Selected Titles in This Series

13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996

12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996

11 Jacques Dixmier, Enveloping algebras, 1996 Printing

10 Barry Simon, Representations of finite and compact groups, 1996

9 Dino Lorenzini, An invitation to arithmetic geometry, 1996

8 Winfried Just and Martin Weese, Discovering modern set theory. I: The basics, 1996

7 Gerald J. Janusz, Algebraic number fields, second edition, 1996

6 Jens Carsten Jantzen, Lectures on quantum groups, 1996

5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995

4 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994

3 William W. Adams and Philippe Loustaunau, An introduction to Grobner bases, 1994

2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity, 1993

1 Ethan Akin, The general topology of dynamical systems, 1993

http://dx.doi.org/10.1090/gsm/003

An Introduction to Grobner Bases

William W.Adams

Philippe Loustaunau

Graduate Studies in Mathematics

Volum e 3

Ivfffl 'J J America n Mathematica l Societ y ^^^

Editorial Board

James E. Humphreys Robion C. Kirby

Lance Small

2000 Mathematics Subject Classification. Primary 13P10.

ABSTRACT. Grobner bases are the primary tool for doing explicit computations in polynomial rings in many variables. In this book we give a leisurely introduction to the subject and its applications suitable for students with a little knowledge of abstract and linear algebra. The book contains not only the theory over fields, but also, the theory in modules and over rings.

Library of Congress Cataloging-in-Publication Data

Adams, William W., 1937-An introduction to Grobner bases/William W. Adams, Philippe Loustaunau.

p. cm. — (Graduate studies in mathematics, ISSN 1065-7339; 3) Includes bibliographical references and index. ISBN 0-8218-3804-0 1. Grobner bases. I. Loustaunau, Philippe, 1958- . II. Title. III. Series.

QA251.3.A32 1994 512/.4—dc20 94-19081

CIP

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to reprint-permissionQams.org.

© 1994 by the American Mathematical Society. All rights reserved. Reprinted with corrections in 1996.

The American Mathematical Society retains all rights except those granted to the United States Government.

Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

This publication was typeset by the authors, with assistance from the American Mathematical Society, using ^v^S-T^gX, the American Mathematical Society's TfeX macro system.

Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 15 14 13 12 11 10 09

To my wife Bizabeth and our daughters Ruth and Sarah

WWA

To my wife Yvonne and our children Bleen, Gareth, and Manon

PL

Contents

Preface ix

Chapter 1. Basic Theory of Grobner Bases 1 1.1. Introduction 1 1.2. The Linear Case 7 1.3. The One Variable Case 10 1.4. Term Orders 18 1.5. Division Algorithm 25 1.6. Grobner Bases 32 1.7. S-Polynomials and Buchberger's Algorithm 39 1.8. Reduced Grobner Bases 46 1.9. Summary 50

Chapter 2. Applications of Grobner Bases 53 2.1. Elementary Applications of Grobner Bases 53 2.2. Hilbert Nullstellensatz 61 2.3. Elimination 69 2.4. Polynomial Maps 79 2.5. Some Applications to Algebraic Geometry 90 2.6. Minimal Polynomials of Elements in Field Extensions 97 2.7. The 3-Color Problem 102 2.8. Integer Programming 105

Chapter 3. Modules and Grobner Bases 113 3.1. Modules 113 3.2. Grobner Bases and Syzygies 118 3.3. Improvements on Buchberger's Algorithm 124 3.4. Computation of the Syzygy Module 134 3.5. Grobner Bases for Modules 140 3.6. Elementary Applications of Grobner Bases for Modules 152 3.7. Syzygies for Modules 161 3.8. Applications of Syzygies 171 3.9. Computation of Horn 183

Vll

V l l l CONTENTS

3.10. Free Resolutions 194

Chapter 4. Grobner Bases over Rings 201 4.1. Basic Definitions 202 4.2. Computing Grobner Bases over Rings 212 4.3. Applications of Grobner Bases over Rings 225 4.4. A Primality Test 237 4.5. Grobner Bases over Principal Ideal Domains 246 4.6. Primary Decomposition in R[x] for R a PID 259

