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Advantages and Disadvantages
of Differential Standard Bases
Alexey Zobnin
Joint research with
M.V. Kondratieva, E.V. Pankratiev and D. Trushin
Department of Mechanics and Mathematics
Moscow State University
e-mail: [email protected]
CADE�2007
February 20, 2007
Contents
1 Introduction
2 Admissible orderings
on di�erential monomials
3 Finiteness criteria
for di�erential standard bases
Contents
1 Introduction
2 Admissible orderings
on di�erential monomials
3 Finiteness criteria
for di�erential standard bases
Di�erential rings
An ordinary di�erential ring (δ-ring)is a commutative ring R with unit
equipped with a derivation operator δ:δ(a + b) = δa + δb; δ(ab) = a δb + b δa.
Any derivative operator can be written as θ = δn.
Θ := {δn | n > 0}.I /R is a di�erential ideal (δ-ideal) if δI ⊂ I.
Let F ⊂ R. Denote by [F ] the minimal δ-ideal in Rcontaining F .
{F} is minimal radical (perfect) di�erential ideal in Rcontaining F .
Di�erential rings
An ordinary di�erential ring (δ-ring)is a commutative ring R with unit
equipped with a derivation operator δ:δ(a + b) = δa + δb; δ(ab) = a δb + b δa.
Any derivative operator can be written as θ = δn.
Θ := {δn | n > 0}.I /R is a di�erential ideal (δ-ideal) if δI ⊂ I.
Let F ⊂ R. Denote by [F ] the minimal δ-ideal in Rcontaining F .
{F} is minimal radical (perfect) di�erential ideal in Rcontaining F .
Di�erential rings
An ordinary di�erential ring (δ-ring)is a commutative ring R with unit
equipped with a derivation operator δ:δ(a + b) = δa + δb; δ(ab) = a δb + b δa.
Any derivative operator can be written as θ = δn.
Θ := {δn | n > 0}.I /R is a di�erential ideal (δ-ideal) if δI ⊂ I.
Let F ⊂ R. Denote by [F ] the minimal δ-ideal in Rcontaining F .
{F} is minimal radical (perfect) di�erential ideal in Rcontaining F .
Di�erential rings
An ordinary di�erential ring (δ-ring)is a commutative ring R with unit
equipped with a derivation operator δ:δ(a + b) = δa + δb; δ(ab) = a δb + b δa.
Any derivative operator can be written as θ = δn.
Θ := {δn | n > 0}.I /R is a di�erential ideal (δ-ideal) if δI ⊂ I.
Let F ⊂ R. Denote by [F ] the minimal δ-ideal in Rcontaining F .
{F} is minimal radical (perfect) di�erential ideal in Rcontaining F .
Di�erenital polynomials
Let F be a δ-�eld.
A ring of di�erential polynomials F{y}is a δ-ring F [y0, y1, y2, . . .] such that δyi = yi+1.
We suppose that charF = 0.We also assume that δF = 0.
A di�erential monomial M is an expression of the formn∏
i=0yαi
i .
wt M :=n∑
i=0αi is the weight of M .
M is the set of all di�erential monomials.
Di�erenital polynomials
Let F be a δ-�eld.
A ring of di�erential polynomials F{y}is a δ-ring F [y0, y1, y2, . . .] such that δyi = yi+1.
We suppose that charF = 0.We also assume that δF = 0.
A di�erential monomial M is an expression of the formn∏
i=0yαi
i .
wt M :=n∑
i=0αi is the weight of M .
M is the set of all di�erential monomials.
Di�erenital polynomials
Let F be a δ-�eld.
A ring of di�erential polynomials F{y}is a δ-ring F [y0, y1, y2, . . .] such that δyi = yi+1.
We suppose that charF = 0.We also assume that δF = 0.
A di�erential monomial M is an expression of the formn∏
i=0yαi
i .
wt M :=n∑
i=0αi is the weight of M .
M is the set of all di�erential monomials.
In�nitely generated δ-ideals
Unlike ordinary polynomial rings,
δ-polynomial rings are not Noetherian.
Example
y2n+1 /∈ [y2
0, y21, . . . , y
2n] for each n > 0,
thus [y20, y
21, y
22, . . .] is in�netely generated.
Ritt-Raudenbush theorem in F{y} is similar
to Hilbert Basis theorem:
The ring F{y} satis�es the descending chain condition
for radical δ-ideals.
In�nitely generated δ-ideals
Unlike ordinary polynomial rings,
δ-polynomial rings are not Noetherian.
Example
y2n+1 /∈ [y2
0, y21, . . . , y
2n] for each n > 0,
thus [y20, y
21, y
22, . . .] is in�netely generated.
