Solving sparse polynomial systems using Gröbner basis

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Solving sparse polynomial systems using Gr¨ obner basis

Mat´ıas R. Bender

Sorbonne Universit´e, CNRS, INRIA, Laboratoire d’Informatique de Paris 6, LIP6, Equipe PolSys, 4 place Jussieu, F-75005, Paris, France´

Joint work with: Jean-Charles Faug`ere & Elias Tsigaridas

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Resum´ e of the talk

Objective

Compute Gr¨obner basis faster by exploiting the sparsity of the supports of the polynomials.

We focus in the mixed case

The polynomials have different supports.

In this talk

Algorithm to compute Gr¨obner basis over semigroup algebras.

Under regularity assumptions, no reductions to zero.

Algorithm and complexity bounds to solve 0-dim. square systems.

Improvements for special cases (mixed multihomogeneous&unmixed).

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Gr¨ obner basics

K[x ] = K[x1,. . ., xn], polynomial ring in n indeterminates over K ⊂ C.

Polynomial →P

ic_ix^α ∈ K[x]. Monomial → x^α, for α ∈ Nⁿ. Monomial ordering <

Total order for monomials in K[x] such that,

The monomial 1 is the smallest: ∀x^α6= 1, 1 < x^α, Compatible with multiplication: for all x^α, x^β, x^γ,

x^α< x^β =⇒ x^αx^γ < x^βx^γ Lexicographical (lex) y < x ,

1 < y < y² < · · · < x < x y < x y² < · · · < x² < x²y < . . . . Degree lexicographical z < y < x ,

1 < z < y < x < z²< y z < y² < x z < x y < x² < . . . . Degree reverse lexicographical order (grevlex) z < y < x ,

1 < z < y < x < z² < y z < x z < y² < x y < x²< . . .

Leading monomial → Biggest monomial (wrt >) with non-zero coefficient. Gr¨obner basis

A subset G ⊂ I is a Gr¨obner basis of the ideal I wrt >, if and only if, for every f ∈ I , there is g ∈ G such that LM_>(g ) divides LM_>(f ).

Mat´ıas BENDER Gr¨obner Basis & Sparse Systems April 2, 2019 2 / 24

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Gr¨ obner basics

K[x ] = K[x1,. . ., xn], polynomial ring in n indeterminates over K ⊂ C.

Polynomial →P

ic_ix^α ∈ K[x]. Monomial → x^α, for α ∈ Nⁿ. Monomial ordering <

Total order for monomials in K[x] such that,

The monomial 1 is the smallest: ∀x^α6= 1, 1 < x^α, Compatible with multiplication: for all x^α, x^β, x^γ,

x^α< x^β =⇒ x^αx^γ < x^βx^γ

Leading monomial → Biggest monomial (wrt >) with non-zero coefficient.

Gr¨obner basis

A subset G ⊂ I is a Gr¨obner basis of the ideal I wrt >, if and only if, for every f ∈ I , there is g ∈ G such that LM_>(g ) divides LM_>(f ).

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Computing Gr¨ obner basis : Lazard’s approach

Compute Gr¨obner basis for (f₁, f₂, f₃) in K[x, y ] wrt Grevlex(x > y ),







f₁ := x + y + 1 f2 := −x + y + 1

f₃ := x²+ x y − y²+ x + y + 1

We homogenize the system over K[x, y , z].







F₁ := x + y + z F₂ := −x + y + z

F3 := x²+ x y − y²+ x z + y z + z² For each d , compute triangular basis for hF1, F2, F3i_d wrt

Grevlex(x > y > z). Degree d = 1,





x y z

F1 1 1 1 F₂ −1 1 1



−→





x y z

F1 1 1 1

F₂+ F₁ 0 2 2





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Computing Gr¨ obner basis : Lazard’s approach







f₁ := x + y + 1 f2 := −x + y + 1

f₃ := x²+ x y − y²+ x + y + 1 We homogenize the system over K[x, y ,z].







