l-G-field R has the natural structure of a left R-module. This set becomes a ring, if we define the product WlW2 as in the semigroup OJ set da = ad + d(a), ga = g(a)g for all d E /l, 9 E G, a E R, and demand the validity of the distributive laws. This ring is said to be a ring of /lG-operators over R and is denoted by FR or simply by F. If A = Eweo aww E F, then the numbers ord A = max{ ord w I aw :f; O}, ordA A = max{ ordA w I aw :f; O}, and orda A = max{ orda w I aw :f; O} are called the order, the /l-order, and the G-order of /l-G-operator A, respectively.

8.3.2. DEFINITION. A left FR-module is said to be a /l-G-R-module. Thus, a vector space M over a /l-G-field R is said to be a /l-G-R-module if the elements of /l U G act on M so that the following conditions are satisfied:

(1) each element I E /l U G is an endomorphism of the additive group of M and 11/2 = 12/1 for all /1 ,12 E /l U Gj

(2) g(ax) = g(a)g(x)j (3) d(ax) = ad(x) + d(a) . Xj

(4) Ix = x

for all x E M, a E R, 9 E G, dE /l (by 1 we mean here the unit of the semigroup 0).

Let DR be a subring of the ring F = FR which consists of the elements of the form EeeEl aeO (ae E R and almost all ae are equal to zero), and let HR denote the subring {Egea agg E FR lag = 0 for almost all 9 E G} of FR. As above, elements of DR are called differential (or /l-) operators over the field R and elements of H R are called G-operators over R. A left DR-module will be called a /l-R-module and a left HR-module will be called a Gi-R-module.

8.3.3. DEFINITION. An ascending chain (F;-)rez (respectively, (Ff)rez) of HR- (respectively, DR-) submodules of the ring F = FR such that F;- = 0 for r < 0 and F;- = {A E F I ordA A ~ r} for all r E N (respectively, :Ff = 0 for r < 0 and Ff = {A E F I orda A ~ r} for r E N) is called a standard /l(respectively, a standard G-) filtration of the ring :F.

By a /l-filtration of a /l-G-R-module M we shall mean an ascending chain (Mr )rez of HR-submodules of M which is an exaustive and separate filtration


of M as a module over the ring F provided with the standard .6.-filtration. The notion of G-filtration of a .6.-G-R-module M is introduced similarly.

A .6.-filtration (Mr )rez of a .6.-G-R-module M is called excellent if all modules Mr are finitely generated over the ring HR and there exists a number TO E &:: such that F~ Mr = Mr+. for all r ~ TO, sEN-

An excellent G-filtration of M is defined similarly (recall that we still consider the group G together with its decomposition (8.3.1.)).

8.3.5. EXAMPLE. Let M be a finitely generated .6.-G-R-module with generators xl, ... ,xp, i.e., M = Ef=1Fxj. Then the sequences (Ef=lF~Xi)jez and (Ef=l F;; Xi) ieZ are excellent .6.- and G-filtrations of the module M, respectively.

8.3.6. THEOREM. Let R be a .6.-G-field and let (Mr)rez be an excellent Gfiltration of a .6.-G-R-module M. Then there exists a numerical polynomial A(t) E Q[t] with the following properties:

(1) A(r) is equal to the differential dimension d(Mr) of the .6.-R-module Mr for all sufficiently large r E &::;

(2) deg A(t) :::; n (n is the rank of the group G, see decomposition (8.3.1)) and

the polynomial A(t) can be written as A(t) = 2n~tn + o(tn), where a E &::. n.

PROOF. Let grG F be the graded ring associated with the ring of .6.-G-operators F = F R provided with the standard G-filtration:

00 00

grG F = E9gr~ F = E9 (F~ /F~1) (F-1 = 0). q=O q=O

Let Xl,"" X2n, Y1, ... , Yk be the canonical images of the elements a1,.·., an, all, ... , a;; 1 , (31, ... , (3k, respectively, in the ring grG F. Obviously, Xl, ... , X2n, Yl, ... ,Yk generate the ring grG F over the ring of .6.-operators D = DR; these elements commute with each other, XiXn+j = 0 (i = 1, ... , n) and yJi = 0 (j = 1, ... , k). Furthermore, xjA = a(A)xj, xn+jA = ai1(A))xn+j (i = 1, ... , n), and yjA =_,Bj(A)¥J (j = 1, ... ,k) where A is an element of the ring D and a1, ... , an, (31, ... ,(3k are the automorphisms of D such that

ai (L aBO) = L aj(aB)O, BeS BeS

,Bj (L aBO) = L (3j(aB)O BeS 8eS

(1::::; i::::; n, l::::;i:::;k)

for each element EBes aBO E D. Obviously, the homogeneous components gr~ F (q EN) of grG F are the left D-modules generated by all elements


such that Ui, vi EN, 0 ~ vi < q (1 ~ i ~ n, 1 ~ j ~ k), E~=l Ui + E~=l vi = q, 1 ~ il, ... ,in ~ 2n and iIJ f: iv for every p. = 1, ... ,n; /I = 1, ... ,n. Below we denote the ring grG:F by Sn, while S~, S~ denote the graded subrings of Sn whose elements can be written in the form that does not include Xn and X2n, respectively.

Since G-filtration (Mr )rEZ of the ~-G-R-module M is excellent, the graded module grG M = Ea~o M(q) (where M(q) = grq M = Mq/Mq_ l for any q E ~)

is finitely generated over the ring grG:F (this fact can be established by the same reasoning which allows to prove a similar statement for differential modules with excellent filtrations (see Theorem 5.1.11».

Since d(Mr) = Eq<r d(M(q) for each r E ~, we only have to establish the existence of a polynomIal h(t) E Q[t] with the following properties:

(1) there exists qo E ~ such that h(q) = d(M(q) for all q E ~, q 2: qo; (2) deg h ~ n - 1 and the polynomial h(t) can be written in the form

where a E~. (Indeed, if such a polynomial h(t) exists, then we can apply Proposition 2.1.6 and obtain that there exists a polynomiall(t) E Q[t] such that

r

l(r) = E d(M(q))) + E h(q)

for all r E ~, r > qo, deg l = deg h + 1 ~ n and the leading coefficient of l (t) is equal to 1. 2"a = 2"a.)

n (n-l)! n!

We prove the existence of the polynomial h(t) by induction on n. If n = 0, then grG M is a finitely generated left D-module, so that M(q) = 0 for all sufficiently large q E ~ and one can put h(t) == O.

Now, suppose that n > O. Let us consider the exact sequence of the graded Sn-modules

o --+ Ker Pn --+ M ~ xnM --+ 0 (8.3.2)

where Pn(Z) = XnZ for all Z E M. Clearly, Pn is additive and Pn(Az) = an (A)Pn(z) for all Z EM, A ED. If (Mi(q)iEZ is an excellent filtration of a finitely generated differential (D-) module M(q), then (p(M;(q) )iEZ is, obviously, an excellent filtration

of the D-module xnM(q) and (Ker Pn n Mi(q)iEZ is an excellent filtration of the D

module Kerpn nM(q) (see Proposition 5.1.15). Let x~q)(t), x~q)(t), x~q)(t) be the

characteristic polynomials associated with the filtrations (M}q);EZ, Pn(Mi(q))iEZ, and (Ker Pn n Mi(q)iEZ, respectively (the existence of such polynomials is stated by Theorem 5.1.11). Since for each i E ~ the sequence

0-+ Ker Pn n Mi(q) -+ Mi(q) P .. IM~·\ Pn(Mi(q) -+ 0

is exact, we have x~q) (t) = x~q) (t) + x~q) (t), hence

d(M(q) = d(Pn(M(q)) + d(Ker Pn n M(q)


for all q E IZ. Furthermore, modules Ker Pn and xnM are annihilated by the elements Xn and X2n, respectively, so that Ker Pn is a graded S~-module and xnM is a graded S~-module. By the inductive hypothesis, for any finitely generated Sn_1-module N = ffiqEz N(q) there exists a polynomial 'l'N(t) E iQ[t) such that

'l'N(q) = d(N(q») for all sufficiently large q E IZ, deg'l'N ::; n - 2, and 'l'N(t) can be

written in the form 'l'M(t) = (:-':;)rtn-2 + o(tn- 2), where bN E IZ.

Let L = ffiqEz L(q) be a finitely generated graded S~-module and Lq) = 0 for all sufficiently small q E IZ. Then in the exact sequence

0-+ Ker Pn n L(q) -+ L(q) p2nl dq » L(q+1) -+ L(q+1) /X2nL(q) -+ 0

(P2n : L -t L is defined by the condition P2n (z) = X2nZ for all z E L). The first and the last D-modules are homogeneous components of the graded Sn_1-modules ffiqEz(Ker P2n n L(q») and ffiqEz (L(q+1) /x2nL(q»), respectively. By the inductive

hypothesis, there exists a polynomial g£(t) = (:~';)1 tn - 2 + o(tn - 2 ) E iQ[t) (h E IZ)

such that gL(q) = d(L(q+1») -d(L(q» for all sufficiently large q E IZ. Since d(L(q» = l:i<q [d(L(i+l) - d(L(i»), there exists a polynomial kL(t) = (:~':{! tn- 1 + o(tn- 1) E

iQ[t) such that kL(t) = d(L(q» for all sufficiently large q E IZ. It is clear that a similar statement holds for a finitely graded S:: -module as well.

Applying the last considerations to the extreme terms of the exact sequence (8.3.2), we conclude that there exist polynomials

where al, a2 E IZ, such that '1'1 (q) = d(Ker Pn n M(q», 'l'2(q) = d(xnM(q» for all sufficiently large q E IZ. Since

we have

d(XiM(q» = d(Mq + aiMq) - d(Mq)

= d(Mq + ail Mq) - d(Mq)

= d(Xn+iM(q» (i = 1, ... , n),

for all sufficiently large q E IZ. By the inductive hypothesis, there exists a polynomial 'l'3(t) E iQ[t) such that 'l'3(q) = d(Ker Pn n M(q) /X2nM(q-l» for all sufficiently large q E IZ and deg '1'3 ::; n - 2. Now, since 'I'd q) = '1'2 (q) + '1'3 (q) for all sufficiently large q E IZ, we conclude that al = a2. Thus, the polynomial

2n a h(t) = 'l'1(t) + 'l'2(t) = 2'1'1(t) + o(tn- l ) = (n _ ~)r-1 + o(tn- 1)

satisfies the statement of our theorem. 0

The analogue of Theorem 8.3.6 for excellent D.-filtrations is presented by the following assertion.


8.3.7. THEOREM. Let R be a D.-G-field and (Mr )rE'lI. an excellent D.-filtration of a D.-G-R-module M. Then there exists a polynomial J-I(t) E Q[tJ with the following properties:

(1) J-I(r) = eSG(Mr) for all sufficiently large r E::Ej (2) deg J-I ~ m and the polynomial J-I(t) can be written in the form

where a E ::E.

PROOF. Let us consider a graded ring

00

grA F = EBgr~ F = EB (F~l) ~ H[X1 , .. . ,Xm ],

qE'lI. q=O

where Xi = eS; + H E grt F (i = 1, ... , m) and H[X1 , ... , XmJ is a polynomial ring in m commuting indeterminates X!, ... , Xm over the ring of G-operators H = HR. As in the proof of Theorem 6.2.5, we obtain that the graded module gr M = EBqE'lI. grq M is finitely generated over H[X1 , ... , XmJ. Since the function eSG(·) is additive in the class ofall finitely generated R-G-modules (see Proposition 8.2.7), we may repeat the standard proof of the Hilbert theorem on characteristic polynomial (the scheme of the proof was also used in the proof of Theorem 6.1.3) to prove the existence of a polynomial k(t) = (m~l)!tm-l + o(tm- 1 ) E Q[tJ (a E ::E) such that k(q) = eSG(grq M) for all sufficiently large q E ::E. Hence, there exists a polynomial J-I(t) = ';:!tm + o(tm ) E Q[tJ such that J-I(r) = Eq<r eSG{rqM) = eSG(Mr) for all suficiently large r E::E. 0 -

8.3.8. DEFINITION. The polynomials l(t) and J-I(t) whose existence is established by Theorems 8.3.6 and 8.3.7 are called, respectively, G- and D.-dimension polynomials of the D.-G-R-module M associated with the corresdonding excellent G- (D.-) filtration of M (and with the given decomposition (8.3.1) of the group G).

8.3.9. EXAMPLE. Let us determine G- and D.-dimension polynomials A:F(t) and J-I:F(t) associated with the standard G- and D.-filtrations of D.-G-R-module F = FR, respectively (the decomposition (8.3.1) of the group G is supposed to be fixed). Since

A:F(r) = d(F:) = Card{g E G I ordg ~ r}

and

J-I:F(r) = eSG(F;-) = Card{O Eel ordO ~ r}

for all sufficiently large r EN, Propositions 2.1. 8 and 2.1. 9 show that

( t+m) J-I:F(t) = m ' (8.3.3)


AT(t) = t, { [ (k ~ ~ ~ 1) + i)-I)" L e + i - V'k-_'~' - qi" -1)]

11=1 1<j'<"'<j,,<k qj,+ ... +q;,,~i

(8.3.4)

where Q = L::=l (qll - 1).

The proofs of the following three lemmas and Proposition 8.3.14 are an easy adaptation of the proofs of similar assertions for differential and inversive difference modules.

8.3.10. LEMMA. Let i : N -+ M be an injective homomorphism of ~- or Gfiltered ~-G-R-modules. If the filtration of M is excellent, then the filtration of N is also excellent.

8.3.11. LEMMA. Let (Mr)rEZ and (M:)rEZ be two excellent ~-(G-) filtrations of a finitely generated ~-G-module M associated with some (may be, different) decompositions of the group G of the form of (8.3.1). Let J.L(t) = ~! tm + o(tm) and J.Ll(t) = ~tm+o(tm) (respectively, A(t) = 2:!bt"+o(t") and Adt) = 2:~lt"+O(t")) be two ~- (G-) dimension polynomials associated with these filtrations. Then a = al and the degree and the leading coefficient of J.L(t) are equal to those of J.Ll (t) (respectively, b = bl and the degree and the leading coefficient of A(t) are equal to those of Al (t)).