Appendix A. Computations and Algorithms 275

Appendix B. Well-ordering and Induction 277

References 279

List of Symbols 283

Index

Preface

We wrote this book with two goals in mind: (i) To give a leisurely and fairly comprehensive introduction to the definition

and construction of Grobner bases; (ii) To discuss applications of Grobner bases by presenting computational

methods to solve problems which involve rings of polynomials. This book is designed to be a first course in the theory of Grobner bases suitable for an advanced undergraduate or a beginning graduate student. This book is also suitable for students of computer science, applied mathematics, and engi­neering who have some acquaintance with modern algebra. The book does not assume an extensive knowledge of algebra. Indeed, one of the attributes of this subject is that it is very accessible. In fact, all that is required is the notion of the ring of polynomials in several variables (and rings in general in a few places, in particular in Chapter 4) together with the ideals in this ring and the concepts of a quotient ring and of a vector space introduced at the level of an undergraduate abstract and linear algebra course. Except for linear algebra, even these ideas are reviewed in the text. Some topics in the later sections of Chapters 2, 3, and 4 require more advanced material. This is always clearly stated at the beginning of the section and references are given. Moreover, most of this material is reviewed and basic theorems are stated without proofs.

The book can be read without ever "computing" anything. The theory stands by itself and has important theoretical applications in its own right. However, the reader will not fully appreciate the power of, or get insight into, the methods introduced in the book without actually doing some of the computations in the examples and the exercises by hand or, more often, using a Computer Algebra System (there are over 120 worked-out examples and over 200 exercises). Com­puting is useful in producing and analyzing examples which illustrate a concept already understood, or which one hopes will give insight into a less well under­stood idea or technique. But the real point here is that computing is the very essence of the subject. This is why Grobner basis theory has become a major research area in computational algebra and computer science. Indeed, Grobner basis theory is generating increasing interest because of its usefulness in pro-

IX

X PREFACE

viding computational tools which are applicable to a wide range of problems in mathematics, science, engineering, and computer science.

Grobner bases were introduced in 1965 by Bruno Buchberger1 [Bu65]. The basic idea behind the theory can be described as a generalization of the theory of polynomials in one variable. In the polynomial ring fe[x], where A: is a field, any ideal J can be generated by a single element, namely the greatest common divisor of the elements of / . Given any set of generators {/i,... , / s } C k[x] for J, one can compute (using the Euclidean Algorithm) a single polynomial d — gcd( / i , . . . , fs) such that J = ( / i , . . . , fs) = (d). Then a polynomial / € k[x] is in J if and only if the remainder of the division of / by d is zero. Grobner bases are the analog of greatest common divisors in the multivariate case in the following sense. A Grobner basis for an ideal J C fc[#i,... ,xn] generates I and a polynomial / G k[x\,... , xn] is in J if and only if the remainder of the division of / by the polynomials in the Grobner basis is zero (the appropriate concept of division is a central aspect of the theory).

This abstract characterization of Grobner bases is only one side of the theory. In fact it falls far short of the true significance of Grobner bases and of the real contribution of Bruno Buchberger. Indeed, the ideas behind the abstract characterization of Grobner bases had been around before Buchberger's work. For example, Macaulay [Mac] used some of these ideas at the beginning of the century to determine certain invariants of ideals in polynomial rings and Hironaka [Hi], in 1964, used similar ideas to study power series rings. But the true significance of Grobner bases is the fact that they can be computed. Bruno Buchberger's great contribution, and what gave Grobner basis theory the status as a subject in its own right, is his algorithm for computing these bases.

Our choice of topics is designed to give a broad introduction to the elemen­tary aspects and applications of the subject. As is the case for most topics in commutative algebra, Grobner basis theory can be presented from a geometric point of view. We have kept our presentation algebraic except in Sections 1.1 and 2.5. For those interested in a geometric treatment of some of the theory we recommend the excellent book by D. Cox, J. Little and D. O'Shea [CLOS]. The reader who is interested in going beyond the contents of this book should use our list of references as a way to access other sources. We mention in particular the books by T. Becker and V. Weispfenning [BeWe] and by B. Mishra [Mi] which contain a lot of material not in this book and have extensive lists of references on the subject.

Although this book is about computations in algebra, some of the issues which might be of interest to computer scientists are outside the scope of this book. For example, implementation of algorithms and their complexity are discussed only briefly in the book, primarily in Section 3.3. The interested reader should consult the references.

Wolfgang Grobner was Bruno Buchberger's thesis advisor.