F₁ := x + y +z F₂ := −x + y +z

F3 := x²+ x y − y²+ xz + yz+z²

For each d , compute triangular basis for hF1, F2, F3i_d wrt

Grevlex(x > y > z). Degree d = 1,





x y z

F1 1 1 1 F₂ −1 1 1



−→





x y z

F1 1 1 1

F₂+ F₁ 0 2 2





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Computing Gr¨ obner basis : Lazard’s approach







f₁ := x + y + 1 f2 := −x + y + 1

f₃ := x²+ x y − y²+ x + y + 1 We homogenize the system over K[x, y , z].







F₁ := x + y + z F₂ := −x + y + z

Grevlex(x > y > z).

Degree d = 1,





x y z

F1 1 1 1 F₂ −1 1 1



−→





x y z

F1 1 1 1

F₂+ F₁ 0 2 2





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Computing Gr¨ obner basis : Lazard’s approach







f₁ := x + y + 1 f2 := −x + y + 1







F₁ := x + y + z F₂ := −x + y + z

Grevlex(x > y > z).

Degree d = 1,





x y z

F1 1 1 1 F₂ −1 1 1





−→





x y z

F1 1 1 1

F₂+ F₁ 0 2 2





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Computing Gr¨ obner basis : Lazard’s approach







f₁ := x + y + 1 f2 := −x + y + 1







F₁ := x + y + z F₂ := −x + y + z

Grevlex(x > y > z).

Degree d = 1,





x y z

F1 1 1 1 F₂ −1 1 1



−→





x y z

F1 1 1 1

F₂+ F₁ 0 2 2





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Computing Gr¨ obner basis : Lazard’s approach for Grevlex

For each degree d , compute triangular basis for hF₁, . . . , F₃i_d: Degree d = 1,





x y z

F₁ 1 1 1 F₂ −1 1 1



−→





x y z

F₁ 1 1 1

F₂+ F₁ 2 2



 Degree d = 2,

z F₁ y F1

x F1

z F₂ y F2

x F2

F₃







x² x y y² x z y z z²

1 1 1

−1 1 1

1 1 −1 1 1 1







Gr¨obner basis of hF₁, F₂, F₃i → {x + y + z, y + z, z²}.

Its dehomogenization (z = 1) is a Gr¨obner basis of hf1, f2, f3i → {1}.

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Computing Gr¨ obner basis : Lazard’s approach for Grevlex





x y z

F₁ 1 1 1 F₂ −1 1 1



−→





x y z

F₁ 1 1 1

F₂+ F₁ 2 2



 Degree d = 2,

z F₁ y F1

x F1

z F₂+ z F₁ y F2+ y F1

(x + y + z) F2− (x − y + z) F₁ F₃− (x −^y₂ + 1)F₁+ (^y₂ − 1)F₂







1 1 1

2 2

−1







Gr¨obner basis of hF₁, F₂, F₃i → {x + y + z, y + z, z²}.

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Computing Gr¨ obner basis : Lazard’s approach for Grevlex





x y z

F₁ 1 1 1 F₂ −1 1 1



−→





x y z F₁ 1 1 1 F₂+ F₁ 2 2



 Degree d = 2,

z F₁ y F1

x F1

z F₂+ z F₁ y F2+ y F1

(x + y + z) F2− (x − y + z) F₁ F₃− (x −^y₂ + 1)F₁+ (^y₂ − 1)F₂







1 1 1

2 2

−1





 Gr¨obner basis of hF₁, F₂, F₃i → {x + y + z, y + z, z²}.

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Computing Gr¨ obner basis : Lazard’s approach for Grevlex





x y z

F₁ 1 1 1 F₂ −1 1 1



−→





x y z

F₁ 1 1 1

F₂+ F₁ 2 2



 Degree d = 2,

z F₁ y F1

x F1

z F₂+ z F₁ y F2+ y F1

(x + y + z) F2− (x − y + z) F₁ F₃− (x −^y₂ + 1)F₁+ (^y₂ − 1)F₂







1 1 1

2 2

−1





 Gr¨obner basis of hF₁, F₂, F₃i → {x + y + z, y + z,z²}.