8.3.12. REMARK. Let R be a ~-G-field and M a finitely generated ~-G-Rmodule. Let J.L(t) and A(t) be, respectively, ~- and G- dimension polynomials associated with some excellent ~- and G-filtrations of M, for instance, with the filtrations described in Example 8.3.5 (each of these filtrations is associated with its own decomposition of G of the form (8.3.1)). Then it follows from Lemma 8.3.11 that the numbers a = ~m J.L(t) , b = ~"A(t)/2", p = degJ.L(t), q = deg A(t), ~p J.L(t) , and ~qA(t) are independent of the choice of such filtrations and, therefore, these numbers characterize the module M itself. (Here ~i I(t) denotes the i-th finite difference of the polynomial I(t): ~/(t) = I(t + 1) - l(t), ~2 I(t) = ~(~/(t)), ... ; it is easy to see that the leading coefficient of the polynomial I(t) E Q[t] of degree d is equal to ~dl(t)/d!') The numbers a = ~mJ.L(t) and b = ~"A(t)/2" will be denoted by a(M) and b(M), respectively.

8.3.13. LEMMA. Let 0 -+ N ~ M -4 F -+ 0 be an exact sequence of finitely generated ~-G-R-modules. Then a(N) + a(F) = a(M), beN) + b(F) = b(M).

Let R be a ~-G-field and M a finitely generated ~-G-R-module. Suppose that rk G = nand H is a free commutative subgroup of G such that rk H = n. Then one may consider R as a ~-H-field and M as a finitely generated ~-H-R-module (indeed, it is easy to see that M is a finitely generated module over the ring :FJ[ of ~-H-operators over R).


8.3.14. PROPOSITION. With the above notation, each of the numbers a(M), b(M) is equal to the maximal number of elements of module M which are linearly independent over the ring:Ff{ (here H is an arbitrary free commutative subgroup of G such that rk H = rk G). This number will be called a Ll-G-dimension of M and denoted by Ll(M).

A subfield L of a Ll-G-field R is called a Ll-G-subfield of R if the restrictions on L of the derivation operators 8 E Ll and automorphisms 9 E G are, respectively, the derivation operators and automorphisms of L. In this case the field R is said to be a Ll-G-extension of the Ll-G-field L. Furthermore, if E ~ R, then the intersection of all Ll-G-subfields of R containing Land E is the unique Ll-G-subfiled of R containing Land E and contained in every Ll-G-subfiled of R containing Land E. It is denoted by L(E). We call E a set of generators ofthe Ll-G-field L(E) over L; we also say that L(E) is a Ll-G-extension of L generated by the set E. If Card E < 00,

E = {TJ1, ... , TJp}, then L(E) is said to be a finitely generated Ll-G-extension of L. Let L be a Ll-G-field of characteristic zero and R = L(T}l, . .. , TJp) a Ll-G-field

extension of L with the finite set of generators TJ = (TJ1, ... , TJp). For each r E N we put 8(r) = {B E 8 I ordB :s r}, G(r) = {g E G I ordg :s r} (the order on o is regarded relative to some fixed decomposition of G in the form (8.3.1)) and consider the ascending chains (R~)rEZ and (R~)rEIZ such that R~ = L(8(r)TJ1 U ... U 8(r)T}p)G, R~ = L(G(r)TJl U· .. U G(r)TJp)A (by L(E)A and L(E)G (E ~ R) we denote, respectively, the differential (Ll-) extension and the G-extension of L with the set of generators E).

8.3.15. DEFINITION. Let R be a Ll-G-extension of a Ll-G-field L of zero characteristic. Elements Zj E R (j E J) are said to be Ll-G-algebraically independent over L if the set of all pairwise distinct elements of the form WZj (w EO, j E J) is algebraically independent over L.

8.3.16. THEOREM. With the above notation, there exist numerical polynomials X~L(t), X~L(t) with the following properties:

(1) X~L(r) = G-trdegL R~ and X~L(r) = Ll-trdegL R~ for all sufficiently large rE ~;

(2) degX~L :s m, degX~L :s n, and the polynomials X~L(t), X~L(t) can be written asXA (t) = 2!...tm+o(tm) andxG (t) = a2ntn+o(tn) where a E~·

'IlL m! 'IlL n! ' , (3) let H be a free commutative subgroup of G such that rk H = rk G; then

the number a is equal to the maximal number of elements of R which are Ll-H -algebraically independent over the field L.

PROOF. Let us prove the existence of a polynomial X~L (t) which satisfies the conditions of the theorem. Let DerL R denote the vector R-space of all R-linear derivations of the field R into itself and let (DerL R)* = HomR(DerL R, R). furthermore, let OL(R) be the vector R-subspace of (DerL R)* generated by the set {dT} I TJ E R}, where dTJ is the element of (DerL R)* such that dTJ(D) = D(T}) for all D E DerL R. It follows from Proposition 3.4.46 and Remark 3.2.43 that DerL R, (DerL R)*, and OdR) may be considered as Ll-G-R-modules on which elements


oED. and 9 E G act as follows:

o(D) = 00 D - Doo,

o(<p) = 0 0 <p - <p 0 0,

g(D) =goDog- 1 ,

g(<p)(D) = g(<p(g-l(D))),

for all D E DerL R, <p E (DerL R)*. In this case od" = doe,,) and gd" = dg(,,) for all 0 E D., 9 E G, " E R.

Let Ol(R)r (r EN) be a R-G-submodule of OL(R) generated by the set {d" I " E R~}. As in Theorem 6.4.11, we obtain that (ot(R)r)rEN is an excellent D.-filtration of the D.-G-R-module OL(R) , oG(Ot(R)r) = G-trdegL R~ for all r E N and if H is a free commutative subgroup of G such that rk H = rk G, then D.G(OL(R)) is equal to the maximal niumber of elements of R which are D.-H -algebraically independent over L. By Theorem 8.3.7, there exists a polynomial X~L(t) E Q[t] such that

X~L (r) = oG(ot(Rr) = G-trdegL R~ for all sufficiently large r E N, deg X~L ~ m, and the polynomial X~L (t) can be written in the form X~L (t) = ~! tm + o(tm) where a = D.(Or(R)).

The proof of the existence of the polynomial X~L (t) which satisfies the conditions of the theorem can be presented in a similar way, if we consider an excellent G-filtration (Or (R)r )rEN, where Or (R)r (r E N) is the differential R-module generated by the set {d7J I 7J E R~}. As above, we can show that d trdegL R~ = d( Or (R)r ) for all r EN. Now the statement follow from Theorem 8.3.6. 0

8.3.17. DEFINITION. The polynomials X~L(t) and X~L(t), whose existence is established by Theorem 8.3.16, will be called, respectively, D.- and G-dimension polynomials of the D.-G-extension R = L("l, ... , 7Jp) associated with the set of generators 7J = ("1, ... , 7Jp) (and with the given decomposition (8.3.1) of the group G). The number a = D.mX~L(t)/m! = D.nX~L(t)/2nn! is called the D.-G-tmnscendence degree of the extension L ~ R and is denoted by D.-G-trdegL R.

Let R be a D.-G-ring, where D. and G are the sets of derivation operators {Ol' ... ,Om} and commutative group considered at the beginning of this section. Let J be arbitrary set and S = R[{Yiw I j E J,w EO}] a polynomial ring in the indeterminates Yjw, (j,w) E J x O. Then each derivation operator 0 ED. and each automorphism 9 E G can be naturally expanded, respectively, to the derivation operator and automorphism of S such that O(Yiw) = Yj6w and 9(Yiw) = Yi,gw for all indeterminates Yiw (j E J, w E 0) and for all 0 E D., 9 E G. Thus, one can consider the ring S as a D.-G-ring (i.e. as a ring with a set of commuting operators D. U G such that elements 0 E D. act as derivations and the elements 9 E G act as automorphisms of S). This D.-G-ring is called a D.-G-polynomial ring over R in the D.-G-indeterminates Yj (j E J); it is denoted by R{Yi I j E J} or by R{Yl' ... , YP h, if J = {I, ... ,p} (p EN). The elements of the ring S = R{Yi Ij E J} are called D.-G-polynomials over R (or "with coefficients in R") in the D.-G-indeterminates (Yi )ieJ' An ideal I of a D.-G-polynomial ring S = R{Yl! ... , Yp} is called a D.-Gideal, if 0(1) E I and g(l) E I for all f E I, 0 E D., 9 E G. Let E be any set


of elements of S. The intersection of all differential ideals containing ~, which is obviously the smallest ~-G-ideal containing ~, is called the ~-G-ideai 0/ S generated by ~ and is denoted by [~]. Set-theoretically, [~] coincides with the ideal of S, generated by the set {w(f) I w E fl, / E ~}. If a ~-G-ideal P = [Ui Ii E A}] is prime, then we say that the system of ~-G-polynomials /1 (I E A) is prime or that the system of algebraic ~-G-equations

(8.3.5)

IS pnme. Let us consider the following special case of the described situation. Let G be

a free commutative group with the set of free generators al, ... , an (so that k = 0 in decomposition (8.3.1)). Then system (8.3.5) is simply a system of algebraic difference-differential equations with the basic set of derivatives ~ = (81 , ... , 8m )

and the basic set of automorphisms u = (aI, ... , an). In this case, the dimension polynomials X~L (t) and X~L (t) can be interpreted as measures of "strength" of the system (8.3.5) relative either to the set of derivations ~ or to the set of automorph isms u (this is an analogue of "measure of strength" of a system of differential equations governing a physical field suggested by A. Einstein [Ei53]).

The problem of computation ofthe polynomials X~R(t), X~R(t) can be solved by the same methods which are used for the calculation of the differential dimension polynomial of a prime system of algebraic differential equations: either by constructing a free resolutions of the appropriate ~-G-R-module of differentials or by constructing a characteristic set of the ~-G-ideal P relative to the corresponding ranking of the set of indeterminates {flYj I 1 ~ j ~ p}.

CHAPTER IX

COMPUTATION OF DIMENSION POLYNOMIALS

9.1. Description of the Program Complex

The first partial implementation by the authors of the described algorithms was made in 1980 [MP80]. The programs were written in the algorithmic language REFAL, run on the computer BESM-6 and were destinated to compute characteristic sets of differential ideals in the ring of differential polynomials.

The second version of the programs [Pa89] was destinated for the computations in differential and difference modules. It was written also in the algorithmic language REFAL and was used on type SM-4 or PDP-ll computers. The existence of compilers from the language REFAL on other computers makes it possible to transfer these programs to such computers as the IBM PC.

The user of the second version had the following opportunities:

- to compute a G-basis of an ideal in a polynomial ring Q[Xl' ... , xn]; - to compute a G-basis of a submodule of a free P-module, where P is a poly-

nomial ring over Q or Q[i]; - to compute a G-basis of a submodule of a free left K[d1 , ... , dn]-module, where

K = Q(Xl, ... , xn), di = 8j8xi; - to compute a G-basis of a submodule of a free left K[Tl, ... , Tn]-module, where

K = Q(Xl, ... , xn ), TiXj = (Xi + l)Ti' TiXj = XjTi when i of. j; - to make a change of variables in generators of a differential or a difference

module; - to compute the Hilbert polynomial (differential or difference dimension poly

nomial).

We note some features of this implementation. The coefficient field is assumed to be the quotient field for one of the following integral domains: Z, Z[i], Z[Xl,"" x n ].

The elements are normed so that their coefficients have no nontrivial common divisor, and the sign of the leading coefficient is fixed. We reduce the elements only while their head term may be reduced (partial reducing); we compute a minimal Gbasis (in each moment the set of head terms of the set of generators is autoreduced).

Later, new versions of programs were written in algorithmic language C++ by A. Astrelin, T. Lonchakov, and I. Prikhodski. They run on IBM PC and give some new possibilities. Recently a new version of the program has been written in algorithmic alnguage MAPLE by V. Mitunin.

Let us note that modern computer algebra systems, for instance, MAPLE allow to make in interactive mode almost all the computations for the examples in the next section.

397


398 IX. COMPUTATION OF DIMENSION POLYNOMIALS

9.2. Computation of Dimension Polynomials for some Systems of Differential Equations

Most of the following examples are taken from the paper [MP80]. In all these examples we suppose that :F is a D.-field of zero characteristic, D. =

{db d2, d3 , dd is the set of derivation operators of:F, 9 is the D.-extension of:F given by the corresponding system of differential equations. We shall assume the ranking to be standard as described in Example 4.1.13. This ranking depends on ordering of differential indeterminates and derivation operators. Sometimes we shall consider the computational complexity as a function of their ordering. It has been noted that the problem of dimension polynomial computations can be divided into two steps. The first step is the most difficult. On this step we consider either a differential ideal in the ring of differential polynomials and compute its characteristic set or a differential submodule in a free differential module if we work in the module of Kahler differentials. On the second step we solve the combinatoric problem described in Section 2.2. If the initial differential equations are linear, then we use the second approach and compute G-bases in polynomial, differential or difference modules, otherwise, we compute a characteristic set of the corresponding D.-ideal in the ring of D.-polynomials.

9.2.1. EXAMPLE. WAVE EQUATION.

Let :F be a D.-field, and 9 = :F(/{), where

d~/{) + d~/{) + d~/{) - d~/{) = O.

The element f(y) = diy + d~y + d~y - d~y forms the characteristic set of the prime D.-ideal generated by f(y) in the ring :F{y}. The same element can be considered as a Grabner basis of the kernel of the epimorphism FP ~ O.T(Q) ~ 0, where FP is the free D.-module with the generator y, and O.T(9) is the module of Kahler differentials. So, we have the free resolvent of O.T(Q):

Since this sequence is an exact sequence of filtered D.-modules (see Definition 5.1.6), the D.-dimension polynomial for this extension is w(t) = 2('t3) - ('12).