PREFACE XI

In Chapter 1 we give the basic introduction to the concept of a Grobner basis and show how to compute it using Buchberger's Algorithm. We are careful to give motivations for the definition and algorithm by giving the familiar examples of Gaussian elimination for linear polynomials and the Euclidean Algorithm for polynomials in one variable. In Chapter 2 we present the basic applications to algebra and elementary algebraic geometry. We close the chapter with three specialized applications to algebra, graph theory, and integer programming. In Chapter 3 we begin by using the concept of syzygy modules to give an improve­ment of Buchberger's Algorithm. We go on to show how to use Grobner bases to compute the syzygy module of a set of polynomials (this is solving diophantine equations over polynomial rings). We then develop the theory of Grobner bases for finitely generated modules over polynomial rings. With these, we extend the applications from the previous chapter, give more efficient methods for com­puting some of the objects from the previous chapter, and conclude by showing how to compute the Hom functor and free resolutions. In Chapter 4 we develop the theory of Grobner bases for polynomial rings when the coefficients are now allowed to be in a general Noetherian ring and we show how to compute these bases (given certain computability conditions on the coefficient ring). We show how the theory simplifies when the coefficient ring is a principal ideal domain. We also give applications to determining whether an ideal is prime and to com­puting the primary decomposition of ideals in polynomial rings in one variable over principal ideal domains.

We give an outline of the section dependencies at the end of the Preface. After Chapter 1 the reader has many options in continuing with the rest of the book. There are exercises at the end of each section. Many of these exercises are computational in nature, some doable by hand while others require the use of a Computer Algebra System. Other exercises extend the theory presented in the book. A few harder exercises are marked with (*).

This book grew out of a series of lectures presented by the first author at the National Security Agency during the summer of 1991 and by the second author at the University of Calabria, Italy, during the summer of 1993.

We would like to thank many of our colleagues and students for their helpful comments and suggestions. In particular we would like to thank Beth Arnold, Ann Boyle, Garry Helzer, Karen Horn, Perpetua Kessy, Lyn Miller, Alyson Reeves, Elizabeth Rutman, Brian Williams, and Eric York. We also want to thank Sam Rankin, Julie Hawks and the AMS staff for their help in the prepa­ration of the manuscript.

Xll PREFACE

Chapter 2 Chapter 3

PREFACE xni

Chapter 4

3.2

4.5

4.6

4.1

4.2

4.3

4.4

| 2.4 | | 3.4 |

References

[AdGo] W.W. Adams and L.J. Goldstein, Introduction to Number Theory, Pren­tice Hall, Englewood Cliffs, NJ, 1976.

[AtMD] M.F. Atiyah and I.G. MacDonald, Introduction to Commutative Alge­bra, Addison-Wesley, Reading, MA, 1969.

[Ba] D. Bayer, The Division Algorithm and the Hilbert Scheme, Ph.D. Thesis, Harvard University, Cambridge, MA, 1982.

[BaSt88] D.A. Bayer and M. Stillman, On the complexity of computing syzygies, J. Symb. Comp. 6 (1988), 135-147.

[BaSt90] D.A. Bayer and M. Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra. User's manual.

[BeWe] T. Becker and V. Weispfenning, Grobner Bases: A Computational Ap­proach to Commutative Algebra, Springer Verlag, Berlin and New York, 1993.

[Bu65] B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. The­sis, Inst. University of Innsbruck, Innsbruck, Austria, 1965.

[Bu79] B. Buchberger, A criterion for detecting unnecessary reductions in the construction of Grobner bases, in EUROSAM'79, An International Sympo­sium on Symbolic and Algebraic Manipulation (E.W. Ng, ed.), Lecture Notes in Comput. Sci., vol. 72, Springer Verlag, Berlin and New York, 1979, 3-21.

[Bu83] B. Buchberger, A note on the complexity of constructing Grobner bases, in EUROCAL'83, European Computer Algebra Conference (J.A. van Hulzen ed.), Lecture Notes in Comput. Sci., vol. 162, Springer Verlag, Berlin and New York, 1983, 137-145.

[Bu85] B. Buchberger, Grobner bases: An algorithmic method in polynomial ideal theory, in Multidimensional Systems Theory (N.K. Bose ed.), Reidel, Dordrecht, 1985, 184-232.

[CGGLMW] B. Char, K. Geddes, G. Gonnet, B. Leong, M. Monogan, and S.M. Watt, Maple V Library Reference Manual, Springer Verlag, Berlin and New York, 1991.