Ritt-Raudenbush theorem in F{y} is similar
to Hilbert Basis theorem:
The ring F{y} satis�es the descending chain condition
for radical δ-ideals.
Testing the membership in δ-ideals
Theorem (Gallo, Mishra, Ollivier, 1994)
The membership problem is undecidable for in�nitely generated
di�erential ideals in rings of di�erential polynomials.
Yu. Blinkov claimed that there exists an algorithm that tests
the membership in arbitrary �nitely generated di�erential ideal.
Algorithmic solutions to membership problem is known for:
1 radical di�erential ideals;
2 isobaric ideals (w.r.t. some weight function);
3 ideals that have �nite di�erential standard basis.
Testing the membership in δ-ideals
Theorem (Gallo, Mishra, Ollivier, 1994)
The membership problem is undecidable for in�nitely generated
di�erential ideals in rings of di�erential polynomials.
Yu. Blinkov claimed that there exists an algorithm that tests
the membership in arbitrary �nitely generated di�erential ideal.
Algorithmic solutions to membership problem is known for:
1 radical di�erential ideals;
2 isobaric ideals (w.r.t. some weight function);
3 ideals that have �nite di�erential standard basis.
Testing the membership in δ-ideals
Theorem (Gallo, Mishra, Ollivier, 1994)
The membership problem is undecidable for in�nitely generated
di�erential ideals in rings of di�erential polynomials.
Yu. Blinkov claimed that there exists an algorithm that tests
the membership in arbitrary �nitely generated di�erential ideal.
Algorithmic solutions to membership problem is known for:
1 radical di�erential ideals;
2 isobaric ideals (w.r.t. some weight function);
3 ideals that have �nite di�erential standard basis.
Testing the membership in δ-ideals
Theorem (Gallo, Mishra, Ollivier, 1994)
The membership problem is undecidable for in�nitely generated
di�erential ideals in rings of di�erential polynomials.
Yu. Blinkov claimed that there exists an algorithm that tests
the membership in arbitrary �nitely generated di�erential ideal.
Algorithmic solutions to membership problem is known for:
1 radical di�erential ideals;
2 isobaric ideals (w.r.t. some weight function);
3 ideals that have �nite di�erential standard basis.
Testing the membership in δ-ideals
Theorem (Gallo, Mishra, Ollivier, 1994)
The membership problem is undecidable for in�nitely generated
di�erential ideals in rings of di�erential polynomials.
Yu. Blinkov claimed that there exists an algorithm that tests
the membership in arbitrary �nitely generated di�erential ideal.
Algorithmic solutions to membership problem is known for:
1 radical di�erential ideals;
2 isobaric ideals (w.r.t. some weight function);
3 ideals that have �nite di�erential standard basis.
Admissible orderings (AO)
De�nition
A linear order ≺ on M is called admissible if:
O1. p ≺M q =⇒ p · s ≺M q · s ∀ p, q, s ∈ M;
O2. 1 4M p ∀ p ∈ M;
O3. the restriction of ≺ to the set of di�erential variables
is an admissible ranking
(in our case there is only one ranking: yi ≺ yi+1).
Previous researchers consider admissible orderings that satisfy
additional strong properties. E.g., G. Carr�a Ferro and
F. Ollivier considered only Lex and DegLex.
Theorem
O1�O3 are su�cient to guarantee that M is well ordered.
Admissible orderings (AO)
De�nition
A linear order ≺ on M is called admissible if:
O1. p ≺M q =⇒ p · s ≺M q · s ∀ p, q, s ∈ M;
O2. 1 4M p ∀ p ∈ M;
O3. the restriction of ≺ to the set of di�erential variables
is an admissible ranking
(in our case there is only one ranking: yi ≺ yi+1).
Previous researchers consider admissible orderings that satisfy
additional strong properties. E.g., G. Carr�a Ferro and
F. Ollivier considered only Lex and DegLex.
Theorem
O1�O3 are su�cient to guarantee that M is well ordered.
Admissible orderings (AO)
De�nition
A linear order ≺ on M is called admissible if:
O1. p ≺M q =⇒ p · s ≺M q · s ∀ p, q, s ∈ M;
O2. 1 4M p ∀ p ∈ M;
O3. the restriction of ≺ to the set of di�erential variables
is an admissible ranking
(in our case there is only one ranking: yi ≺ yi+1).
Previous researchers consider admissible orderings that satisfy
additional strong properties. E.g., G. Carr�a Ferro and
F. Ollivier considered only Lex and DegLex.
Theorem
O1�O3 are su�cient to guarantee that M is well ordered.