This example can be used in order to show that the D.-dimension polynomial is not invariant with respect to D.-birational equivalence of D.-extensions. In particular, the D.-field extension 9/:F is D.-birational equivalent to the D.-extension 92 = :F(/{)O,/{)1,/{)2,/{)3,/{)4}, where

/()i = di/{)O, 1 ~ i ~ 4,

d1/{)1 + d2/{)2 + d3 /{)3 + d4 /{)4 = O.

In this case the filtration on 92 is obtained from the filtration on 9 by shifting it by one, i.e. the D.-dimension polynomial of the extension 92/:F is equal to W2(t) = Wl (t + 1) = 2et3) + Ct2) + t + 2.

Theorem 5.7.4 implies that the D.-dimension polynomial w(t) = 2et3 ) - et2) is minimal for this extension.

9.2. COMPUTATION OF DIMENSION POLYNOMIALS FOR SOME SYSTEMS 399

9.2.2. EXAMPLE. DIRAC'S EQUATION.

The Dirac equation has the form

where

(! 0 0

~i) (I 0

'/= 0 -2 "'12 = 0

0 o ' 1 0 0 0 0

G 0 0

n' ~=nD-"'14 = 1 0 0 -1 0 0 -1

In coordinate form we have the system

d41/Jl

0 ~1)

G 0

1 "'13 = 0

0 o ' 0 0 0 -i

i = v'-T.

-id31/J3-(id1 + d2)1/J4 = 0,

d41/J2-(id1 - d2)1/J3 +id3 1/J4 = 0,

id3 1/Jl +(id1 + d2)1/J2 -d41/J3 = 0,

(id1 - d2)1/Jl -id31/J2 -d41/J4 = o.

-2

D' 0 0 0

As in the previous example, we can consider the differential module over the field lQl[i](Xl, ... , X4), whose generators 1/Jl, 1/J2, 1/J3, 1/J4 satisfy the Dirac equation.

For computation in this case we may use the computation program of the Grabner bases of polynomial modules over polynomial rings with coefficients in IE [i]. Let F, G, U, V denote the differential indeterminates and X, Y, Z, T the derivation operators. Then the input has the following form

T*F-i*Z*U-i*X*V-Y*V

T*G-i*X *U + Y*U +i*Z* V

i*Z*F+i*X*G+Y*G-T*U i*X *F- Y* F- i* Z *G-T* V.

After computation, as a result, we obtain the Grabner basis

T*G-i*X*U + Y*U +i*Z*V

T*F-i*Z*U-i*X*V-Y*V

Z * F+X * G- i*Y * G+i*T* U

X*F+i*Y*F-Z*G+i*T*V

X2 * V + y2 * V + Z2 * V + T2 * V

X2 * U + y2 * U + Z2 * U + T2 * U

X2 * G + y2 * G + Z2 * G + i * X * T * U - Y * T * U - i * Z * T * V.

The ~-dimension polynomial for this system is w (t) = 4 e~3) .


9.2.3. EXAMPLE. CHANGE OF VARIABLES.

Let the generators 'Pl, 'P2, 'P3, 'P4 of a differential module over Q( Xl, ... , X4) satisfy the system of differential equations

d4'Pl -d3'P3-d2'P4 = 0,

d4'P2-dl'P3+d3'P4 = 0,

d3'Pl+d2'P2-d4'P3 = 0,

d l 'Pl-d3'P2 -d4'P4 = 0,

(9.2.1)

where dj = a/aXj. Since this example is obtained from the previous one by a linear change of the derivation operatots, the ~-dimension polynomial for this module is also w(t) = 4e~3).

Let e = 'Pl + X4'P2· Then e, 'P3, 'P4 generate the same differential module, because

'Pl = e - X4 d4e + X4 d3'P3 + X4 d2'P4 + X~dl 'P3 - X~d3'P4; 'P2 = d4e - d3'P3 - d2'P4 - X4 dl 'P3 + X4 d3'P4·

Substituting these values into system (9.2.1), we obtain the following system

d~e - X4dld4'P3 - d3d4'P3 + X4d3d4'P4 - d2d4'P4 - 2dl 'P3 + 2d3'P4 = 0;

d2d4e - X4d3d4e - X4 dl d2'P3 + X~dld3'P3 - d2d3'P3 + x4d~'P3 - d~'P4

+ 2X4d2d3'P4 - X~d~'P4 + d3e - d4'P3 = 0;

-X4dld4e - d3d4e + x~di'P3 + 2X4dld3'P3 + d~'P3 + X4 dl d2'P4

-X~dld3'P4 + d2d3'P4 - x4d~'P4 + dle - d4'P4 = o. The Grabner basis in these generators is

d~e - X4 dl d4'P3 - d3d4'P3 + X4 d3d4'P4 - d2d4'P4 - 2d l 'P3 + 2d3'P4;

d2d4e - X4 d3d4e - X4 dl d2'P3 + X~dld3'P3 - d2d3'P3 + x4d~'P3 - d~'P4

+ 2X4d2d3'P4 - X~d~'P4 + d3e - d4'P3;

-X4dl d4e - d3d4e + x~di'P3 + 2X4dld3'P3 + d~'P3 + X4 dl d2'P4

-X~dld3'P4 + d2d3'P4 - x4d~'P4 + dle - d4'P4;

-dl d2e - d~e + X4dld4'P3 + d3d4'P3 + d2d4'P4 - X4 d3d4'P4;

dl d2'P4 + d~'P4 - d~'P4;

dl d2'P3 + d~'P3 - d~'P3;

and the ~-dimension polynomial is w(t) = 4C~3) - 1. Thus, the dimension polynomiaI4C~3) is not minimal for the Dirac equations.

9.2.4. EXAMPLE. EULER EQUATIONS FOR IDEAL FLUID.

Consider the following system of nonlinear equations

dl Vl + d2V2 + d3V3 = 0;

(d4 + Vl d l + V2d2 + V3d3)Vl + d l V4 = 0;

(d4 + v1d l + V2d2 + V3d3)V2 + d2V4 = 0;

(d4 + v1d l + V2d2 + V3d3)V3 + d3V4 = o.


Since the system is nonlinear, we shall compute the coherent autoreduced set of the corresponding differential ideal. If we suppose that Vi > V2 > V3 > V4 then this set consists of five 6-polynomials:

dlVl + d2V2 + d3V3;

V2d2Vl + V3d3Vl + d l V4 - vld2V2 - vl d3V3;

vldl V2 + V2d2V2 + V3d3V2 + d4V2 + d2V4;

vldl V3 + V2 d2V3 + V3d3V3 + d4V3 + d3V4;

d~V4 + d~V4 + d~V4 - dlvld2V2 - dlvld3V3 + 2d2vldlV2 + 2d3vldlV3 + (d~V2)2 + 2d2V3d3V2 + (d3V3?'

which can be easily found with paper and pencil. The reader can verify that the corresponding differential ideal is prime and the set obtained is the characteristic set of this ideal.

If we change the order of variables, assuming V4 > V3 > V2 > Vl, then the calculations are essentially more tedious.

The 6-dimension polynomial in this case is equal to w(t) = 4('t3).

9.2.5. EXAMPLE. MAXWELL EQUATIONS FOR EMPTY SPACE.

Consider 6-equations

'Pij=-'Pji,1:::;i:::;j:::;4,

4

Ui = E dj'Pij = 0, 1:::; i :::; 4, j=l

Vikl = dl'Pik + di'Pkl + dk'P/i = 0, i i- k i- Ii-i.

(9.2.2)

(9.2.3)

(9.2.4)

Using (9.2.2), we can assume that we compute the Grobner basis in the free differential or polynomial module with the generators 'P12, 'P13, 'P14, 'P23, 'P24, 'P34· The Grobner basis consists of polynomials (9.2.3) and (9.2.4) and their linear combinations: d3Ul - d2 V123; d4Ul - d2 V124 - d3 V134 ; d3U2 + d l V123 - d2U3; dtiU2 - d2U4 + d l V124 - d3 V234 ; d4U3 - d3U4 + dl V134 + d2 V234 . We obtain the set of the leaders: {dl 'P12, d2'P12, d 3'P12, d 4'P12, d l 'P13, d 4'P13, d~'P13, d l 'P14, d~'P14' dr'P23, d4'P23, dr'P24, dr'P34}.

The free resolvent of f!:F(9) has the form

o --+ Fi --+ FJ --+ F~ --+ f!F(9) --+ o.

A-dimension polynomial is equal to

A-type of this extension is equal to 3, and its typical 6-dimension is equal to 4.


9.2.6. EXAMPLE. EQUATIONS FOR ELECTROMAGNETIC FIELD GIVEN BY THE

POTENTIAL.

The electromagnetic field may be defined by the ~-equations for its potential: 9 = :F{1/J), where 1/Jj, i = 1,2,3,4 satisfy the following system of ~-equations

4

Uj = I)dJ1/Jj - djdj1/Jj) = 0, j=1

4

B = 'Ldj1/Jj = O. j=1

This system is equivalent to the system

4

A j ='L d]1/Jj=O, j=1

4

B = 'Ldj1/Jj = O. j=1

Again, we can make the computation in the differential or in the polynomial module. The Grobner basis consists of the polynomials U1 , A2 , A3 , A4 , B. We have the set of the leaders: {d~1/Jl' df1/J2, df1/J3, df1/J4, d11/JtJ·

The free resolvent of [2.1'(9) has the form

0-+ Ff -+ F; EB Ft -+ F~ -+ [2.1'(9) -+ O.

~-dimension polynomial is equal to

w(t) = 6 C ~ 3) - C ~ 2) - (t + 1).

The ~-type of this extension is equal to 3, and its typical ~-dimension is equal to 6.

9.2.7. EXAMPLE. COMPARISON OF TWO METHODS OF DEFINING OF ELECTRO

MAGNETIC FIELDS.

For the extensions 92 = :F{cp) and 93 = :F(1/J) considered in Examples 9.2.5 and 9.2.6, we have the monomorphism a of filtered fields, deg a = 1, such that a (CPij ) = dj 1/Jj -di1/Jj. If we identify 92 with its image under a, then we have a tower of extensions :F C 92 C 93, and corresponding sequence of modules. Identifying CPij with their images under a we can find the Grobner basis. It consists of the following elements:

d11/Jl + d21/J2 + d31/J3 + d41/J4;

dj1/Ji-di1/Jj-CPij, l~i<j~4;

(dr + d~)1/J2 + d2d31/J3 + d2d41/J4 - d1CP12;

(dr + d~ + d~)1/J3 + d3d41/J4 - d1CP13 - d2CP23;

(dr + d~ + d~ + d~)1/J4 - d1CP14 - d2CP24 - d3CP34.


The free resolvent of Og2(Q3) has the form

o -t Ff -t Fl -t Fj -t F2 -t Og2 (Q3) -t O.

~-dimension polynomial is equal to

[AL94)

[AS96)

[Akr88)

References

Adams William W. and Loustaunau Philippe, An Introduction to Grabner Bases, Amer. Math. Soc., 1994. Adolphson Alan, Sperber Steven, Differential modules defined by systems of equations, Rend. Sem. Mat. Univ. Padova 95 (1996),37-57. Akritas A.G., Elements of Computer Algebra with Applications, Wiley, New York, 1988.

[ACLT95) Alberti M.A., Carra-Ferro G., Lammoglia B., Torelli M., The dimension method in

[A196)

[Ao91)

[Ap95)

[Ar92)

[AM69)

[AT96)

[Bab62)

[Ba84)

(BaS5)

[Ba87)

(BW93)

[BR94)

[Be71)

[Ber96)

[Bia62)

[Bj79) [BLOP95)

[Bou59) [Bou61) [Bou68)

[Bou70a) [Bou70b) [Bou54)

elementary and differential geometry, Ann. Math. Artificial Intelligence 13 (1995), no. 1-2,47-71. Alwash M. A-M., Periodic solutions of a quartic differential equation and Grabner basis, J. Comput. Appl. Math. 75 (1996), no. 1,67-76. Aoki T., Characteristic sets of differential operators of infinite order, In: "Geometrical and algebraic aspects in several complex variables (Cetrano, 1989), 1-11. Sem. Conf. 8" (El Rende, ed.), 1991. Apel Joachim, A Grabner approach to involutive bases, J. Symbolic Comput. 19 (1995), no. 5, 441-457. Aripov R.G., Generators of a differential field of invariant differential rational functions of paths for actions of real orthogonal groups in en, Uzbek. Mat. Zh. (1992), no. 1, 3-7. (Russian) Atiyah M. F., Macdonald. I.G., Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass., 1969. Attardi Giuseppe, Traverso Carlo, Strategy-accurate Buchberger algorithms. Parallel symbolic computation, J. Symbolic Comput. 21 (1996), no. 4-6,411-425. Babbitt A.E., Finitely generated pathological extensions of difference fields, Trans. Amer. Math. Soc. 102 (1962), no. 1,63-81. Balaba LN., The dimensional polynomials of the extension of difference fields, Vestnik Mosk. Univ., Ser I (1984), no. 2, 31-35. (Russian) ___ , Computation of the dimensional polynomial of a simple principal difference ideal, Vestnik Mosk. Univ., Ser I (1985), no. 2, 16-20. (Russian) ___ , Finitely generated extensions of difference fields, Dep. VINITI no 6632-B87, 1987. (Russian) Becker T., Weispfenning V., Grabner Bases. A Computational Approach to Commutative Algebra, Springer-Verlag, New York, 1993. Bekbaev U.D., Rakhimov I.S., Invariants and orbits of linear partial differential equations under linear transformations of indeterminates, Uzbek Mat. Zh. (1994), no. 3, 54-58. Bentsen Irving, The existence of solutions of abstract partial difference polynomials, Trans. Amer. Math. Soc. 158 (1971), no. 2, 373-397. Berger Robert W., Prefinite differential modules, Ann. Univ. Sarav., Ser. Math. 7 (1996), no. 2, i-iv and 37-49. Bialynicki-Birula A., On Galois theory of fields with operators, Amer. J. Math. 84 (1962),89-109. Bjork J.E., Rings of Differential Operators, North Holland, 1979. Boulier F., Lazard D., Ollivier F., Petitot M., Representation for the radical of a finitely generated differential ideal, Proceedings of ISSAC 1995, ACM Press, 1995, pp. 158-166. Bourbaki N., Elements de Mathematique. Algebre. Chap. 9, Hermann, Paris, 1959. ___ , Elements de Mathematique. Algebre commutative., Hermann, Paris, 1961-65. ___ , Elements de Mathematique. Groups et algebres de Lie. Chap. 4-6, Hermann, Paris, 1968. ___ , Elements de Mathematique. Algebre. Chap. 1-3, Hermann, Paris, 1970. ___ , Elements de Mathematique. Algebre. Chap. 4-6, Hermann, Paris, 1970. ___ , Elements de Mathematique. Theory des Ensembles, Hermann, Paris, 1954-1956.