[Coh] H. Cohen, A Course in Computational Algebraic Number Theory, Springer Verlag, Berlin and New York, 1993.

[CoTr] P. Conti and C. Traverso, Buchberger algorithm and integer programming,

279

280 REFERENCES

in Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes AAECC'9 (H.F. Mattson, T. Mora, and T.R.N Rao eds.), Lecture Notes in Comput. Sci., vol. 539, Springer verlag, Berlin and New York, 1991, 130-139.

[CLOS] D. Cox, J. Little, and D. O'Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer Verlag, Berlin and New York, 1992.

[Cz] S.R. Czapor, Grobner basis methods for solving algebraic equations, Ph.D. Thesis, University of Waterloo, Canada, 1988.

[EHV] D. Eisenbud, C. Huneke, and W. Vasconcelos, Direct methods for primary decomposition, Invent. Math. 110 (1992), 207-235.

[FGLM] J.C. Faugere, P. Gianni, D. Lazard, and T. Mora, Efficient computation of zero-dimensional Grobner bases by change of ordering, J. Symb. Comp. 16 (1993), 329-344.

[Ga] D. Gale, The Theory of Linear Economic Models, McGraw-Hill, New York, 1960.

[GTZ] P. Gianni, B. Trager, and G. Zacharias, Grobner bases and primary de­composition of polynomial ideals, J. Symb. Comp. 6 (1988), 149-167.

[GMNRT] A. Giovini, T. Mora, G. Niesi, L. Robbiano, and C. Traverso, uOne sugar cube, please" or selection strategies in the Buchberger algorithm, in Proc. International Symposium on Symbolic and Algebraic Computation ISSAC'91 (S.M. Watt ed.), ACM Press, New York, 1991, 49-54.

[GiNi] A. Giovini and G. Niesi, Co Co A user manual, 1990. [Go] L. J. Goldstein, Abstract Algebra, a First Course, Prentice Hall, Englewood

Cliffs, NJ, 1973. [He] I.N. Herstein, Topics in Algebra, Blaisdell, New York, 1964. [Hi] H. Hironaka, Resolution of singularities of an algebraic variety over a field

of characteristic zero, Ann. Math. 79 (1964), 109-326. [Him] T.W. Hungerford, Algebra, Springer Verlag, Berlin and New York, 1974. [Huy] D.T. Huynh, A superexponential lower bound for Grobner bases and

Church-Rosser commutative Thue systems, Inf. Contr. 68 (1986), 196-206. [JeSu] R.D. Jenks and R.S. Sutor, AXIOM: The Scientific Computation System,

Springer Verlag, Berlin and New York, 1992. [Lak] Y.N. Lakshman, On the complexity of computing Grobner bases for zero-

dimensional polynomials ideals, Ph.D. Thesis, Rensselaer Polytechnic Insti­tute, Troy, NY, 1990.

[Lan] S. Lang, Algebra, Addison-Wesley, Reading, MA, 1970. [Laz85] D. Lazard, Ideal bases and primary decomposition: Case of two vari­

ables, J. Symb. Comp. 1 (1985), 261-270. [Laz91] D. Lazard, Systems of algebraic equations (algorithms and complexity),

Computational Algebraic Geometry and Commutative Algebra (D. Eisenbud and L. Robbiano, eds.), Cambridge University Press, London, 1993.

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[Mac] F.S. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531-555.

[MaMe] E.W. Mayr and A.R Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals. Adv. Math 46 (1982), 305-329.

[Mi] B. Mishra, Algorithmic Algebra, Springer Verlag, Berlin and New York, 1993.

[M688] H.M. Moller, On the construction of Grobner bases using syzygies, J. Symb. Comp. 6 (1988), 345-359.

[M690] H.M. Moller, Computing syzygies a la Gauss Jordan, in Effective Meth­ods in Algebraic Geometry (T. Mora and C. Traverso, eds.), Progress in Mathematics 94, Birkhauser Verlag, Basel, 1990, 335-345.

[MoMo] H.M. Moller and T. Mora, New constructive methods in classical ideal theory, J. Algebra 100 (1986), 138-178.

[MoRo] T. Mora and L. Robbiano, The Grobner fan of an ideal, J. Symb. Comp. 6 (1988), 183-208.