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Complexity of Lazard’s algorithm

Complexity depends on maximal degree. In generic coordinates,

→ Castelnuovo-Mumford (CM) regularity of I .

Regular sequence

(F1, . . . , Fm) is a regular seq. ⇔ ∀k ≤ m, Fk is regular in K[x]/hF¹, . . . , Fk−1i.

Macaulay bound

If F1, . . . , Fmregular sequence → CM regularity =Pm

i =1deg (fi) − m + 1

Drawback: Many rows reduce to zero

z F1

y F1

x F1

z F2

y F2+ y F1

(x + y + z) F2− (x − y + z) F1

F3− (x −^y₂+ 1)F1+ (^y₂− 1)F2







1 1 1

−1 1 1

2 2

0 0 0 0 0 0

−1







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Regular sequence

Macaulay bound

If F1, . . . , Fmregular sequence → CM regularity=Pm

i =1deg (fi) − m + 1

z F1

y F1

x F1

z F2

y F2+ y F1

(x + y + z) F2− (x − y + z) F1

F3− (x −^y₂+ 1)F1+ (^y₂− 1)F2







1 1 1

−1 1 1

2 2

0 0 0 0 0 0

−1







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Regular sequence

Macaulay bound

If F1, . . . , Fmregular sequence → CM regularity =Pm

i =1deg (fi) − m + 1

z F1

y F1

x F1

z F2

y F2+ y F1

(x + y + z) F2− (x − y + z) F1

F3− (x −^y₂+ 1)F1+ (^y₂− 1)F2







1 1 1

−1 1 1

2 2

0 0 0 0 0 0

−1







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F5 criterion to detect trivial syzygies

(x − y + z)

| {z }

F2

F₁− (x + y + z)

| {z }

F1

F₂ = 0 ←→ Trivial syzygy (F2, −F1, 0)

Trivial syzygy

Syzygy of (F₁, . . . , F_m) →

(H₁, . . . , H_m) ∈ K[x]^m such that X

i

H_iF_i = 0.

Trivial → H_m = · · · = H_k+1 = 0, H_k 6= 0, and H_k ∈ hF₁, . . . , F_k−1i.

Trivial syzygy

Syzygy of (F₁, . . . , F_m) → (H₁, . . . , H_m) ∈ K[x]^m such that P

iH_iF_i = 0. Trivial → H_m = · · · = H_k+1= 0, H_k 6= 0, and H_k ∈ hF₁, . . . , F_k−1i. F5 criterion

If x^α ∈ LM_>(hF₁, . . . , F_k−1i), then the row x^αF_k reduces to zero. F5 criterion : Optimality

If (F1, . . . , Fm) is a regular sequence =⇒

F5 criterion detects all the reductions to zero.

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F5 criterion to detect trivial syzygies

(x − y + z)

| {z }

F2

F₁− (x + y + z)

| {z }

F1

F₂ = 0 ←→ Trivial syzygy (F₂, −F₁, 0)

Trivial syzygy

Syzygy of (F₁, . . . , F_m) → (H₁, . . . , H_m) ∈ K[x]^m such thatP

iH_iF_i = 0.

Trivial → Hm = · · · = Hk+1 = 0, Hk 6= 0, and H_k ∈ hF₁, . . . , Fk−1i.

F5 criterion

If x^α ∈ LM_>(hF₁, . . . , F_k−1i), then the row x^αF_k reduces to zero. F5 criterion : Optimality

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F5 criterion to detect trivial syzygies

(x − y + z)

| {z }

F2

F₁− (x+ y + z)

| {z }

F1

Trivial syzygy

Syzygy of (F1, . . . , Fm) → (H1, . . . , Hm) ∈ K[x]^m such thatP

iH_iF_i = 0.

Trivial → H_m = · · · = H_k+1 = 0, H_k 6= 0, and H_k ∈ hF₁, . . . , F_k−1i.