Di�erential standard bases (DSB)Carr�a Ferro and Ollivier's de�nition
The set S ⊂ I is called a di�erential standard basis of an ideal Iw.r.t. ≺, if ΘS is an (in�nite) algebraic Gr�obner basis
of the ideal I in the ring F [y, y1, y2, . . .].
Di�erential reduction w.r.t. �nite or parametric DSB
of a δ-ideal solves the membership problem.
Examples
Linear ideals always have �nite DSB.
The ideal [y2 + y + 1] has �nite DSB {y2 + y + 1, y1}.Ideals [yn] have �nite DSB {yn} w.r.t. DegRevLex ordering.Previous researchers have supposed
that the ideal [yn] has no �nite DSB!
Di�erential standard bases (DSB)Carr�a Ferro and Ollivier's de�nition
The set S ⊂ I is called a di�erential standard basis of an ideal Iw.r.t. ≺, if ΘS is an (in�nite) algebraic Gr�obner basis
of the ideal I in the ring F [y, y1, y2, . . .].
Di�erential reduction w.r.t. �nite or parametric DSB
of a δ-ideal solves the membership problem.
Examples
Linear ideals always have �nite DSB.
The ideal [y2 + y + 1] has �nite DSB {y2 + y + 1, y1}.Ideals [yn] have �nite DSB {yn} w.r.t. DegRevLex ordering.Previous researchers have supposed
that the ideal [yn] has no �nite DSB!
Di�erential standard bases (DSB)Carr�a Ferro and Ollivier's de�nition
The set S ⊂ I is called a di�erential standard basis of an ideal Iw.r.t. ≺, if ΘS is an (in�nite) algebraic Gr�obner basis
of the ideal I in the ring F [y, y1, y2, . . .].
Di�erential reduction w.r.t. �nite or parametric DSB
of a δ-ideal solves the membership problem.
Examples
Linear ideals always have �nite DSB.
The ideal [y2 + y + 1] has �nite DSB {y2 + y + 1, y1}.Ideals [yn] have �nite DSB {yn} w.r.t. DegRevLex ordering.Previous researchers have supposed
that the ideal [yn] has no �nite DSB!
Di�erential standard bases (DSB)Carr�a Ferro and Ollivier's de�nition
The set S ⊂ I is called a di�erential standard basis of an ideal Iw.r.t. ≺, if ΘS is an (in�nite) algebraic Gr�obner basis
of the ideal I in the ring F [y, y1, y2, . . .].
Di�erential reduction w.r.t. �nite or parametric DSB
of a δ-ideal solves the membership problem.
Examples
Linear ideals always have �nite DSB.
The ideal [y2 + y + 1] has �nite DSB {y2 + y + 1, y1}.Ideals [yn] have �nite DSB {yn} w.r.t. DegRevLex ordering.Previous researchers have supposed
that the ideal [yn] has no �nite DSB!
Examples of �nite and parametric DSB (1)
DegRevLex ordering
Ideals [(y + c)n], n > 1
(y + c)n.
The ideal [yy1]y0 y1;
y20 yk, k > 2.
Ideals [yn1 + y0], n > 3
nn yn−10 yn+k+(−1)n
n∏l=1
(n(n−2)+(n−1)(k−l)
)yk, k > 0;
(nk − n− k) y1 yk + n y0 yk+1, k > 1.
Examples of �nite and parametric DSB (1)
DegRevLex ordering
Ideals [(y + c)n], n > 1
(y + c)n.
The ideal [yy1]y0 y1;
y20 yk, k > 2.
Ideals [yn1 + y0], n > 3
nn yn−10 yn+k+(−1)n
n∏l=1
(n(n−2)+(n−1)(k−l)
)yk, k > 0;
(nk − n− k) y1 yk + n y0 yk+1, k > 1.
Examples of �nite and parametric DSB (1)
DegRevLex ordering
Ideals [(y + c)n], n > 1
(y + c)n.
The ideal [yy1]y0 y1;
y20 yk, k > 2.
Ideals [yn1 + y0], n > 3
nn yn−10 yn+k+(−1)n
n∏l=1
(n(n−2)+(n−1)(k−l)
)yk, k > 0;
(nk − n− k) y1 yk + n y0 yk+1, k > 1.
Examples of �nite and parametric DSB (2)
Lex ordering
The ideal [y21 + y0]
y21 + y0;
2 y0 y2 + y0;
4 y32 + 4 y2
2 + y2;
2 y2 y3 + y3;
y2k, k > 3.
Ideals [yn1 + y0], n > 3
yn1 + y0;
n y0 y2 − y21;
n yn−21 y2
2 + y2;
y3 − n(n− 2) yn−31 y3
2.