[NZM] I. Niven, H.S. Zuckerman, and H.L. Montgomery, An Introduction to the Theory of Numbers, 5th ed., Wiley, New York, 1991.

[RoSw] L. Robbiano and M. Sweedler, Subalgebra bases, in Proc. Commutative Algebra Salvador (W. Burns and A. Simis eds.), Lecture Notes in Math., vol. 1430, Springer Verlag, Berlin and New York, 1988, 61-87.

[Schre] F.O. Schreyer, Die Berechnung von Syzygien mit dem verallgemeinerten Weierstrasschen Divisionsatz, Diplomarbeit, Hamburg, 1980.

[Schri] A. Schrijver, Theory of Linear and Integer Programming, Wiley, Chich­ester, NY, 1986.

[Se] A. Seidenberg, Constructions in algebra, Trans. Amer. Math. Soc. 197 (1974), 272-313.

[ShSw] D. Shannon and M. Sweedler, Using Grobner bases to determine al­gebra membership, split surjective algebra homomorphisms determine bira-tional equivalence, J. Symb. Comp. 6 (1988), 267-273.

[Sz] G. Szekeres, A canonical basis for the ideals of a polynomial domain, Am. Math. Mon. 59 (1952), 379-386.

[Wo] S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, Addison-Wesley, Reading, MA, 1991.

[Za] G. Zacharias, Generalized Grobner bases in commutative polynomial rings, Bachelor's Thesis, MIT, Cambridge, MA, 1978.

List of Symbols

N natural numbers Z ring of integers Z n ring of integers modulo n Q field of rational numbers M field of real numbers C field of complex numbers T n set of power products in the variables x\,... , xn

k a field kn affine space k[x\,... ,xn] ring of polynomials in the variables sci, . . . , x n with

coefficients in the field k {fii • • • > fs) ideal (submodule) generated by / i , . . . , / s

/ = g (mod J) / congruent to g modulo / k[x\,... , xn]/I quotient ring of k[x\,... , xn] by the ideal J / -f- / coset of / modulo J V(S) variety in kn defined by the set of polynomials S V{fi,... , fa) variety in kn defined by the polynomials / i , . . . , fs

I(V) ideal in k[x\,... , xn] defined by V C kn

(S) ideal generated by the polynomials in the set S gcd greatest common divisor lcm least common multiple dim*; (L) dimension of the A;-vector space L l t ( / ) leading term of / lp(/) leading power product of / lc(/) leading coefficient of / X <* <rai . . . T a «

f — > h f reduces to h modulo G in one step

f —>+ h f reduces to h modulo G

283

284 LIST OF SYMBOLS

f ^ h U(S)

S(f,g) NGU)

VK(S) k

inj I: J im(4>) ker(<j>) k[h,... k[F] K\Xi, . . . , Xn)

k(a) k(ai,... ,an)

M/N

fs]

6 1 , . . . , en

[fi ' • • [A -Syz( / i , . . . Syz(F)

M/) Fee [F\G] *S ann(M) F®G Hom(M, AT) (U) S-XA AP

Aa

fs ] fs] fs)

,OLn

f reduces to h and h = f — Xg leading term ideal (submodule) defined by S S-polynomial of / and g normal form of / with respect to G variety in Kn denned by S C k[xi,... ,xn] algebraic closure of the field k radical of the ideal J intersection of J and J ideal quotient of J by J image of the map 0 kernel of the map (j) A:-algebra generated by the polynomials / 1 , . . . , fs

fc-algebra generated by the polynomials in the set F rational function field field extension of k obtained by adjoining a field extension of k obtained by adjoining a\,... set of column vectors with entries in the ring A quotient module of M by N isomorphic to standard basis for the free module Am

1 x 5 matrix whose entries are polynomials / 1 , . . m x 5 matrix whose columns are vectors / l 5 . . . s v z v gy module of the matrix [ f\ • • • fs ] syzygy module of the matrix F leading monomial of the vector / direct sum of the matrices F and G concatenation of the matrices F and G transpose of the matrix S annihilator of the module M tensor product of the matrices F and G set of all A-module homomorphisms between M and N module generated by the columns of the matrix U localization of the ring A at the multiplicative set S localization of the ring A at the prime ideal P localization of the ring A at the set {1, g, g2, # 3 , . . . }