405

406

[Bou80}

[Bu65}

[Bu70]

[Bu76}

[Bu79}

[BL82}

[Bui92}

[Bui93}

[Bui94a}

[Bui94b} [Bui95}

[CF87}

[CF89a}

[CF89b}

[CF90} [CF94}

[CF97}

[CFS94}

[CE56} [Cas882}

[Cass72}

[Cass80}

[Cass89}

[CNP95}

[Ch087}

[Chen96}

REFERENCES

___ , Eliments de Mathematique. Algebre Homologique. Chap. 10, Masson, Paris, 1980. Buchberger B., Ein Algorithmus zum zufJinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph. Thesis, Univ. Innsbruck, 1965. ___ , Ein algorithmisches /( riterium jiir die Losbarkeit eines algebraischen Gleichungssystems, Aequationes Math. 4 (1970), no. 3, 374-383. ___ , A theoretical basis for the reduction of polynomial to canonical form, ACM SIGSAM Bull. 10 (1976), no. 3, 19-29. ___ , A criterion for detecting unnecessary reduction in the construction of Grobner bases, Lect. Notes Comput. Sci. 42 (1979),3-21. Buchberger B. and Loos R., Algebraic simplification, Computer Algebra. Symbolic and Algebraic Computation (B. Buchberger, G.E. Collins and R. Loos, eds.), SpringerVerlag, Vienna, 1982, pp. 11-34. Buium Alexandru, Differential Algebraic Groups of Finite Dimension, Lect. Notes Math., 1506, Springer, Berlin, 1992. ___ , Geometry of differential polynomial functions. I. Algebraic groups, Amer. J. Math. 115 (1993), no. 6, 1385-1444. ___ , Geometry of differential polynomial functions. II. Algebraic curves, Amer. J. Math. 116 (1994), no. 4, 785-818. ___ , Differential Algebra and Diophantine Geometry, Hermann, Paris, 1994. ___ , Geometry of differential polynomial functions. III. Moduli spaces, Amer. J. Math. 117 (1995), no. 1, 1-73. Carra-Ferro Guiseppa, Some properties of the lattice points and their application to differential algebra, Comm. Algebra 15 (1987), no. 12, 2625-2632. ___ , Grobner bases and differential algebra, Collection: Applied algebra, algebraic algorithms and error-correcting codes (Menorca, 1987), Lect. Notes Comput. Sci. 356 (1989), 129-140. ___ , Some remarks on differential dimension, Collection: Applied algebra, algebraic algorithms and error-correcting codes (Rome, 1988), Lect. Notes Comput. Sci. 357 (1989), 152-163. ___ , /(olchin schemes, J. Pure Appl. Algebra 63 (1990), no. 1,13-27. ___ , An extension of a procedure to prove statements in differential geometry, J. Automat. Reason. 12 (1994), no. 3, 351-358. ___ , Differential Griibner bases in one variable and in the partial case. Algorithms and software for symbolic analysis of nonlinear systems, Math. Comput. Modelling 25 (1997), no. 8-9, 1-10. Carra Ferro Giuseppa, Sit William, On term-ordering and rankings, In: Computational Algebra (Fairfax, VA, 1993), Lect. Notes Pure Appl. Math., 151, M. Dekker, New York, 1994, pp. 31-37. Cartan H., Eilenberg S., Homological Algebra, Princeton Univ. Press, 1956. Casorati F., II calcolo delle differenze finite, Annali di Matematica Pura ed Appiicata, Series II 10 (1880-1882), 10-43. Cassidy Phyllis J., Differential algebraic groups, Amer. J. Math. 94 (1972), no. 3, 891-954. ___ , Differential algebraic group structures on the plane, Proc. Amer. Math. Soc. 80 (1980), no. 2, 210-214. ___ , The classification of the semisimple differential algebraic groups and the linear semisimple differential algebraic Lie algebras, J. Algebra 121 (1989), no. 1, 169-238. Cerlienco L., Mureddu M., Piras F., Combinatorial-algebraic techniques in Griibner bases theory, Sem. Lothar. Combin. 34 (1995), Art B34e, approx 15pp (electronic). Chou Shang-Ching, Mechanical Geometry Theorem Proving, D. Reidel, Dordrecht, 1987. Chen Caoyu, Derivations on differential operator algebra and Weyl algebra, Chinese Ann. Math. Ser. B17 (1996), no. 2, 199-212.

[Co71J

[CoR48J

[CoR49J [CoR52aJ [CoR52bJ

[CoR55J

[CoR56J

[CoR65J [CoR69J [CoR70]

[CoR73]

[CoR76J

[CoR79]

[CoR80]

[CoR83] [CoR86a] [CoR86b]

[Cou95]

[CL092]

[CJ72]

[Cr96]

[Cz95]

[DST88]

[Del93]

[Diop93]

[Dzh79a]

[Dzh79b]

[Dzh79b]

[Ei53]

[ERV96]

REFERENCES 407

Cohn P.M., Free Rings and Their Relations, Academic Press, London, New York, 1971. Cohn R.M., Manifolds of difference polynomials, Trans. Amer. Math. Soc. 64 (1948), 133-172. ___ , Inversive difference fields, Bull. Amer. Math. Soc. 55 (1949),597-603. ___ , Extensions of difference fields, Amer. J. Math. 74 (1952),507-530. ___ , On extensions of difference fields and the resolvents of prime difference ideals, Proc. Amer. Math. Soc. 3 (1952), 178-182. ___ , Finitely generated extensions of difference fields, Proc. Amer. Math. Soc. 6 (1955),3-5. ___ , An invariant of difference field extensions, Proc. Amer. Math. Soc. 7 (1956), 656-661. Cohn R.M., Difference Algebra, Interscience, New York, 1965. ___ , Systems of ideals, Canadian J. Math. 21 (1969),783-807. ___ , A difference-differential basis theorem, Canadian J. Math. 22 (1970), no. 6, 1224-1237. ___ , Types of singularity of components of difference polynomial, Aequat. Math. 9 (1973), no. 2-3, 236-241. ___ , The general solution of a first order differential polynomial, Proc. Amer. Math. Soc. 55 (1976), no. 1, 14-16. ___ , Specializations of differential kernels and the Ritt problem, J. Algebra 61 (1979), no. 1, 256-268. ___ , The Greenspan bound for the order of differential systems, Proc. Amer. Math. Soc. 79 (1980), no. 4, 523-526. ___ , Order and dimension, Proc. Amer. Math. Soc. 87 (1983), no. 1, 1-6. ___ , Valuations and the Ritt problem, J. Algebra 10l (1986), no. 1, 1-15. ___ , Solutions in the general solution of second order algebraic differential equations, Amer. J. Math. 108 (1986), no. 3, 505-523. Coutinho S.C., A Primer of Algebraic D-modules. London Mathematical Society, Student Text 93, Cambridge University Press, 1995. Cox David, LottIe John, and O'Shea Donald, Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, SpringerVerlag, New York, 1992. Cozzens John, Johnson Joseph L., Some applications of differential algebra to ring theory, Proc. Amer. Math. Soc. 31 (1972),354-356. Crew Richard, The differential Galois theory of regular singular p-adic differential equations, Math. Ann. 305 (1996), no. 1, 45-64. Czichowski Gunter, A note on Grobner bases and integration of rational functions, J. Symbolic Comput. 20 (1995), no. 2, 163-167 .. Davenport J.H., Siret Y., Tournier E., Computer Algebra. Systems and Algorithms for Algebraic Computation, Academic Press, London, 1988. Delaleau Emmanuel, Sur les derivee de l'entree en representation et commande des systemes non linea ires, These de doctorat, Universute de Paris-Sud, 1993. Diop S., Closedness of morphisms of differential algebraic sets. Applications to system theory, Forum Math. 5 (1993),33-47. Dzhavadov G.A., Characteristic Hilbert polynomial for differential-difference modules and its applications, Dep. VINITI, no. 1992-79 (1979). (Russian) ___ , Differential-difference rings and modules of finite generated type and differential-difference Jacobsohn radical, Dep. VINITI, no. 1993-79 (1979). (Russian) ___ , Algebraic differential-difference equations, Dep. VINITI, no. 3234-79 (1979). (Russian) Einstein A., Appendix II. Generalization of gravitation theory, The Meaning of Relativity, 4th edn., Princeton, 1953, pp. 133-165. Elias Juan, Rossi M.E., Valla Giuseppe, On the coefficients of the Hilbert polynomial, J. Pure and Appl. Algebra 108 (1996), no. 1,35-60.

408

[Ev73]

[Ev84]

[Fa96]

[FI89] [Fr63]

[FN96]

[Ga85]

[GM91a]

[GM91b]

[GM091]

[Gao89]

[GCL92]

[Ger97]

[GB96]

[GB97]

[GL96]

[Goo75]

[Goo78]

[Gou72] [Gre59]

[GP]

[Gri91]

[Gr57]

[GD71]

[Hac87]

[HC96]

REFERENCES

Evanovich Peter, Algebraic extensions of difference fields, Trans. Amer. Math. Soc. 179 (1973), 1-22. ___ , Finitely generated extensions of partial difference fields, Trans. Amer. Math. Soc. 281 (1984), no. 2, 795-811. Farto Jose-Miguel, Multiplicity of the solutions of a differential polynomial, J. Pure and App!. Algebra 108 (1996), no. 2, 203-218. Fleiss M., Automatique et corps differentiels, Forum Math. 1 (1989),227-238. Franke C., Picard- Vessiot theory of linear homogeneous difference equations, Trans. Amer. Math. Soc. 108 (1963), no. 3, 491-515. Furuya Mamoru, Niitsuma Hiroshi, Notes on differential ideals of Laskerian rings, SUT. J. Math. 32 (1996), no. 2, 133-139. Galligo A., Some algorithmic questions on ideals of differential operators, Lect. Notes Comput. Sci. 204 (1985),413-421. Gallo G., Mishra B., Efficient algorithms and bounds for Wu-Ritt characteristic sets, Progress in Math. Effective Methods in Algebraic Geometry 94 (1991), 119-142. ___ , Wu-Ritt characteristic sets and their complexity, In: Discrete and Commutative Geometry. (Papers from DIMACS special year), 6. Amer. Math. Soc. and the Assoc. of Compo Machinery, 1991. Gallo G., Mishra B., Ollivier F., Some constructions in rings of differential polynomials, Lect. Notes Comput. Sci. 539 (1991), 171-182. Gao Xiao Shan, Minimal characteristic bases of differential polynomial ideals, Kexue Tongbao 34 (1989), no. 6, 405-407. (Chinese) Geddes K.O., Czapor St.R., and Labahn G., Algorithms for Computer Algebra, Kluwer Academic Pub!., Dordrecht, 1992. Gerdt V.P., Grabner Bases and Inllolutille Methods for Algebraic and Differential Equations, Mathematics and Computers in Modelling 25 (1997), no. 8/9, 75-90. Gerdt V.P. and Blinkov Yu.A., Inllolutille Bases of Polynomial Ideals, Preprint- Nr. 01/1996, Naturwissenschaftlich-Theoretisches Zentrum, Universitiit Leipzig, 1996; Preprint JINR E5-97-3, Dubna, 1997, To appear in "Mathematics and Computers in Simulation" . ___ , Minimal Inllolutille Bases, Preprint JINR E5-97-4, Dubna, 1997, To appear in "Mathematics and Computers in Simulation". Gonciulea Nicolae, Lakshmibai Venkatramani, Grabner bases and standard monomial bases, C. R. Acad. Sci. Paris, Ser.I, Math. 322 (1996), no. 3, 255-260. Goodearl K.R, Global dimension of differential operator rings. II, Trans. Amer. Math. Soc. 209 (1975),65-85. ___ , Global dimension of differential operator rings. III, J. London Math. Soc. 17 (1978), no. 3, 397-409. Gould H.W., Combinatorian Indentities, Margantown W.Va, 1972. Greenspan B., A bound for the orders of the components of a system of algebraic difference equations, Pacific J. Math. 9 (1959),473-486. Greuel G.-M., Pfister G., Adllances and improllements in the theory of standard bases and syzygies, Arch. Math (Basel) 66 (1996), no. 2, 163-176. Grigorev D.Yu., Complexity of the recognition of the irreducibility of a system of linear ordinary differential equations, Zap. Nauchn. Sem. Leningrad. Otde!. Mat. Inst. Steklov. (LOMI) 192 (1991), no. 5, 60-68, 174. Grothendieck A., Sur quelques points d'algebre homologique, Tohoky Math. J. 9 (1957), 119-221. Grothendieck A., Dieudonne J., EGA I, Elements de Geometrie Algebrique, I, Grundlehren, vo!' 166, Springer-Verlag, Heidelberg, 1971. Hackbush W., Elliptic Differential Equations. Theory and Numerical Treatment, Springer, New York, 1987, pp. 311. Hadjiev D., Callialp F., On a differential analog of the prime-radical and properties of the lattice of radical differential ideals in a&sociatille differential rings, Turkish J. Math. 20 (1996), no. 4, 571-582.

