Advantages and Disadvantages of Differential Standard...

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: al_zobnin@mail.ru

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

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

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

wt M :=n∑

i=0αi is the weight of M .

M is the set of all di�erential monomials.

i=0yαi

wt M :=n∑

i=0yαi

wt M :=n∑

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

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

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.

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.

De�nition

O1. p ≺M q =⇒ p · s ≺M q · s ∀ p, q, s ∈ M;

O2. 1 4M p ∀ p ∈ M;

Theorem

De�nition

O1. p ≺M q =⇒ p · s ≺M q · s ∀ p, q, s ∈ M;

O2. 1 4M p ∀ p ∈ M;

Theorem

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

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.

DegRevLex ordering

(y + c)n.

y20 yk, k > 2.

n∏l=1

(n(n−2)+(n−1)(k−l)

)yk, k > 0;

(nk − n− k) y1 yk + n y0 yk+1, k > 1.

DegRevLex ordering

(y + c)n.

y20 yk, k > 2.

n∏l=1

(n(n−2)+(n−1)(k−l)

)yk, k > 0;

(nk − n− k) y1 yk + n y0 yk+1, k > 1.

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.

yn1 + y0;

n y0 y2 − y21;

n yn−21 y2

2 + y2;

y3 − n(n− 2) yn−31 y3

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.

Lex ordering

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.

Lex ordering

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

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}:

...αk

)≺lex M

...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

...αk

)≺lex M

...βk

)⇐⇒ yα0

0 . . . yαkk ≺ yβ0

0 . . . yβkk .

Theorem

Monomial matrices

...αk

)≺lex M

...βk

)⇐⇒ yα0

0 . . . yαkk ≺ yβ0

0 . . . yβkk .

Theorem

Matrix speci�cation of admissible orderings (1)

(0 11 0

0 0 11 0

0 0 0 1

1 01 0

0 0 0 0 1

1 01 0

, . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 . . .. . . 1 . . .. . . 1 . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . . . .

1 . . . . . .1 . . . . . .

(0 11 0

0 0 11 0

0 0 0 1

1 01 0

0 0 0 0 1

1 01 0

, . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 . . .. . . 1 . . .. . . 1 . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . . . .

1 . . . . . .1 . . . . . .

(0 11 0

0 0 11 0

0 0 0 1

1 01 0

0 0 0 0 1

1 01 0

, . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 . . .. . . 1 . . .. . . 1 . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . . . .

1 . . . . . .1 . . . . . .

(0 11 0

0 0 11 0

0 0 0 1

1 01 0

0 0 0 0 1

1 01 0

, . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 . . .. . . 1 . . .. . . 1 . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . . . .

1 . . . . . .1 . . . . . .

(0 11 0

0 0 11 0

0 0 0 1

1 01 0

0 0 0 0 1

1 01 0

, . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 . . .. . . 1 . . .. . . 1 . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . . . .

1 . . . . . .1 . . . . . .

(0 11 0

0 0 11 0

0 0 0 1

1 01 0

0 0 0 0 1

1 01 0

, . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 . . .. . . 1 . . .. . . 1 . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . . . .

1 . . . . . .1 . . . . . .

DegLex

(1 10 1

1 1 10 0 1

1 1 1 10 0 0 1

1 01 0

1 1 1 1 10 0 0 0 1

1 01 0

, . . . .

1 1 1 . . . 1 1 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 1 . . .

. . . 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 . . . . . .1 . . . . . .

DegLex

(1 10 1

1 1 10 0 1

1 1 1 10 0 0 1

1 01 0

1 1 1 1 10 0 0 0 1

1 01 0

, . . . .

1 1 1 . . . 1 1 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 1 . . .

. . . 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 . . . . . .1 . . . . . .

DegLex

(1 10 1

1 1 10 0 1

1 1 1 10 0 0 1

1 01 0

1 1 1 1 10 0 0 0 1

1 01 0

, . . . .

1 1 1 . . . 1 1 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 1 . . .

. . . 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 . . . . . .1 . . . . . .

DegLex

(1 10 1

1 1 10 0 1

1 1 1 10 0 0 1

1 01 0

1 1 1 1 10 0 0 0 1

1 01 0

, . . . .

1 1 1 . . . 1 1 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 1 . . .

. . . 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 . . . . . .1 . . . . . .

DegLex

(1 10 1

1 1 10 0 1

1 1 1 10 0 0 1

1 01 0

1 1 1 1 10 0 0 0 1

1 01 0

, . . . .

1 1 1 . . . 1 1 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 1 . . .

. . . 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 . . . . . .1 . . . . . .

DegLex

(1 10 1

1 1 10 0 1

1 1 1 10 0 0 1

1 01 0

1 1 1 1 10 0 0 0 1

1 01 0

, . . . .

1 1 1 . . . 1 1 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 1 . . .

. . . 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 . . . . . .1 . . . . . .

DegRevLex

1 1 1 . . . 1 1 1 . . .−1 . . . . . .

−1 . . . . . .−1 . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . −1 . . .. . . −1 . . .. . . −1 . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

This matrix is not canonical.

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 10 1

1 1 11 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

, . . . .

DegRevLex

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 10 1

1 1 11 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

, . . . .

DegRevLex

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 10 1

1 1 11 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

, . . . .

0 1 2 . . . k − 1 k k + 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 1 . . .

. . . 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 . . . . . .1 . . . . . .

1 . . . . . .

(Wt+Deg)InvRevLex

1 2 3 . . . k k + 1 k + 2 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . −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-�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

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

Corollary

For a δ-lexicographic admissible ordering

the condition is both necessary and su�cient.

Finiteness criteria

Corollary

Finiteness criteria

Corollary

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

Key role of Lex:

Colloraries

Key role of Lex:

�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. . .

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. . .

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.

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.

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.

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