• 1 Js

,fs

Index

affine algebra, 84, 180 space, 2

algebra, 232 affine, 84, 180 First Isomorphism Theorem, 80 homomorphism, 79, 92, 232 membership problem, 39

algebraic closure, 62 element, 97 geometry, 90

algebraically closed field, 62 algorithm, 275

crit2, 130 division

module case, 145 multivariable case, 28 one variable case, 12 ring case, 207

Euclidean, 14 Grobner basis

Buchberger's, field case, 43 Buchberger's, module case, 149 improved Buchberger's, 129 PID case, 250 ring case, 216 ring case using Moller's techniqi

219 primality test, 244

annihilator, 177, 217 ascending chain condition, 6

basis for A m / M , 155 for k[xi,... , x n ] / / , 58 in Am, 114 standard in Am, 114

Bayer, 102 Buchberger, viii, 39, 113, 124, 131

algorithm, ix, 43

algorithm for modules, 149 improved algorithm, 129 Theorem, 40, 48

CoCoA, 275 column vector, 113 commutative ring, 1 computation, 275 Computer Algebra System, vii, 275 confluence relation, 37 congruence, 5 Conti, 105 coset, 5

field case multivariable case, 57 one variable case, 15

module case, 155 ring case, 227

cost function, 105 cri t l , 128, 152, 258 crit2, 128, 168, 224

deglex, 20 degrevlex, 20 determine, 5, 53 Dickson's Lemma, 23 direct sum of matrices, 174 division

for vectors, 141 multivariable case, 25, 26, 27 one variable case, 11

division algorithm field case

multivariable case, 28 one variable case, 12

module case, 145 ring case, 207

effective, 53 effective coset representatives, 226

285

286 INDEX

elimination Gauss-Jordan, 2 ideal

field case, 70 ring case, 229

module, 156 order, 69, 229

Euclidean Algorithm, 14 exact sequence, 185

short, 184 explicitly given module, 117 Ext functor, 194, 199

Faugere, 68 field

algebraic closure, 62 algebraically closed, 62

First Isomorphism Theorem algebra case, 80 module case, 117

free module, 114 free resolution, 195

Gauss-Jordan elimination, 2 generating set, 3, 114

finite, 6 Gianni, 68, 237 global dimension, 196 graph, 102 graph of a map, 91 greatest common divisor, 13, 15, 71 Grobner basis

algorithm Buchberger's, field case, 43 Buchberger's, module case, 148 improved Buchberger's, 129 PID case, 250 ring case, 216 ring case using Moller's technique,

219 for ideal, field case, 32, 34

minimal, 47 reduced, 48 universal, 50 using syzygies, 121

for ideal, ring case, 201, 208 minimal, 211 minimal strong, 251, 254, 262 strong, 251

for module, 147 reduced, 150

Hensel's Lemma, 267 Hilbert

Basis Theorem, 5, 6 Nullstellensatz, 62

Horn, 183 homogeneous

ideal, 49 component, 22 polynomial, 22, 38 syzygy, 121, 212

homogenization, 38 homological algebra, 185 homomorphism

algebra, 79, 92, 232 image of, 80 kernel of, 80, 232 module, 116

ideal, 3 field case

elimination, 70 Grobner basis for, 32, 34 homogeneous, 49 intersection, 70, 172, 175 leading term, 32 membership problem, 5, 53 monomial, 23 of relations, 80 quotient, 72 quotient for modules, 157, 177, 182

irreducible, 261 leading term, 32, 206 monomial, 23 P- primary, 261 primary, 260 prime, 61, 237, 259 principal, 13, 246 radical of, 62, 259

INDEX 287

ring case elimination, 229 Grobner basis for, 201, 208 intersection, 230 leading term, 206 membership problem, 225 of relations, 232 quotient, 217, 231

saturation of, 238 zero-dimensional, 64, 262

image of a homomorphism, 80 implicitization, 91 induction, 277 integer programming, 105 inter-reduced, 131 intersection

of ideals field case, 70, 172, 175 ring case, 230

of modules, 157, 175, 176, 178 inverse in k[xi,... , x n ] / 7 , 59, 182 irreducible ideal, 261 isomorphic varieties, 94 isomorphism, module, 116