Ideals [(y1 + c)2 + y0], c 6= 0
y21 + 2 c y1 + y0 + c2 = (y1 + c)2 + y0;
2 y0 y2 + c y1 + y0 + c2 = y0 (2 y2 + 1) + c y1 + c2;
2 c y3 + y2 (2 y2 + 1)2.
Examples of �nite and parametric DSB (3)
Lex ordering
Ideals [yn2 + y0], n > 5
yn2 + y0;
n y0 y3 − y1 y2;
n(n− 1) yn−22 y3
3 + y2 y3 − y1 y4;
n(n− 1) yn−32 y3
3 + n yn−22 y3 y4 + y3.
n(n− 1) yn−32 y2
3 y4 + n yn−22 y2
4 + y4.
n(n− 1)2(n− 3) yn−52 y5
3 + n(n− 1)(4n− 7) yn−42 y3
3 y4 ++n(3n− 4) yn−3
2 y3 y24 − y5.
Ideals [fmn], where fmn = (y1 + 1)m − yn, m, n > 1
[fmn] has �nite lex DSB ⇐⇒ either m = 1, or m - n.
Examples of �nite and parametric DSB (3)
Lex ordering
Ideals [yn2 + y0], n > 5
yn2 + y0;
n y0 y3 − y1 y2;
n(n− 1) yn−22 y3
3 + y2 y3 − y1 y4;
n(n− 1) yn−32 y3
3 + n yn−22 y3 y4 + y3.
n(n− 1) yn−32 y2
3 y4 + n yn−22 y2
4 + y4.
n(n− 1)2(n− 3) yn−52 y5
3 + n(n− 1)(4n− 7) yn−42 y3
3 y4 ++n(3n− 4) yn−3
2 y3 y24 − y5.
Ideals [fmn], where fmn = (y1 + 1)m − yn, m, n > 1
[fmn] has �nite lex DSB ⇐⇒ either m = 1, or m - n.
Contents
1 Introduction
2 Admissible orderings
on di�erential monomials
3 Finiteness criteria
for di�erential standard bases
Monomial matrices
Any monomial ordering can be speci�ed by an m× (k + 1)monomial matrix M with real entries and lexicographically
positive columns such that KerQ M = {0}:
M
( α0
...αk
)≺lex M
( β0
...βk
)⇐⇒ yα0
0 . . . yαkk ≺ yβ0
0 . . . yβkk .
A set of monomial matrices {Mk} is called concordant if Mk−1
can be obtained from Mk by deleting the rightmost column and
then by deleting a row of zeroes, if it exists.
Theorem
Any admissible ordering on di�erential monomials can be
speci�ed by a concordant set of monomial matrices or,
equivalently, by an in�nite monomial matrix.
Monomial matrices
Any monomial ordering can be speci�ed by an m× (k + 1)monomial matrix M with real entries and lexicographically
positive columns such that KerQ M = {0}:
M
( α0
...αk
)≺lex M
( β0
...βk
)⇐⇒ yα0
0 . . . yαkk ≺ yβ0
0 . . . yβkk .
A set of monomial matrices {Mk} is called concordant if Mk−1
can be obtained from Mk by deleting the rightmost column and
then by deleting a row of zeroes, if it exists.
Theorem
Any admissible ordering on di�erential monomials can be
speci�ed by a concordant set of monomial matrices or,
equivalently, by an in�nite monomial matrix.
Monomial matrices
Any monomial ordering can be speci�ed by an m× (k + 1)monomial matrix M with real entries and lexicographically
positive columns such that KerQ M = {0}:
M
( α0
...αk
)≺lex M
( β0
...βk
)⇐⇒ yα0
0 . . . yαkk ≺ yβ0
0 . . . yβkk .
A set of monomial matrices {Mk} is called concordant if Mk−1
can be obtained from Mk by deleting the rightmost column and
then by deleting a row of zeroes, if it exists.
Theorem
Any admissible ordering on di�erential monomials can be
speci�ed by a concordant set of monomial matrices or,
equivalently, by an in�nite monomial matrix.
Matrix speci�cation of admissible orderings (1)
Lex
(1),
(0 11 0
),
0 0 11 0
1 0
,
0 0 0 1
1 01 0
1 0
,
0 0 0 0 1
1 01 0
1 01 0
, . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 . . .. . . 1 . . .. . . 1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . . . .
1 . . . . . .1 . . . . . .
.
Matrix speci�cation of admissible orderings (1)
Lex
(1),
(0 11 0
),
0 0 11 0
1 0
,
0 0 0 1
1 01 0
1 0
,
0 0 0 0 1
1 01 0
1 01 0
, . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 . . .. . . 1 . . .. . . 1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . . . .
1 . . . . . .1 . . . . . .
.
Matrix speci�cation of admissible orderings (1)
Lex
(1),
(0 11 0
),
0 0 11 0
1 0
,
0 0 0 1
1 01 0
1 0
,
0 0 0 0 1
1 01 0
1 01 0
, . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 . . .. . . 1 . . .. . . 1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . . . .
1 . . . . . .1 . . . . . .
.
Matrix speci�cation of admissible orderings (1)
Lex
(1),
(0 11 0
),
0 0 11 0
1 0
,
0 0 0 1
1 01 0
1 0
,
0 0 0 0 1
1 01 0
1 01 0
, . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 . . .. . . 1 . . .. . . 1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . . . .
1 . . . . . .1 . . . . . .
.
Matrix speci�cation of admissible orderings (1)
Lex
(1),
(0 11 0
),
0 0 11 0
1 0
,
0 0 0 1
1 01 0
1 0
,
0 0 0 0 1
1 01 0
1 01 0
, . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 . . .. . . 1 . . .. . . 1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . . . .
1 . . . . . .1 . . . . . .
.
Matrix speci�cation of admissible orderings (1)
Lex
(1),
(0 11 0
),
0 0 11 0
1 0
,
0 0 0 1
1 01 0
1 0
,
0 0 0 0 1
1 01 0
1 01 0
, . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 . . .. . . 1 . . .. . . 1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . . . .
1 . . . . . .1 . . . . . .
.
Matrix speci�cation of admissible orderings (2)
DegLex
(1),
(1 10 1
),
1 1 10 0 1
1 0
,
1 1 1 10 0 0 1
1 01 0
,
1 1 1 1 10 0 0 0 1
1 01 0
1 0
, . . . .
1 1 1 . . . 1 1 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 1 . . .
. . . 1 . . .
. . . 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 . . . . . .1 . . . . . .
.
Matrix speci�cation of admissible orderings (2)
DegLex
(1),
(1 10 1
),
1 1 10 0 1
1 0
,
1 1 1 10 0 0 1
1 01 0
,
1 1 1 1 10 0 0 0 1
1 01 0
1 0
, . . . .
1 1 1 . . . 1 1 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 1 . . .
. . . 1 . . .
. . . 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 . . . . . .1 . . . . . .
.
Matrix speci�cation of admissible orderings (2)
DegLex
(1),
(1 10 1
),
1 1 10 0 1
1 0
,
1 1 1 10 0 0 1
1 01 0
,
1 1 1 1 10 0 0 0 1
1 01 0
1 0
, . . . .
1 1 1 . . . 1 1 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 1 . . .
. . . 1 . . .
. . . 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 . . . . . .1 . . . . . .
.
Matrix speci�cation of admissible orderings (2)
DegLex
(1),
(1 10 1
),
1 1 10 0 1
1 0
,
1 1 1 10 0 0 1
1 01 0
,
1 1 1 1 10 0 0 0 1
1 01 0
1 0
, . . . .
1 1 1 . . . 1 1 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 1 . . .
. . . 1 . . .
. . . 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 . . . . . .1 . . . . . .
.
Matrix speci�cation of admissible orderings (2)
DegLex
(1),
(1 10 1
),
1 1 10 0 1
1 0
,
1 1 1 10 0 0 1
1 01 0
,
1 1 1 1 10 0 0 0 1
1 01 0
1 0
, . . . .
1 1 1 . . . 1 1 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 1 . . .
. . . 1 . . .
. . . 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 . . . . . .1 . . . . . .
.
Matrix speci�cation of admissible orderings (2)
DegLex
(1),
(1 10 1
),
1 1 10 0 1
1 0
,
1 1 1 10 0 0 1
1 01 0
,
1 1 1 1 10 0 0 0 1
1 01 0
1 0
, . . . .
1 1 1 . . . 1 1 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 1 . . .
. . . 1 . . .
. . . 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 . . . . . .1 . . . . . .
.
Matrix speci�cation of admissible orderings (3)
DegRevLex
1 1 1 . . . 1 1 1 . . .−1 . . . . . .
−1 . . . . . .−1 . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . −1 . . .. . . −1 . . .. . . −1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
This matrix is not canonical.
Matrix speci�cation of admissible orderings (3)
DegRevLex
Non-canonical matrix:
1 1 1 . . . 1 1 1 . . .−1 . . . . . .
−1 . . . . . .−1 . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . −1 . . .. . . −1 . . .. . . −1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Canonical matrix:
1 1 1 . . . 1 1 1 . . .1 1 . . . 1 1 1 . . .