[Ha77]

[HP95]

[He26]

[Herz35]

[In80]

[In81]

[Ja890] [Jan20]

[Jan21]

[Jan29]

[Jo69a]

[Jo69b] [Jo69c]

[Jo71]

[Jo74]

[Jo77]

[Jo78]

[Jo82]

[JS79]

[KV96]

[Ka57] [Ka69] [Kash77] [Katz72]

[Katz90]

[Keig73]

[Keig78]

[Keig82]

[Keig83]

[Keig97]

REFERENCES 409

Hartshorne R., Algebraic Geometry, Springer-Verlag, New York, Heidelberg, Berlin, 1977. Hendriks Peter, Put Marius, Galois action on solutions oj a differential equations, J. Symbolic Comput. 19 (1995), no. 6, 559-576. Herman G., Die Frage der endlichen vielen Schritte in der Theorie der Polynomideale, Math. Ann. B. 96 (1926),736-788. Herzog Fritz, Systems oj algebraic mixed difference equations, Trans. Amer. Math. Soc. 37 (1935),286-300. Infante R.P., Strong normality and normality Jor difference fields, Aequat. Math. 20 (1980), no. 2-3, 159-165. ___ , On the Galois theory oj difference fields, Aequat. Math. 22 (1981), no. 2-3, 194-207. Jacobi C.G.J., Gesammelte Werke, vol. 5, Berlin, 1890, pp. 191-216. Janet M., Sur les syst'emes d'equations aux derivees partielles, J. de Math. (8) 3 (1920),65-151. ___ , Sur Ie systemes aux derivees partielles comprenant autant d'equations que de Jonctions inconnues" C. R. Acad. Sci. Paris 172 (1921),1637-1639. ___ , Ler;ons sur Ie Systemes d'Equations aux Derivees Partielles, Gauthier-Villars, Paris, 1929. Johnson Joseph L., Differential dimension polynomials and a Jundamental theorem on differential modules, Amer. J. Math. 91 (1969),239-248. ___ , Kiihler differentials and differential algebra, Ann. of Math. 89 (1969),92-98. ___ , A notion oj Krull dimension Jor differential rings, Comment. Math. Helv. 44 (1969),207-216. ___ , Extensions oj differential modules over Jormal power series rings, Amer. J. Math. 93 (1971), 731-74l. ___ , K iihler differentials and differential algebra in arbitrary characteristic, Trans. Amer. Math. Soc. 192 (1974), 201-208. ___ , A notion oj regularity Jor differential local algebras, In: Contributions to algebra (collection of papers dedicated to Ellis Kolchin), Academic Press, New York, 1977,pp.211-232. ___ , Systems oj n partial differential equations in n unknown Junctions; the conjecture oj M. Janet, Trans. Amer. Math. Soc. 242 (1978),329-334. ___ , Order Jor systems oj differential equations and a generalization oj the notion oj differential ring, J. Algebra 78 (1982),91-119. Johnson Joseph, Sit William, On the differential transcendence polynomials oj finitely generated differential field extensions, Amer. J. Math. 101 (1979), no. 6,1249-1263. Kamoi Yuuji, Vogel Wolfgang, On Grobner bases and Buchsbaum algebras, Arch. Math. (Basel) 67 (1996), no. 6, 457-464. Kaplansky 1., An Introduction to Differential Algebra, Hermann, Paris, 1957. ___ , Fields and Rings, Chicago, 1969. Kash F., Moduln und Ringe, Stuttgart, 1977. Katz N., Algebraic solutions oj differential equations, Invent. Math. 18 (1972), no. 1-2, 1-119. ___ , Exponential Sums and Differential Equations. Ann. oj Math. Stud., No. 124, Princeton Univ. Press, Princeton, NJ, 1990. Keigher W.F., Prime differential ideals in differential rings, Contributions to algebra, Academic Press, New York, 1973, pp. 239-250. ___ , Quasi-prime ideals in differential rings, Houston J. Math. 4 (1978), no. 3, 379-388. ___ , Differential rings constructed Jrom quasi-prime ideals, J. Pure Appl. Algebra 26 (1982), 191-20l. ___ , On the structure presheaJ oj a differential ring, J. Pure Appl. Algebra 27 (1983), 163-172. ___ , On the ring oj Hurwitz series, Comm. Algebra 25 (1997), no. 6, 1845-1859.

410

[Kh92]

[Kh95]

[KN95]

[KoI39]

[KoI42a]

[KoI42b] [KoI44] [KoI47]

[KoI48]

[KoI52]

[KoI53] [KoI55] [KoI59]

[KoI60]

[KoI64]

[KoI65]

[KoI68]

[KoI73]

[KoI80]

[KoI85] [KoI92]

[KR39]

[Kon88]

[Kon89]

[KLMP92]

[KMP82]

[KP88]

[KP90]

REFERENCES

Khovanskii A.G., Newton poly top, Hilbert polynomial, and sums of finite sets, Funct. Anal. and Applic. 26 (1992), no. 4, 57-63. ___ , Sums of finite sets, orbits of commutative semigroups and Hilbert functions, Funct. Anal. and Applic. 29 (1995), no. 4, 36-50. Kishimoto Kazuo, Nowicki Andrzey, On the image of derivations, Comm. Algebra 23 (1995), no. 12,4557-4562. Kolchin E.R., On the basis theorem for infinite systems of differential polynomials, Bull. Amer. Math. Soc. 45 (1939),923-926. ___ , On the basis theorem for differential systems, Trans. Amer. Math. Soc. 52 (1942), 115-127. ___ , Extensions of differential fields. I, Ann. of Math. 43 (1942),724-729. ___ , Extensions of differential fields. II, Ann. of Math. 45 (1944), 358-36l. ___ , Extensions of differential fields. III, Bull Amer. Math.Soc. 53 (1947), 397-40l. ___ , Algebraic matrix groups and the Picard- Vessiot theory of homogeneous linear ordinary differential equations, Ann. of Math. 49 (1948), 1-42. ___ , Picard- Vessiot theory of partial differential fields, Proc. Amer. Math. Soc. 3 (1952),596-603. ___ , Galois theory of differential fields, Amer. J. Math. 15 (1953),753-824. ___ , On the Galois theory of differential fields, Amer. J. Math. 11 (1955),868-894. ___ , Rational approximation to solutions of algebraic differential equations, Proc. Amer. Math. Soc. 10 (1959),238-244. ___ , Le thioreme de la base finie pour les polyniimes differentiels, Seminaire Dubreil- Pisot 14 (1960-1961), no. 7, 15. ___ , The notion of dimension in the theory of algebraic differential equations, Bull. Amer. Math. Soc. 10 (1964),570-573. ___ , Singular solutions of algebraic differential equations and a lemma of Arnold Shapiro, Suppl. 2, Topology 3 (1965),309-318. ___ , Some problems in differential algebra, Proceeding of Moscow International Congress of Mathematics, Moscow, 1968, pp. 269-216. ___ , Differential Algebra and Algebraic Groups, Academic Press, New York - London, 1973. ___ , On universal extensions of differential fields, Pacific J. Math. 86 (1980), 139-143. ___ , Differential Algebraic Groups, Academic Press, Orlando, 1985. ___ , A problem on differential polynomials, Part 2, Contemporary Mathematics 131 (1992),449-462. Kolchin E.R and Ritt J.F, On certain ideals of differential polynomials, Bull. Amer. Math. Soc. 45 (1939),895-898. Kondrat'eva M.V., Description of the set of minimal differential dimension polynomials, Vestnik Mosk. Univ., ser I (1988), no. 1,35-39. (Russian) ___ , The minimal dimension polynomial of an extension of a field that is defined by a system of linear differential equations, Math. Zametki 45 (1989), no. 3, 80-86. (Russian) Kondrat'eva M.V., Levin A.B., Mikhalev A.V., and Pankrat'ev E.V., Computation of dimension polynomials, Internat. J. of Algebra and Comput. 2 (1992), no. 2,111-137. Kondrat 'eva M.V., A.B., Mikhalev A.V., and Pankrat'ev E.V., On Jacobi's bound for systems of differential polynomials, In: Algebra (Collections of papers) (A.I. Kostrikin, ed.), Moscow Univ. Press, Moscow, 1982, pp. 79-85. (Russian) Kondrat'eva M.V., Pankrat'ev E.V., Algorithms of computation of characteristic Hilbert polynomials, Packets of Applied Programs. Analytic Transformations, Nauka, Moscow, 1988, pp. 129-146. (Russian) ___ , A recursive algorithm for the computation of Hilbert polynomial, Lect. Notes Comput. Sci. 318 (1990), Springer-Verlag, 365-315.

[KPS85]

[Kov69]

[Kov71]

[Kov86]

[Land70]

[Land72]

[Lang71] [Lat88]

[LMP85]

[Laz83]

(Le78]

[Le80a]

[Le80b]

[Le82]

[Le85a]

[Le85b]

[LM87]

[LM88a]

[LM88b]

[LM89a]

[LM89b]

[LM91a]

[LM91b]

[LM92a]

[LM92b]

REFERENCES 411

Kondrat'eva M.V., Pankrat'ev E.V., Serov R.E., Computations in differential and difference modules, Proceedings of the Intemat. Conf. on the Analytic Computations and their Applications in Theoret. Physics, Dubna, 1985 .. pp. 208-213. (Russian) Kovacic Jerald, The inverse problem in the Galois theory oj differential fields, Ann. of Math. 89 (1969), 583-608. ___ , On the inverse problem in the Galois theory oj differential fields, Ann. of Math. 93 (1971),269-284. ___ , An algorithm Jor solving second order linear homogeneous differential equations, J. Symbolic Computation 2 (1986),3-43. Lando B., Jacobi's bound Jor the order oj system of firat order differential equations, Trans. Amer. Math. Soc. 152 (1970), no. I, 119-135. ___ , Jacobi's bound Jor first order difference equations, Proc. Amer. Math. Soc. 32 (1972), no. 1,8-12. Lang S., Algebra, Addison-Wesley, Mass., 1971. Latyshev V.N., Constructive Ring Theory. Standard Bases, Moscow Univ. Press, Moscow, 1988. Latyshev V.N., Mikhalev A.V., and Pankrat'ev E.V., Construction oj canonical simplifiers in modules over rings oj polynomials, Visnik Kiiv Univ., Ser. Mat., Mech. 27 (1985),65-67. (Ukrainian) Lazard D., Griibner bases, Gaussian elimination and a resolution oj system oj algebraic equations, Lect. Notes Comput. Sci. 162 (1983), 146-156. Levin A.B., Characteristic polynomials of filtered difference modules and extensions oj difference fields, Uspehi Math. Nauk 33 (1978), no. 3, 177-178 (Russian); English trans!', Russian Math. Surveys 33 (1978), no. 3, 165-166. ___ , Characteristic polynomials oj inversive difference modules and some properties oj inversive difference dimension, Uspehi Math. Nauk 35 (1980), no. I, 201-202 (Russian); English trans!., Russian Math. Surveys 35 (1980), no. 1,217-218. ___ , Characteristic polynomiala oj difference modules and some properties oj diJJerence dimension, Dep. VINITI, no. 2175-80 (1980). (Russian) ___ , Type and dimension of inverse difference vector spaces and difference algebras, Dep. VINITI 6.04.1982. no. 1606-82. (Russian) ___ , Characteristic polynomials of A-modules and finitely generated A-field extensions, Dep. VINITI, no. 334-85 (1985). (Russian) ___ , Inversive difference modules and problems of solvability oj systems oj linear difference equations, Dep. VINITI, no. 335-85 (1985). (Russian) Levin A.B., Mikhalev A.V., A differential dimension polynomial and the strength oj a system of differential equations, Computable invariants in the theory of algebraic systems (collection of papers), Novosibirsk, 1987, pp. 58-66. (Russian) ___ , Difference-differential dimension polynomials, Dep. VINITI, no. 6848-B88 (1988). (Russian) ___ , Dimension polynomials oj filtered G-modules, Dep. VINITI, no. 6849-B88 (1988). (Russian) ___ , Dimension polynomials oj filtered G-modules and finitely generated oj G-field extensions, Algebra (collection of papers) (A.I. Kostrikin, ed.), Moscow Univ. Press, Moscow, 1989, pp. 74-94. ___ , Dimension polynomials oj differential modules, Abelian Groups and Modules 9 (1989), 51-67. (Russian) ___ , Type and dimension oj finitely generated vector G-spaces, Vestnik Mosk. Univ., Ser I (1991), no. 4, 72-74 (Russian); English trans!., Moscow Univ. Math. Bul!. 46 (1991), no. 4, 51-52. ___ , Dimension polynomials oj difference-differential modules and oj differencedifferential field extensions, Abelian Groups and Modules 10 (1991),56-82. (Russian) ___ , Dimension polynomials oj filtered differential G-modules and extensions oj differential G-fields., Contemporary Mathematics 131, Part 2 (1992),469-489. ___ , Type and dimension of finitely generated G-algebras., Contemporary Mathematics 184 (1995),275-280.