kernel of a homomorphism, 80, 232

Lakshman, 77 Lazard, 68, 201, 254, 259 leading coefficient

multi-variable case, 21, 202 one variable case, 10 vector case, 143

leading monomial, 143 leading power product, 21, 202 leading term

ideal, 32, 206 module, 147 mul t ivar ia te case, 21, 202 one variable case, 10 vector case, 143

least common multiple of polynomials, 71 of vectors, 147

lex, 19

linear equations are solvable, 204, 262 local ring, 238 localization, 89 long division, 10

membership problem algebra, 39 ideal over field, 5, 53 ideal over ring, 225 module, 153

minimal Grobner basis, field case, 47 Grobner basis, ring case, 211 strong Grobner basis, 251 with respect to F , 205

minimal polynomial, 97 module, 113

elimination, 156 explicitly given, 117 First Isomorphism Theorem, 117 free, 114 Grobner basis for, 147 homomorphism, 116 ideal quotient, 157, 177, 182 intersection, 157, 175, 176, 178 isomorphism, 116 leading term module, 147 membership problem, 153 Noetherian, 116 normal form, 155 quotient, 116 reduction in, 143 S-polynomial, 148 syzygy, 118

Moller, 170, 201, 216, 250 monic polynomial, 13 monoid, 37 monomial

ideal, 23 in modules, 141 leading, 143

Mora, 23, 68 multiplicative set, 238

Noetherian

288 INDEX

module, 116 ring, 6, 259

normal form polynomial case

over a field, 57 over a ring, 227

vector case, 155 normal selection strategy, 130 Nullstellensatz, 62

order, 18 degree lexicographic (deglex), 19 degree reverse lexicographic

(degrevlex), 20 elimination, 69 induced by a matrix, 166 lexicographic (lex), 19 position over term (POT), 142 term, 18, 140 term over position (TOP), 142 total, 18, 277 well-ordering, 18, 277

P-primary ideal, 261 parametrized variety, 91 polynomial, 1

elementary symmetric, 25 homogeneous, 22, 38 minimal, 97 monic, 13 normal form, 57, 227 reduced, 27 square free, 75 symmetric, 25, 88

position over term (POT), 142 power product, 1, 18

leading, 21, 202 presentation, 117, 119, 195 primality test, 244 primary

component, 261 decomposition, 261 ideal, 260

prime component, 261

ideal, 61, 237, 259 principal ideal, 13, 246 Principal Ideal Domain (PID), 13, 246 projection map, 90 pullback, 182

quotient module, 116 ring, 5

radical of an ideal, 62, 259 reduced

Grobner basis ideal case over fields, 48 module case, 150

polynomial over a field, 27 totally, 227 vector, 144

reduction linear case, 8 module case, 143 multivariable case over a field, 25, 26,

27 multivariable case over a ring, 203, 205 one variable case, 11 strong, 252, 258

remainder module case, 144 multivariable case over a field, 27 one variable case over a field, 11

ring commutative, 1 local, 238 localization, 89 Noetherian, 6, 259 of fractions, 238 quotient, 5

Robbiano, 23, 39 row echelon form, 2

S-basis, 247 S-polynomial

module case, 148 multivariable case over a field, 40, 121,

124

PID case, 249 ring case, 213

SAGBI basis, 39 saturated set, 213, 226 saturation of a set, 213 saturation of an ideal, 238 Schreyer, 165 Seidenberg, 78 Shannon, 82, 88 square free polynomial, 75 standard basis, 32 strong

Grobner basis, 251 minimal Grobner basis, 251, 254, reduction, 252, 258

subalgebra, 39 submodule, 114 Sweedler, 39, 82, 88 symmetric polynomial, 25, 88 syzygy

and Grobner bases, 121 applications of, 171 homogeneous, 121, 212 of a matrix, 161

module of, 161 of vectors over a field, 118

module of, 118, 134 of vectors over a ring, 212

module of, 232, 246 Szekeres, 254

tensor product of matrices, 189 term

module case, 140 order

module case, 140 polynomial case, 18

polynomial case, 1, 18 term over position (TOP), 142 three color problem, 102 total order, 18, 277 totally reduced, 227 Trager, 237 transpose matrix, 176 Traverso, 105

INDEX 289

universal Grobner basis, 50

variety, 2, 61, 90 isomorphic, 94 parametrized, 91

vector space, 1

well-ordering, 18, 277

Zacharias, 201, 226, 237 Zariski closure, 90 zero divisor, 61 zero-dimensional ideal, 64, 262