1 . . . 1 1 1 . . .. . . 1 1 1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 1 1 . . .. . . 1 1 . . .. . . 1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(1),
(1 10 1
),
1 1 11 1
0 0 1
,
1 1 1 1
1 1 11 1
0 0 0 1
,
1 1 1 1 1
1 1 1 11 1 1
1 10 0 0 0 1
, . . . .
Matrix speci�cation of admissible orderings (3)
DegRevLex
Non-canonical matrix:
1 1 1 . . . 1 1 1 . . .−1 . . . . . .
−1 . . . . . .−1 . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . −1 . . .. . . −1 . . .. . . −1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Canonical matrix:
1 1 1 . . . 1 1 1 . . .1 1 . . . 1 1 1 . . .
1 . . . 1 1 1 . . .. . . 1 1 1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 1 1 . . .. . . 1 1 . . .. . . 1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(1),
(1 10 1
),
1 1 11 1
0 0 1
,
1 1 1 1
1 1 11 1
0 0 0 1
,
1 1 1 1 1
1 1 1 11 1 1
1 10 0 0 0 1
, . . . .
Matrix speci�cation of admissible orderings (3)
DegRevLex
Non-canonical matrix:
1 1 1 . . . 1 1 1 . . .−1 . . . . . .
−1 . . . . . .−1 . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . −1 . . .. . . −1 . . .. . . −1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Canonical matrix:
1 1 1 . . . 1 1 1 . . .1 1 . . . 1 1 1 . . .
1 . . . 1 1 1 . . .. . . 1 1 1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 1 1 . . .. . . 1 1 . . .. . . 1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(1),
(1 10 1
),
1 1 11 1
0 0 1
,
1 1 1 1
1 1 11 1
0 0 0 1
,
1 1 1 1 1
1 1 1 11 1 1
1 10 0 0 0 1
, . . . .
Matrix speci�cation of admissible orderings (4)
WtLex
0 1 2 . . . k − 1 k k + 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 1 . . .
. . . 1 . . .
. . . 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 . . . . . .1 . . . . . .
1 . . . . . .
(Wt+Deg)InvRevLex
1 2 3 . . . k k + 1 k + 2 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . −1 . . .
. . . −1 . . .
. . . −1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
−1 . . . . . .−1 . . . . . .
−1 . . . . . .
WtInvLex
0 1 2 . . . k − 1 k k + 1 . . .1 . . . . . .
1 . . . . . .1 . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 . . .. . . 1 . . .. . . 1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Wt+Deg)RevLex
1 2 3 . . . k k + 1 k + 2 . . .−1 . . . . . .
−1 . . . . . .−1 . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . −1 . . .. . . −1 . . .. . . −1 . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
δ-stability
AO ≺ is called strictly δ-stable if
M ≺ N =⇒ lm≺ δM ≺ lm≺ δN.
Examples: Lex, DegLex è WtLex.
AO ≺ is called δ-stable, if for all nontrivial monomials M and N
M ≺ N =⇒ lm≺ δM 4 lm≺ δN.
Example: WtDegInvRevLex.
δ-stability
AO ≺ is called strictly δ-stable if
M ≺ N =⇒ lm≺ δM ≺ lm≺ δN.
Examples: Lex, DegLex è WtLex.
AO ≺ is called δ-stable, if for all nontrivial monomials M and N
M ≺ N =⇒ lm≺ δM 4 lm≺ δN.
Example: WtDegInvRevLex.
δ-lexicographic and β-orderings
AO ≺ is δ-lexicographic if lm≺ δM = lmlex δM for all M 6= 1.
Examples: Lex, DegLex and WtLex, and also any δ-stableordering.
If, conversely, all summands in δkM are compared
antilexicographically, then ≺ is called a β-ordering.
Examples: DegRevLex, (Wt+Deg)RevLex.
lm≺ δ2ny2 = y2n is a β-monomial w.r.t. a β-ordering.
lm≺ δ2ny2 = yy2n w.r.t. a δ-stable AO.
δ-lexicographic and β-orderings
AO ≺ is δ-lexicographic if lm≺ δM = lmlex δM for all M 6= 1.
Examples: Lex, DegLex and WtLex, and also any δ-stableordering.
If, conversely, all summands in δkM are compared
antilexicographically, then ≺ is called a β-ordering.
Examples: DegRevLex, (Wt+Deg)RevLex.
lm≺ δ2ny2 = y2n is a β-monomial w.r.t. a β-ordering.
lm≺ δ2ny2 = yy2n w.r.t. a δ-stable AO.
δ-lexicographic and β-orderings
AO ≺ is δ-lexicographic if lm≺ δM = lmlex δM for all M 6= 1.