412

[LW96]

[MacI6]

[Mac27]

[Mag94]

[MaI95] [MaI97]

[Mak95]

[Man58]

[Man65]

[Mans91] [Mat70] [Matz96]

[MP73]

[MP80]

[MP84]

[MP87]

[MP89]

[Mi93] [MS96a]

[MS96b]

[Mod92]

[MM86]

[MoI95]

[Mora94]

[MM83]

[Morr78]

[Morr87] [Mou96]

REFERENCES

Liu Hai Xia, Wang Ming Sheng, Rings of differential operators on curves and modules over them, Acta Math. Sinica 39 (1996), no. 2, 280-285. Macaulay F.S., The Algebraic Theory of Modular Systems, Cambridge Tracts in Mathematics and Mathematical Physics, no 19, Cambridge Univ. Press, London, 1916. ___ , Some properties of enumeration in the theory of modular system, Proc. London Math. Soc. (2) 26 (1927),531-555. Magid Andy R., Lectures on Differential Galois Theory, University Lecture Series, vol. 7, AMS, American Mathematical Society, Providence, RI, 1994. Maloo A. K., Maximally differential ideals, J. Algebra 176 (1995), no. 3, 806-823. ___ , Differential simplicity and the module of derivations, J. Pure Appl. Algebra 115 (1997), no. I, 81-85. Makarov E.K., On the structure of characteristic sets of solutions of linear completely solvable differential equations, Differential Equations 31 (1995), no. 2, 203-208. Manin Yu.I., Algebraic curves over fields with differentiation, Izv. Akad. Nauk SSSR, Mat. Ser., 737-756 (Russian); English transl., AMS transl. series 37 (1964), no. 2, 59-78. ___ , Moduli fuchsiani, Ann. Scuola Norm. Sup. di Pisa (3) 19 (1965), no. 1,113-126. Mansfield E., Differential Grabner bases, Ph. D. Thesis, University of Sidney, 1991. Matsumura H., Commutative Algebra, W.A. Benjamin Co., New York, 1970. Matzuk Jerzy, Maximal ideals of skew polynomial rings of automorphism type, Comm. Algebra 24 (1996), no. 3, 907-917. Mikhalev A.V. and Pankrat'ev E.V., Differential modules, Modules III, Inst. Mat. Sibirsk. Otdel Akad. Nauk SSSR, Novosibirsk, 1973, pp. 14-21. (Russian) ___ , Differential dimension polynomial of a system of differential equations, In: Algebra (collection of papers) (A.I. Kostrikin, ed.), Moscow Univ., Moscow, 1980, pp. 57-67. (Russian) ___ , Computation of the differential dimension polynomial with the help of the computer, In: "Teorija i praktika avtomatizirovannykh sistem analiticheskikh preobrazovanij, Vilnus, 1984, pp. 50-53. (Russian) ___ , Differential and difference algebra, In: "Itogi nauki i tekhniki VINITI AN SSSR. Algebra. Topologija. Geometrija.", vol. 25, VINITI, Moscow, 1987, pp. 67-139, (Russian); English transl. in J. Soviet Math. 45 (1989), no.!. ___ , Computer Algebra. Computations in Differential and Difference Algebra, Moscow Univ. Press, Moscow, 1989. (Russian) Mishra B., Algorithmic Algebra, Springer-Verlag, New York, 1993. Mitschi C., Singer M.F., Connected linear groups as differential Galois groups, J. Algebra 184 (1996), no. 1,333-361. ___ , On Rami's solution of the local inverse problem of differential Galois theory, J. Pure Appl. Algebra 110 (1996), no. 2, 185-194. Modak M.R., Combinatorial meaning of the coefficients of Hilbert polynomial, Proc Indian Acad. Sci. 102 (1992), no. 2, 93-123. Moller H.M. and Mora F., New constructive methods in classical ideal theory, J. Algebra 100 (1986), 138-178. Moloney James John, The prime spectrum of commutative differential algebras, J. Func. Anal. 133 (1995), no. 2, 495-500. Mora T., An Introduction to Commutative and Noncommutative Grabner Bases, Theoretical Computer Science, Iss.l, vol. 134, 1994, pp. 131-173. Mora F., Moller H.M., The computation of the Hilbert function, Proc. EUROCAL 83, vol. 162, Springer-Verlag, 1983, pp. 157-167. Morrison S.D., Extensions of differential places, Amer. J. Math. 100 (1978), no. 2, 245-261. ___ , Continuous derivations, J. Algebra 110 (1987), no. 2, 468-479. Moulin Ollagnier Jean, Algorithms and methods in differential algebra. Algorithmic complexity of algebraic and geometric models. (Creteil, 1991,), Theoret. Compo Sci. 157 (1996), no. 1, 115-127.

[Mu95]

[Ni96]

[No94]

[No96]

[01196]

[0Iliv91a]

[0Iliv91b]

[Ov82]

[Pa70] [Pa71]

[Pa72a]

[Pa72b]

[Pa73]

[Pa89]

[PZ96]

[PK96]

[PZ96]

[PS92)

[Po78]

[Po83]

[Po94]

[Rau34]

[RR39]

[RM67]

[Rio68] [Riql0]

REFERENCES 413

Muller Eric F., Differential fields and differentiable functions of algebraic numbers, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 2, 341-350. Nishioka Keiji, Differential field extensions with no movable algebraic branches, Hokkaido Math. J. 25 (1996), no. 3, 453-463. Nowicki A., Rings and fields of constant6 for derivations in characteristic zero, J. Pure Appl. Algebra 96 (1994), no. 1, 47-55. ___ , On the nonexistence of rational first integrals for systems of linear differential equations, Linear Algebra Appl. 235 (1996), 107-120. Ollagnier J.M., Algorithms and methods in differential algebra, Th. Compo Sci. 157 (1996), 115-127. Ollivier Fran<;ois, Standard bases of differential ideals, Applied algebra, algebraic algorithms and error-correcting codes (Tokyo, 1990), vol. 508, Springer, Berlin, 1991, pp. 304-321. ___ , Canonical bases: relations with standard bases, finiteness conditions and application to tame automorphisms, Effective methods in algebraic geometry (Castiglioncello, 1990), Progr. Math., 94, Birkhiiuser, Boston, MA, 1991, pp. 379-400. Ovsiannikov L.V., Group Analysis of Differential Equations (Russian); English translation, Academic Press, New York, 1982, pp. 416. Pankrat'ev E.V., On fuchsian modules, Matern. Zametki 8 (1970), no. 6, 789-810. ___ , The inverse Galois problem for a differential field of converging power series over an algebraically closed field with a p-adic norm, Uspehi. Mat. Nauk 26 (1971), no. 4, 241-242. (Russian) ___ , The inverse Galois problem for the extensions of difference fields, Algebra i Logika 11 (1972),87-118 (Russian); English transl., Algebra and Logic 11 (1972), 51-69. ___ , The inverse Galois problem for extensions of difference fields, Uspehi. Mat. Nauk 27 (1972), no. 1,249-250. (Russian) ___ , Fuchsian difference modules, Uspehi. Mat. Nauk 28 (1973), no. 3,193-194. (Russian) ___ , Computations in differential and difference modules, Symmetries of partial differential equations, Part III, 1989; Acta Appl. Math., vol. 16, 1989, pp. 167-189. Parfionov G.N., Zapatrin R.R., Dual structures in non-commutative differential algebra, Pure Math. Appl. 7 (1996), no. 1-2, 165-174. Park Chan-Bong, Kim Mi-Kyung, A note on Hilbert - Samuel polynomials, Bull. Honam. Math. Soc. 13 (1996), 47-55. Pauer Franz, Zampieri Sandro, Grabner bases with respect to generalized term orders and their application to the modelling problem, J. Symbolic Comput. 21 (1996), no. 2, 155-168. Pillay A., Sokolovic' Z., Superstable differential fields, J. Symbolic Logic 57 (1992), no. 1,97-108. Pommaret J.-F., Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, London, 1978. ___ , Differential Galois Theory, Math. and Appl., vol. 15, Gordon and Breach Sci. Publ., New York, 1983. ___ , Partial Differential Equations and Group Theory. New Perspectives for Applications, Mathematics and its Applications, 293, Kluwer Academic Publishers Group, Dordrecht, 1994. Raudenbush H.W., Ideal theory and algebraic differential equations, Trans. Amer. Math. Soc. 36 (1934),361-368. Raudenbush H.W., Ritt J.F., Ideal theory and algebraic difference equations, Trans. Amer. Math. Soc. 46 (1939),445-453. Richtmyer Robert D., Morton K.W., Difference Methods for Initial- Value Problems, Second Edition, Interscience Publishers, 1967. Riordan J., Combinatorian Identities, John Willey Sons, New York, 1968. Riquier F., Les Systemes d'Equations aux Derivees Partielles, Gauthier-Villars, Paris, 1910.

414

[Rit3l]

[Rit32]

[Rit34] [Rit35a]

[Rit35b] [Rit38]

[Rit39]

[Rit41] [Rit45]

[Rit50]

[RD33]

[Ro75]

[Rt96]

[RuRe97]

[Sa82]

[Sch91]

[Sch92]

[Se56]

[Se58]

[Se68]

[Se74] [Sib82]

[Sin86]

[Si75a]

[Si75a]

[Si75b]

[Si78]

[Si88]

[Si89]

REFERENCES

Ritt J.F., Systems of algebraic differential equations, Proc. Nat. Acad. Sci. U.S.A. 17 (1931),366-368. ___ , Differential Equations from the Algebraic Standpoint, Amer. Math. Soc., Colloq. publication, vol. 14, New York, 1932. ___ , Algebraic difference equations, Bull. Amer. Math. Soc. 40 (1934),303-308. ___ , Jacobi's problem on the order of system of differential equation, Ann. of Math. 36 (1935),303-312. ___ , Systems of algebraic differential equations, Ann. of Math. 36 (1935),293-302. ___ , On certain points in the theory of algebraic differential equations, Amer. J. Math. 60 (1938),1-43. ___ , On the intersections of algebraic differential manifolds, Proc. Nat. Acad. Sci. U.S.A. 25 (1939),214-215. ___ , Complete difference ideals, Amer. J. Math. 63 (1941),681-690. ___ , On the manifolds of partial differential polynomials, Ann. of Math. 46 (1945), 102-112. ___ , Differential Algebra, Amer. Math. Soc., Colloq. Publication, vol. 33, New York,1950. Ritt J.F., Doob J.L., Systems of algebraic difference equations, Amer. J. Math. 55 (1933), 505-514. Rosenlicht M., Differential extension fields of exponential type, Pacific J. Math. 57 (1975), 289-300. Rtveliashvili E., On Krull dimension of Ore extensions, Georgian Math. J. 3 (1996), no. 3, 263-274. Rust C.J., Reid G.J., Ranking of partial derivatives, Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, ACM Press, 1997, pp.9-16. Sachkov V.N., An Introduction in Combinatorian Methods of Discrete Mathematics, Nauka, Moscow, 1982. (Russian) Schwarz F., Monomial orderings and Grobner bases, ACM SIGSAM Bulletin 25 (1991), 10-23. ___ , Reduction and completion algorithms for partial differential equations, Proceedings of ISSAC 1992, ACM Press, New York, 1992, pp. 49-56. Seidenberg A., An elimination theory for differential algebra, Univ. California Publications Math. (N.S.) 3 (1956),31-65. , Abstract differentail algebra and the analytic case, I, Proc. Amer. Math. Soc. 9 (1958), 159-164. ___ , Abstract differentail algebra and the analytic case, II, Proc. Amer. Math. Soc. 23 (1968),689-691. ___ , Constructions in algebra, Trans. Amer. Math. Soc. 197 (1974),273-313. Sibirsky K.S., Introduction to Algebraic Invariant Theory of Differential Equations, Stiintsa, Kishinev, 1982. (Russian) Singer M., Algebraic relations among solutions of linear differential equations, Trans. Amer. Math. Soc. 295 (1986), 753-763. Sit W., Typical differential dimension of the intersection of linear differential algebraic groups, J. Algebra 32 (1974), no. 3,476-487. ___ , Well ordering of certain numerical polynomials, Trans. Amer. Math. Soc. 212 (1975), no. I, 37-45. ___ , Differential algebraic subgroups of 8L(2) and strong normality in simple extensions, Amer. J. Math. 97 (1975), no. 3. ___ , Differential dimension polynomials of finitely generated extensions, Proc. Amer. Math. Soc. 68 (1978), no. 3, 251-257. ___ , On Goldman's algorithm for solving first order multinomials autonomous systems, Lect. Notes Comput. Sci. 357 (1988),386-395. ___ , Some comments on term-ordering in Grobner basis computations, ACM SIGSAM Bulletin 23 (1989), no. 2, 34-38.

[Si92]

[Sp69j

[Staf78) [Stan78) [Sta96] [Str39)

[Stu96)

[Th031]

[TS90]

[T076]

[Tou95)

[Tr96j

[Um96a)

[Um96b]

[VdP95)

[VdW67)

[VdW71j

[Wa96a)

[Wa96b)

[Wu78)

[Wu84)

[ZS58] [ZS60] [Zu65j

[CAPR91)

REFERENCES 415

___ , Algorithm for solving parametric linear systems, J. Symbolic Computation 13 (1992),353-394. Spencer D.C., Overdetermined systems of linear partial differential equations, Bull. Amer. Math. Soc. 75 (1969), 179-239. Stafford J.T., Structure of Weyl algebra, J. London Math. Soc.(2) 18 (1978),429-442. Stanley R., Hilbert Functions of Graded Algebras, Adv. in Math. 28 (1978), 57-83. ___ , Combinatorics and Commutative Algebra, 2-nd ed., Birkhauser, Berlin, 1996. Strodt W.C., Irreducible systems of algebraic differential equations, Trans. Amer. Math. Soc. 45 (1939),276-297. Sturmfels B., Griibner Bases and Convex Polytopes, AMS, University Lecture Series, vol. 8, Providence, 1996. Thomas J.M., Matrices of integers ordering derivatives, Trans. Amer. Math. Soc. 33 (1931),389-410. Tikhonov A.N., Samarskii A.A., Equations of Mathematical Physics (Russian); English translation, Dover Publications Inc., New York, 1990, pp. 765. Tomasovic T.S., A Generalized Jacobi Conjecture for Arbitrary Systems of Algebraic Differential Equations, Ph. D. dissertation, Columbia University, 1976. Touzet F., Differential equations admitting Liouvillian Solutions, C. R. Acad. Sci., Ser.I- Math. 321 (1995),1409-1412. Traverso Carlo, Hilbert functions and the Buchberger algorithm, J. Symbolic Comput. 22 (1996), no. 4, 355-376. Umemura Hiroshi, Galois theory of algebraic and differential equations, Nagoya Math. J. 144 (1996),1-58. ___ , Differential Galois theory of infinite dimension, Nagoya Math. J. 144 (1996), 59-135. Van der Put Marius, Differential equations in characteristic p, Special issue in honour of Frans Oort, Compositio Math. 97 (1995), no. 1-2, 227-251. Van der Warden B.L., Algebra II. Funfte Auf/age, Springer-Verlag, Berlin, Heidelberg, New York, 1967. ___ , Algebra I. Achte Auf/age der modernen Algebra, Springer-Verlag, Berlin, Heidelberg, New York, 1971. Wang Dongming, An elimination method for differential polynomial systems, Systems Sci. Math. Sci. 9 (1996), no. 3, 216-228. ___ , Solving polynomial equations: characteristic sets and triangular systems. Symbolic computation, new trends and developments (Lille, 1993), Math. Comput. Simulation 42 (1996), no. 4-6, 339-351. Wu W.-T., On the decision problem and the mechanization of theorem proving in elementary geometry, Scientia Sinica 21 (1978),157-179. ___ , Basic principles of mechanical theorem proving in geometries, Journal of Sys. Sci. and Math. Sci. 4(3) (1984),207-235. Zariski 0., Samuel P., Commutative Algebra (Vol. I), Van Nostrand, Princeton, 1958. ___ , Commutative Algebra (Vol. II), Van Nostrand, Princeton, 1960. Zuckerman I., A new measure of a partial differential field extension, Pacific J. Math. 15 (1965), no. 1,357-371. Shirkov D.V., Rostovtsev V.A., and Gerdt V.P. (eds.), IV International Conference on Computer Algebra in Physical Research. Dubna, USSR, 22-26 May 1990, World Scientific, Singapore, 1991.