Examples: Lex, DegLex and WtLex, and also any δ-stableordering.
If, conversely, all summands in δkM are compared
antilexicographically, then ≺ is called a β-ordering.
Examples: DegRevLex, (Wt+Deg)RevLex.
lm≺ δ2ny2 = y2n is a β-monomial w.r.t. a β-ordering.
lm≺ δ2ny2 = yy2n w.r.t. a δ-stable AO.
δ-�xed orderings
AO ≺ is called δ-�xed if
∀ f ∈ F{y} \ F ∃M ∈ M; ∃ k0, r ∈ N :
lm≺ δkf = Myr+k äëÿ âñåõ k > k0.
Example: any δ-lexicographic ordering.
β-orderings are not δ-�xed!
Concordance with quasi-linearity
A polynomial f ∈ F{x} \ F is called ≺-quasi-linear,if deg lm≺ f = 1.
Example: f = y1 + y20 is quasi-linear w.r.t. Lex,
but not DegLex.
AO ≺ is concordant with quasi-linearity, if the derivative of
any ≺-quasi-linear polynomial is also ≺-quasi-linear.
Examples: Lex, DegLex, DegRevLex,and also any δ-lexicographic orderingand any δ-stable ordering.
Concordance with quasi-linearity
A polynomial f ∈ F{x} \ F is called ≺-quasi-linear,if deg lm≺ f = 1.
Example: f = y1 + y20 is quasi-linear w.r.t. Lex,
but not DegLex.
AO ≺ is concordant with quasi-linearity, if the derivative of
any ≺-quasi-linear polynomial is also ≺-quasi-linear.
Examples: Lex, DegLex, DegRevLex,and also any δ-lexicographic orderingand any δ-stable ordering.
Relations between classes of orderings
strictly δ-stable:Lex, DegLex, WtLex
⊂δ-stable:
WtDegInvRevLex,DegWtInvRevLex⋂ ⋂
δ-lexicographic ⊂ concordantwith quasi-linearity⋂
↓
δ-�xed su�cient condition
↓ ↓
necessarycondition
β-orderings:DegRevLex,
(Wt+Deg)RevLex
Contents
1 Introduction
2 Admissible orderings
on di�erential monomials
3 Finiteness criteria
for di�erential standard bases
Finiteness criteria
Let I / F{y} be a proper δ-ideal.
Necessary condition: for a δ-�xed AO ≺
I has a �nite ≺-DSB ⇒ I contains
an ≺-quasi-linear polynomial
Su�cient condition:
for a concordant with quasi-linearity AO ≺
I has a �nite ≺-DSB ⇐ I contains
an ≺-quasi-linear polynomial
Corollary
For a δ-lexicographic admissible ordering
the condition is both necessary and su�cient.
Finiteness criteria
Let I / F{y} be a proper δ-ideal.
Necessary condition: for a δ-�xed AO ≺
I has a �nite ≺-DSB ⇒ I contains
an ≺-quasi-linear polynomial
Su�cient condition:
for a concordant with quasi-linearity AO ≺
I has a �nite ≺-DSB ⇐ I contains
an ≺-quasi-linear polynomial
Corollary
For a δ-lexicographic admissible ordering
the condition is both necessary and su�cient.
Finiteness criteria
Let I / F{y} be a proper δ-ideal.
Necessary condition: for a δ-�xed AO ≺
I has a �nite ≺-DSB ⇒ I contains
an ≺-quasi-linear polynomial
Su�cient condition:
for a concordant with quasi-linearity AO ≺
I has a �nite ≺-DSB ⇐ I contains
an ≺-quasi-linear polynomial
Corollary
For a δ-lexicographic admissible ordering
the condition is both necessary and su�cient.
Colloraries
Generalisations of Carr�a Ferro's theorems:
Let ≺ be δ-�xed.If degrees of monomials in f1, . . . , fn are greater than 1
then [f1, . . . , fn] has no �nite DSB w.r.t. ≺.
Let ≺ be strictly δ-stable. The reduced DSB of [f ] w.r.t. ≺consists of f itself ⇐⇒ f is ≺-quasi-linear.
Key role of Lex:
A DSB w.r.t. a δ-�xed AOis �nite
⇒ Lex DSB is also �nite .
Colloraries
Generalisations of Carr�a Ferro's theorems:
Let ≺ be δ-�xed.If degrees of monomials in f1, . . . , fn are greater than 1
then [f1, . . . , fn] has no �nite DSB w.r.t. ≺.
Let ≺ be strictly δ-stable. The reduced DSB of [f ] w.r.t. ≺consists of f itself ⇐⇒ f is ≺-quasi-linear.