Index

Artinian module, 12 ring, 13

additive conservative system, 34 admissible elements, 70

transformation, 298 vectors, 70 ordering, 164

algebra of difference polynomials, 137 algebra of u-polynomials, 137

of inversive difference polynomials in u'indeterminates, 160

of u'-polynomials in u'-indeterminates, 160

of A-u-~' -polynomials, 176 ascending chain condition, 3

chain condition for submodules, 12 associated graded module, 29

graded ring, 28 ideal,8

automorphism, 4 autoreduced Grobner basis, 202

set, 165, 182, 201, 232

basic rectangle, 333 set, 135

of derivations, 125 of operators, 123

u'-field, 168 u-field, 150

basis, 162, 180 bijective mapping, 2

Cartesian product A x B of sets, 1 canonical isomorphism of conservative sys

terns, 33 characteristic function of a NP-graded mod

ule, 212 Hilbert polynomial, 211 polynomial, 225, 346, 383

of an excellently NP-filtered module, 219 of a module, 286 of a u'-R-module, 294

set, 147, 167, 186,234 coherent autoreduced set, 244, 321 complete system of u-overfields, 154 composition of mappings, 2

series, 14 condition of minimality, 3 confluent reduction relation, 196 conservative mapping, 32

system, 32 coprime ideals, 7

u-ideals, 150 uO-ideals, 163

417

defining difference ideal, 141 ideal, 132, 142 A-u-~' -ideal, 178 u' -ideal, 161

degree, 63, 124 of a differential polynomial, 128

derivation, 37, 181 operator, 125 over k, 38

derivative, 125, 181 descending chain condition for submodules,

12 difference algebra, 143

algebraically dependent family, 137 algebraically independent family, 137 dimension, 289 extension, 135 field, 135 homomorphism, 135 ideal,135 indeterminates, 137 operators, 157 overring, 135 ring, 135 Ritt's ring, 144 subfield, 135 subring, 135 type, 289, 307 vector R-space, 158 R-module, 158

differential dimension, 229, 238 dimension polynomial of a prime A-ideal,

238 polynomial of the system, 238

module, 134 operator, 388 order, 181 polynomial ring, 128 polynomials, 128 ring, 125 transcendence degree, 131 type, 229, 258

differentially algebraic extension, 131 differential-difference homomorphism, 190

operator, 189 ring, 174 vector space, 189 R-module, 189

dimension of a module, 356 of a ring, 15 polynomial, 57, 225, 286, 294,346,383

of a system in finite differences, 320 of a finitely generated u-extension, 306

418

of a finitely generated 17' -extension, 311 of a prime system of algebraic 17'-equa

tions, 319 of a prime 17'-ideal, 316

discrete filtration, 27 disjoint 17-ideals, 150

17'-ideals, 163 divisible conservative system, 34 domain of a relation, 1

embedded prime ideal, 9 endomorphism, 4 epimorphism, 4 equivalence relation, 1 equivalent basic set, 297 essential disjoint divisors, 150

prime divisors, 149, 163, 180 separated components, 155

exact sequence, 223 excellent filtration, 218, 219, 223, 228, 285,

290,344,351,357,380,388 excellently filtered ~-R-module, 223

filtered 17-R-module, 285 filtered 17' -R-module, 290

exhaustive filtration, 27

family of 17'-indeterminates, 159 field extension, 5 filtered differential R-module, 223

ring, 26 G-A-module, 380 R-module, 27 17-R-module, 284

filtration, 290, 344 associated with a system of generators, 27 of a field extension, 228 of a module, 27 of a ring, 26

finite filtration, 223, 228, 290, 344, 380 filtration of a 17-R-module, 284

finitely filtered ~-R-module, 223 filtered 17-R-module, 284

17'-R-module, 290 generated extension, 124

field extension, 5 ideal,4 G-ring extension, 377 G-A-algebra, 378 ~-G-extension, 394

flat module, 11 free filtered R-module, 30

filtered ~-17-E:'-R-module, 347 function, 2

Grobner representation, 195 basis, 195

generic zero, 132, 142, 162, 178 good filtered ~-R-module, 223

filtered 17-R-module, 285

INDEX

17*-R-module, 290 filtration, 223, 228, 285, 290, 344, 380

graded difference R-module, 281 left A-module, 21 ring, 19 ring associated with a filtered ring R, 28 subring,20 grR-module associated with a filtered R

module, 29 17-R-module, 281

Hilbert function, 210 function of a filtered module, 29

of a filtered ring, 29 of a graded module, 25

polynomial, 25 head,28,29 homogeneous component, 19, 20

element, 19, 212 of degree i, 19, 212

ideal,20 linear differential equation, 268 subring,20

homogenization, 287 homomorphism of filtered modules of degree

p, 27 of filtered D-modules, 223 of filtered ~-R-modules, 223 of graded modules of degree p, 22 of G-A-modules, 380

image, 2 of an homomorphism, 4

independent derivation operators, 131 index set of the family, 2 indexed set, 2

family, 2 indexing, 3 induction condition, 3 initial, 145, 164, 182, 231

subset, 108 injective mapping, 2 integral closure, 17

dependence, 16 element, 16 extension, 16

integrated ranking, 145 inverse image of a set, 2 inversive closure, 138, 177

difference dimension, 296 dimension of the prime 17* -ideal, 317 homomorphism, 172 ring, 135 type, 296, 314 vector space, 172 R-module, 172 operator, 171 polynomials, 160

~-17-e* -ring, 177

irreducible components, 153, 169 of a variety, 153

element, 194 Grabner basis, 202 ideal,14 representation, 149,152, 180 variety, 168 [-variety, 152, 168

isolated prime ideal, 9 set of prime ideals, 9

isomorphism, 4

Jacobi conjecture, 274 Jacobson radical of a ring, 7 Jordan-Holder series, 14

Kolchin polynomial, 57, 65 Kolchin's differential dimension polynomial,

228 kernel of an homomorphism, 4

leader, 145, 164, 182, 191, 194, 231 leading coefficient, 60, 194

total coefficient, 63 least common multiple, 67 length,325

of a chain, 14 of a module, 15

linear differential polynomial, 130 ideal,130 order,3 a·-ideal, 320

linearly ordered set, 3 local ring, 7

~-a-e:· -K-algebra, 370 ~-a-e:·-K-algebra of finitely generated

type, 372 localization, 10 locally confluent, 196

mapping, 2 matrix of a transformation, 298 maximal element, 3

ideal,7 module, 12

minimal differential dimension polynomial, 268

element, 3 Grabner basis, 202 module, 12 polynomial of a a-extension, 316

of a a·-extension, 316 primary decomposition of an ideal, 8 prime ideal, 9

minimizing coefficients, 99 module of derivations, 37

of differentials, 38 of Kahler differentials, 39

monomial, 191

INDEX

monomorphism, 4 multiple, 181 multiplicative set, 6 multiplicatively closed set, 6

Nakayama's lemma, 7 Noetherian conservative system, 35

module, 12 ring, 13

natural epimorphism, 324 normal reduction process, 194

G-representation, 196 normalized matrix, 81 normal-form algorithm, 194 numerical polynomial, 45

419

order, 124, 133, 157, 158, 171, 191,350,351, 357, 388

of a differential polynomial, 128 of a G-operator, 379 of an element, 379

ordered pair, 1 ordinary differential ring, 125

a-ring, 135 ortant, 107, 164 overfield, 5 overring,5

parametric subring, 367 partial differential ring, 125

reduction process, 195 remainder, 232 a-ring, 135

partially ordered set, 3 reduced, 231

partition of a set, 54 perfect closure, 143, 162, 179

conservative system, 34 ideal,9 ~-ideal, 123 a-ideal, 143 a·-ideal,162 ~-a-e:· -ideal, 179

positive filtration, 26, 27 positively filtered module, 27 primary decomposition of an ideal, 8

ideal,8 prime components of an ideal, 129

difference ideal, 137 ideal,5 system of algebraic a·-equations, 318

of ~-G-polynomials, 396 G-ideal, 377 ~-ideal, 123 a·-ideal, 137

proper subvariety, 152 [-subvariety, 168 transform, 164

pseudo-locally confluent, 196

420

quasiprime ideal, 128

Ritt's algebra, 126 inversive difference ring, 162 t1-U-&* -ring, 180

radical,6 ideal,9

range of a relation, 1 ranking, 145, 191, 193, 231

orderly, 193 reduced t1-U-E* -polynomial, 182

t1-polynomial, 231 reducible element, 194

Grobner basis, 202 variety, 152, 168

reduction of a u'-polynomial, 323 relation, 194

reflexive closure, 1, 137 ideal,137 relation, 1 t1-U-E' -ideal, 176

regular order, 181 relation, 1 remainder, 232 ring extension, 5

of constants, 125, 135, 175 of difference operators over a difference

ring, 158 of differential-difference operators, 189 of fractions, 10 of generalized polynomials, 192 of inversive difference operators, 171 of G-operators, 379 of t1-G-operators, 388 of t1-operators over R, 124 of u· -operators, 171

separant, 182, 231 separated filtration, 27

varieties, 154 separating basis, 38 sequence, 2 set of generators, 5, 123, 124, 394

of terms, 192 of G-generators, 377,378

simply generated extension, 124 skew derivation, 43

polynomial ring, 43 solution, 132, 141, 238 standard filtration, 26, 351

grading, 20 ranking, 145, 193 G-filtration, Z-dimensional polynomial, 113 t1-filtration, 351, 388

subindexing, 3 substitution, 161, 178 subvariety, 152, 168 superfluous row, 58

INDEX

surjective mapping, 2 symmetric closure, 2

relation, 1 system of algebraic u'-equations, 319 u· -operator, 318

terms, 145, 164, 181 total coefficient, 63

order, 3 totally ordered set, 3 transform, 135, 164 transitive closure, 2

relation, 1 translation, 135, 181 trivial filtration, 26, 27 type, 355

of a ring R over a family of ideals, 361 typical difference dimension, 289

difference transcendence degree, 307 differential dimension, 229 inversive difference dimension, 296, 317

difference transcendence degree, 314 G-dimension, 386, t1-dimension, 258 t1-u-e:' -dimension, 349 u-transcendence degree, 307

universal system of u-overfields, 151 system of u* -overfields, 168

upper bound, 3

value, 2 of a difference polynomial at a family, 141

variety, 151 defined by a set 4.i over F, 168 over a u*-field F, 168

vector t1-U-E* -space, 189 u· -R-space, 172

Weyl algebra, 134 weak Jacobi conjecture, 274

Jacobi number, 273 well-order, 4

zero, 238

G-algebraically dependent, 378 independent, 378

G-basis, 195 G-dimension, 386

polynomials, 395 G-extension, 378 G-field, 378 G-homomorphism, 377, 380 G-ideal, 377 G-module over a G-ring, 379, G-operator, 379, 388 G-order, 388 G-overfield, 378 G-polynomial, 378

algebra, 378 G-quotient field, 378 G-representation, 195 G-ring,377

extension, 377 of quotients, 379

G-subfield, 378 G-subring, 377 G-transcendence degree, 387 G-type, 386 G-A-algebra, 378 G-A-module, 379 Gi-R-module, 388

k-derivation,38

m-basis, 144, 162, 180

NP-filtered ring 215 D-module, 216

NP-filtration, 216 associated with a choice of generators, 216

NP-graded module, 212 ring, 211

nth suspension of a graded module, 24

R-module,4

S-polynomial, 244 s-dimensional points, 161 s-tuples, 161

Z-dimensional polynomial, 113

A-leader, 146, 183

iJ-derivation, 43

b.-algebraic element, 131 b.-algebraically dependent family, 131

independent family, 131 b.-automorphisms, 123 b.-dimension polynomials, 392 b.-endomorphisms, 123 b.-epimorphisms, 123 b.-extension, 13 b.-field, 123 b.-finitely generated extension, 131 b.-homomorphism, 123 b.-ideal, 123 b.-isomorphism, 123 b.-monomorphism, 123 b.-order, 350, 351, 388 b.-overfield, 123 b.-overring, 123 b.-polynomial,124

ring, 124, 128 b.-ranking, 262 b.-ring, 123, 125 b.-subfield, 123 b.-subring, 123

INDEX 421

b.-transcendence basis, 131 b.-transcendental element, 131 b.-G-algebraically independent elements, 394 b.-G-dimension, 394 b.-G-extension, 394 b.-G-ideal, 395 b.-G-operator, 388 b.-G-polynomial,395

ring, 395 b.-G-subfield, 394 b.-G-transcendence degree, 395 b.-G-R-module, 388 b.-R-homomorphisms, 124 b.-R-module, 124,388 b.-R-submodules, 124 b.-O"-e*-K-algebra, 361 b.o-ranking, 262 b.-e-order, 350, 351 b.-e"-ring, 174 b.-O"-order, 350,351 b.-O"" _eo -ring, 177 b.-O"-ring, 174

[-subvariety, 168 [-variety, 150, 168 e-order, 350, 351

filt-basis of a free filtered R-module, 30

E-components, 34

b.-O"-e"-algebraic element, 176 b.-O"-e" -algebraically dependent element, 186

dependent family, 176 b.-O"-e* -dimension, 349 b.-O"-e* -epimorphism, 190 b.-O"-e* -factor-ring, 176 b.-O"-e* -field of quotients, 177

of rational fractions in b.-O"-e" -indetermi-nates, 177

b.-O"-e* -homomorphism, 190 b.-O"-e" -independent family, 176 b.-O"-e· -indeterminates, 176 b.-O"-e* -isomorphism, 190 b.-O"-e* -linearly dependent elements, 349

independent elements, 349 b.-O"-e"-monomorphism, 190 b.-O"-e* -operator, 189 b.-O"-e* -polynomials, 176 b.-O"-e*-ring, 174 b.-O"-e* -transcendence basis, 187

degree, 188 b.-O"-e*-transcendental element, 176 b.-O"-e* -type, 349 b.-O"-e" -R-algebra, 179 b.-O"-e* -R-module, 189

0" "-algebra over R, 162 O""-algebraic element, 159 O""-algebraically dependent element, 169

422

dependent family, 159 independent family, 159

q*-equivalent, 161 q*-extension, 159 q* -factor-ring, 159 q*-field of rational fractions in q*-indeter-

minates, 161 q* -homomorphism, 112 q*-ideal,131 q*-linearly dependent elements, 291

independent elements, 291 q * -operator, 111 q*-order, 333 q*-overfield,159 q*-overring of Ro generated by the set of

q* -generators, 159 q*-polynomial,160 q*-quotient field, 160 q* -ring, 135 q*-trancendence basis, 110

degree of Gover F, 110 q*-transcendental element, 159 q* -type, 311 q*-R-algebra, 162 q*-e-variety, 168

q-algebra, 143 q-algebraic element, 138

extension, 151 q-algebraically dependent family, 131

independent family, 131 q-equivalent points, 142 q-extension, 135 q-field, 135

of rational fractions in q-indeterminates, 140

q-homomorphism, 135 q-ideal, 135 q-indeterminates, 131 q-order, 350, 351 q-overring, 135 q-polynomial reduced with respect to a set

of q-polynomial, 146 reduced with respect to a q-polynomial,

146 q-quotient field, 139

ring, 131 q-Ritt's ring, 144 q-subfield, 135 q-subring, 135 q-transcendence basis, 156

basis of a q-extension, 156 degree of rover:F, 156

q-transcendental element, 138 q-e-filtration, 351 q-e-order, 350, 351

INDEX

Other Mathematics and Its Applications titles of interest:

P.H. Sellers: Combinatorial Complexes. A Mathematical Theory of Algorithms. 1979, 200 pp. ISBN 90-277-1000-7

P.M. Cohn: Universal Algebra. 1981,432 pp. ISBN 90-277-1213- 1 (hb), ISBN 90-277-1254-9 (pb)

J. Mockor: Groups of Divisibility. 1983,192 pp. ISBN 90-277-1539-4

A. Wwarynczyk: Group Representations and Special Functions. 1986,704 pp. ISBN 90-277-2294-3 (Pb), ISBN 90-277-1269-7 (hb)

I. Bucur: Selected Topics in Algebra and its Interrelations with Logic, Number Theory and Algebraic Geometry. 1984,416 pp. ISBN 90-277-1671-4

H. Walther: Ten Applications of Graph Theory. 1985, 264 pp. ISBN 90-277-1599-8

L. Beran: Orthomodular Lattices. Algebraic Approach. 1985, 416 pp. ISBN 90-277-1715-X

A. pazman: Foundations of Optimum Experimental Design. 1986, 248 pp. ISBN 90-277-1865-2

K. Wagner and G. Wechsung: Computational Complexity. 1986, 552 pp. ISBN 90-277-2146-7

A.N. Philippou, G.E. Bergum and A.F. Horodam (eds.): Fibonacci Numbers and Their Applications. 1986,328 pp. ISBN 9O-277-2234-X

C. Nastasescu and F. van Oystaeyen: Dimensions of Ring Theory. 1987,372 pp. ISBN 90-277-2461-X

Shang-Ching Chou: Mechanical Geometry Theorem Proving. 1987,376 pp. ISBN 90-277-2650-7

D. Przeworska-Rolewicz: Algebraic Analysis. 1988,640 pp. ISBN 90-277-2443-1

C.T.J. Dodson: Categories, Bundles and Spacetime Topology. 1988,264 pp. ISBN 90-277-2771-6

V.D. Gappa: Geometry and Codes. 1988, 168 pp. ISBN 90-277-2776-7

A.A. Markov and N.M. Nagomy: The Theory of Algorithms. 1988,396 pp. ISBN 90-277-2773-2

E. Kratzel: Lattice Points. 1989,322 pp. ISBN 90-277-2733-3

A.M.W. Glass and W.Ch. Holland (eds.): Lattice-Ordered Groups. Advances and Techniques. 1989,400 pp. ISBN 0-7923-0116-1

N.E. Hurt: Phase Retrieval and Zero Crossings: Mathematical Methods in Image Reconstruction. 1989,320 pp. ISBN 0-7923-0210-9

Du Dingzhu and Hu Guoding (eds.): Combinalorics, Computing and Complexity. 1989, 248 pp. ISBN 0-7923-0308-3


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J. Martinez (ed.): Ordered Algebraic Structures. 1989,304 pp. ISBN 0-7923-0489-6

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E. Goles and S. Martinez: Neural and Automata Networks. Dynamical Behavior and Applications. 1990, 264 pp. ISBN 0-7923-0632-5

A. Crumeyrolle: Orthogonal and Symplectic Clifford Algebras. Spinor Structures. 1990, 364 pp. ISBN 0-7923-0541-8

S. Albeverio, Ph. Blanchard and D. Testard (eds.): Stochastics, Algebra and Analysis in Classical and Quantum Dynamics. 1990,264 pp. ISBN 0-7923-0637-6

G. Karpilovsky: Symmetric and G-Algebras. With Applications to Group Representations. 1990, 384 pp. ISBN 0-7923-0761-5

J. Bosak: Decomposition of Graphs. 1990,268 pp. ISBN 0-7923-0747-X

J. Adamek and V. Trnkova: Automata and Algebras in Categories. 1990,488 pp. ISBN 0-7923-0010-6

A.B. Venkov: Spectral Theory of Automorphic Functions and Its Applications. 1991,280 pp. ISBN 0-7923-0487-X

M.A. Tsfasman and S.G. Vladuts: Algebraic Geometric Codes. 1991,668 pp. ISBN 0-7923-0727-5

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V.K. Kharchenko: Automorphisms and Derivations of Associative Rings. 1991,386 pp. ISBN 0-7923-1382-8

A. Yu. 0lshanskii: Geometry of Defining Relations in Groups. 1991, 513 pp. ISBN 0-7923-1394-1

F. Brackx and D. Constales: Computer Algebra with USP and REDUCE. An Introduction to Computer-Aided Pure Mathematics. 1992, 286 pp. ISBN 0-7923-1441-7

N.M. Korobov: Exponential Sums and their Applications. 1992, 2lO pp. ISBN 0-7923-1647-9

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E. Goles and S. Martinez: Statistical Physics. Automata Networks and Dynamical Systems. 1992,208 pp. ISBN 0-7923-1595-2

M.A. Frumkin: Systolic Computations. 1992, 320 pp. ISBN 0-7923-1708-4

J. Alajbegovic and J. Mockor: Approximation Theorems in Commutative Algebra. 1992, 330 pp. ISBN 0-7923-1948-6


LA. Faradzev, A.A. Ivanov, M.M. Klin and A.J. Woldar: Investigations in Algebraic Theory of Combinatorial Objects. 1993,516 pp. ISBN 0-7923-1927-3

I.E. Shparlinski: Computational and Algorithmic Problems in Finite Fields. 1992, 266 pp. ISBN 0-7923-2057-3

P. Feinsilver and R. Schott: Algebraic Structures and Operator Calculus. Vol. I. Representations and Probability Theory. 1993,224 pp. ISBN 0-7923-2116-2

A.G. Pinus: Boolean Constructions in Universal Algebras. 1993,350 pp. ISBN 0-7923-2117-0

V.V. Alexandrov and N.D. Gorsky: Image Representation and Processing. A Recursive Approach. 1993,200 pp. ISBN 0-7923-2136-7

L.A. Bokut' and G.P. Kukin: Algorithmic and Combinatorial Algebra. 1994, 384 pp. ISBN 0-7923-2313-0

Y. Bahturin: Basic Structures of Modem Algebra. 1993, 419 pp. ISBN 0-7923-2459-5

R. Krichevsky: Universal Compression and Retrieval. 1994,219 pp. ISBN 0-7923-2672-5

A. Elduque and H.C. Myung: Mutations of Alternative Algebras. 1994, 226 pp. ISBN 0-7923-2735-7

E. Gales and S. Martfuez (eds.): Cellular Automata, Dynamical Systems and Neural Networks. 1994, 189pp. ISBN 0-7923-2772-1

A.G. Kusraev and S.S. Kutateladze: Nonstandard Methods of Analysis. 1994,444 pp. ISBN 0-7923-2892-2

P. Feinsilver and R. Schott: Algebraic Structures and Operator Calculus. Vol. II. Special Functions and Computer Science. 1994, 148 pp. ISBN 0-7923-2921-X

V.M. Kopytov and N. Ya. Medvedev: The Theory of Lattice-Ordered Groups. 1994, 400 pp. ISBN 0-7923-3169-9

H. Inassaridze: Algebraic K-Theory. 1995,438 pp. ISBN 0-7923-3185-0

C. Mortensen: Inconsistent Mathematics. 1995, 155 pp. ISBN 0-7923-3186-9

R. Ablamowicz and P. Lounesto (eds.): Clifford Algebras and Spinor Structures. A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919-1992).1995,421 pp.

ISBN 0-7923-3366-7

W. Bosma and A. van der Poorten (eds.), Computational Algebra and Number Theory. 1995,336 pp. ISBN 0-7923-3501-5

A.L. Rosenberg: Noncommutative Algebraic Geometry and Representations of Quantized Algebras. 1995,316 pp. ISBN 0-7923-3575-9

L. Yanpei: Embeddability in Graphs. 1995.400 pp. ISBN 0-7923-3648-8

B.S. Stechkin and V.L Baranov: Extremal Combinatorial Problems and Their Applications. 1995.205 pp. ISBN 0-7923-3631-3


Y. Fong, H.E. Bell, W.-F. Ke, G. Mason and G. Pilz (eds.): Near-Rings and Near-Fields. 1995, 278 pp. ISBN 0-7923-3635-6

A. Facchini and C. Menini (eds.): Abelian Groups and Modules. (Proceedings of the Padova Conference, Padova, Italy, June 23-July 1, 1994). 1995, 537 pp. ISBN 0-7923-3756-5

D. Dikranjan and W. Tholen: Categorical Structure of Closure Operators. With Applications to Topology, Algebra and Discrete Mathematics. 1995,376 pp.

ISBN 0-7923-3772-7

A.D. Korshunov (ed.): Discrete Analysis and Operations Research. 1996,351 pp. ISBN 0-7923-3866-9

P. Feinsilver and R. Schott: Algebraic Structures and Operator Calculus. Vol. III: Representations of Lie Groups. 1996,238 pp. ISBN 0-7923-3834-0

M. Gasca and C.A. Micchelli (eds.): Total Positivity and Its Applications. 1996,528 pp. ISBN 0-7923-3924-X

W.O. Wallis (ed.): Computational and Constructive Design Theory. 1996,368 pp. ISBN 0-7923-4015-9

F. Cacace and G. Lamperti: Advanced ReiationalProgramming. 1996,410 pp. ISBN 0-7923-4081-7

N.M. Martin and S. Pollard: Closure Spaces and Logic. 1996,248 pp. ISBN 0-7923-4110-4

A.D. Korshunov (ed.): Operations Research and Discrete Analysis. 1997,340 pp.

W.O. Wallis: One-Factorizations. 1997,256 pp.

G. Weaver: Henkin-Keisler Models. 1997,266 pp.

ISBN 0-7923-4334-4

ISBN 0-7923-4323-9

ISBN 0-7923-4366-2

V.N. Kolokoltsov and V.P. Maslov: Idempotent Analysis andlts Applications. 1997,318 pp. ISBN 0-7923-4509-6

J.P. Ward: Quaternions and Cayley Numbers. Algebra and Applications. 1997,250 pp. ISBN 0-7923-4513-4

E.S. Ljapin and A.E. Evseev: The Theory of Partial Algebraic Operations. 1997, 245 pp. ISBN 0-7923-4609-2

S. Ayupov, A. Rakhimov and S. Usmanov: Jordan, Real and Lie Structures in Operator Algebras. 1997,235 pp. ISBN 0-7923-4684-X

A. Khrennikov: Non-ArchimedeanAnalysis: Quantum Paradoxes, Dynamical Systems and BiologicalModels. 1997,389 pp. ISBN 0-7923-4800-1

G. Saad and M.J. Thomsen (eds.): Nearrings, Nearfields and K-Loops. (Proceedings of the Conference on Nearrings and Nearfields, Hamburg, Germany. July 30-August 6, 1995). 1997, 458 pp. ISBN 0-7923-4799-4

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Differential and Difference Dimension Polynomials