Key role of Lex:
A DSB w.r.t. a δ-�xed AOis �nite
⇒ Lex DSB is also �nite .
Colloraries
Generalisations of Carr�a Ferro's theorems:
Let ≺ be δ-�xed.If degrees of monomials in f1, . . . , fn are greater than 1
then [f1, . . . , fn] has no �nite DSB w.r.t. ≺.
Let ≺ be strictly δ-stable. The reduced DSB of [f ] w.r.t. ≺consists of f itself ⇐⇒ f is ≺-quasi-linear.
Key role of Lex:
A DSB w.r.t. a δ-�xed AOis �nite
⇒ Lex DSB is also �nite .
�Improved Ollivier process�
Input: F ⊂ F{y};≺ � δ-�xed and concordant with quasi-linearity AO.
Output: Reduced ≺-DSB of [F ] (if it is �nite).
G := F;
H := ∅;
s := maxf∈F ord f;k := 0;repeat
Gold := ∅;
while G 6= Gold do
H := Diff Complete (G, s + k);Gold := G;
G := ReducedGr�obnerBasis (H,≺);end do;
k := k + 1;until ContainsQuasiLinear (G,≺);return DiffAutoreduce (G,≺);
Finite DSB w.r.t. Lex and DegLex
Proposition
The following are equivalent:
the ideal I / F{y} has �nite lex DSB;
I contains a lex-quasi-linear polynomial;
factoralgebra F [y, y1, y2, . . .]/I is �nitely generated.
Proposition
The following are equivalent:
the ideal I / F{y} has �nite deglex DSB;
I contains a linear polynomial.
Powers of quasi-linear polynomials
Theorem
Let k > 0. There exists an AO ≺(k) such that
∀ lex-quasi-linear polynomial f of order k and ∀ n > 1[fn] has a �nite ≺(k)-DSB consisting only of fn.
This AO can be speci�ed by the matrix
y0 y1 y2 . . . yk−1 yk yk+1 yk+2 . . . y2k−1 y2k y2k+1 . . .
1 2 3 . . . k k + 1 k + 2 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . .−1
−1−1
. . .−1
−1−1
1. . .
11
1
.
Powers of quasi-linear polynomials
Theorem
Let k > 0. There exists an AO ≺(k) such that
∀ lex-quasi-linear polynomial f of order k and ∀ n > 1[fn] has a �nite ≺(k)-DSB consisting only of fn.
This AO can be speci�ed by the matrix
y0 y1 y2 . . . yk−1 yk yk+1 yk+2 . . . y2k−1 y2k y2k+1 . . .
1 2 3 . . . k k + 1 k + 2 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . .−1
−1−1
. . .−1
−1−1
1. . .
11
1
.
Finite DSB w.r.t. β-orderings
Proposition
The ideal [yn], n > 1, has �nite ≺-DSB {yn} ⇐⇒⇐⇒ ≺ is a β-ordering.
The ideal [yn] has no �nite DSB w.r.t. any δ-�xed ordering.
Conjecture
Let ≺ be a β-ordering that is concordant with quasi-linearity.
Then:
A proper δ-ideal Ihas �nite ≺-DSB
⇐⇒
∃ ≺-quasi-linearpolynomial f :
either I 3 f ,
or I = [fn]for some n > 1.
Finite DSB w.r.t. β-orderings
Proposition
The ideal [yn], n > 1, has �nite ≺-DSB {yn} ⇐⇒⇐⇒ ≺ is a β-ordering.
The ideal [yn] has no �nite DSB w.r.t. any δ-�xed ordering.
Conjecture
Let ≺ be a β-ordering that is concordant with quasi-linearity.
Then:
A proper δ-ideal Ihas �nite ≺-DSB
⇐⇒
∃ ≺-quasi-linearpolynomial f :
either I 3 f ,
or I = [fn]for some n > 1.
Integral properties of δ-ideals [yn] (a conjecture)
Conjecture
Let f be a homogeneous and isobaric polynomial.
If the derivative δf is in [yn], then f is also in [yn].
For any �xed triple (n, deg f, wt f) this conjecturecan be reduced to some linear algebra problem
and can be checked directly using computer algebra system.
It is checked that for su�ciently large n, deg f and wt fthis conjecture is true.
Integral properties of δ-ideals [yn] (a conjecture)
Conjecture
Let f be a homogeneous and isobaric polynomial.
If the derivative δf is in [yn], then f is also in [yn].
For any �xed triple (n, deg f, wt f) this conjecturecan be reduced to some linear algebra problem
and can be checked directly using computer algebra system.
It is checked that for su�ciently large n, deg f and wt fthis conjecture is true.