F5 criterion

Ifx^α ∈ LM_>(hF₁, . . . , F_k−1i), then the rowx^αF_k reduces to zero.

z F1

y F1

x F1

z F2

y F2

x F2

F3







1 1 1

−1 1 1

1 1 −1 1 1 1







→







1 1 1

−1 1 1

2 2

0 0 0 0 0 0

−1







F5 criterion : Optimality

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F5 criterion to detect trivial syzygies

(x − y + z)

| {z }

F2

F₁− (x + y + z)

| {z }

F1

Trivial syzygy

Syzygy of (F₁, . . . , F_m) → (H₁, . . . , H_m) ∈ K[x]^m such thatP

iH_iF_i = 0.

Trivial → Hm = · · · = Hk+1 = 0, Hk 6= 0, and H_k ∈ hF₁, . . . , Fk−1i.

F5 criterion

If x^α ∈ LM_>(hF₁, . . . , F_k−1i), then the row x^αF_k reduces to zero.

F5 criterion : Optimality

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Computing Gr¨ obner bases

Lazard’s approach: For each d , compute triangular basis of (hF1, . . . , Fmi)d.

For each k ≤ m, (Jk)d = triangular basis of hF1, . . . , Fkid

(J_k)_d := Gaussian Elim.((J_k−1)_d ∪ {x^αF_k : deg (x^α) = d − deg (F_k)} )

F5 criterion:

If Fk is regular (non-zero divisor) in K[x]/(F¹, . . . , Fk−1) ⇒

Trivial reductions to zero in (J_k)d ←→ polynomials in (J_k−1)d −deg (F_k) .

(J _k ) _d

If (F₁, . . . , F_m) is a regular sequence =⇒ we skip every reduction to zero.

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Computing Gr¨ obner bases

Lazard’s approach: For each d , compute triangular basis of (hF1, . . . , Fmi)d. For each k ≤ m, (Jk)d = triangular basis of hF1, . . . , Fkid

F5 criterion:

^F5 ^Gauss.

Lazard’s algorithm → Computes Gr¨obner basis using linear algebra.

In generic coordinates, complexity → Castelnuovo-Mumford regularity.

Generic coordinates destroy sparsity. Complexity unknown.

F5 criterion → Avoids trivial reductions to zero.

If the system is a regular sequence.

Generic sparse systems are not regular sequences.

F5 avoids every redundant computations.

F5 always misses reductions to zero.

Castelnuovo-Mumford regularity → Macaulay bound.

Castelnuovo-Mumford regularity unknown.

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Summing up

Computing Gr¨obner basis for sparse systems

X Lazard’s algorithm → Computes Gr¨obner basis using linear algebra.

In generic coordinates, complexity → Castelnuovo-Mumford regularity.

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Summing up

X In generic coordinates, complexity → Castelnuovo-Mumford regularity.

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Summing up

X F5 criterion → Avoids trivial reductions to zero.

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Summing up

X If the system is a regular sequence.

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Summing up

X If the system is a regular sequence.

X F5 avoids every redundant computations.

X Castelnuovo-Mumford regularity → Macaulay bound.

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Sparse systems

Newton polytope of f =P

αc_αX^α −→ Convex hull of {α : c_α 6= 0}.

Sparse system −→ Newton polytopes of the polynomials are “small”.

1 + xy + x²y + x²y²+ x³y = 1+ 0 ·x+ 0 ·y+ 0 ·x²+xy + 0 ·y²+ 0 ·x³+x²y + 0 ·xy²+ 0 ·y³+

0 ·x⁴+x³y +x²y²+ 0 ·xy³+ 0 ·y⁴

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Sparse systems

Unmixed sparse system −→ Polynomials with equal Newton polytope.

Mixed sparse system −→ Different Newton polytope.

1+xy +x²y +x²y²+x³y 1 + xy + xy²+ xy³ 1+x +xy +x²y +x²y²

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Sparse systems

Unmixed sparse system −→ Polynomials with equal Newton polytope.

Mixed sparse system

| {z }

This talk!

−→ Different Newton polytope.

1+xy +x²y +x²y²+x³y 1 + xy + xy²+ xy³ 1+x +xy +x²y +x²y²

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Previous work (Non-exhaustive!)

Toric varieties

[Demazure, 1970], [Hochster, 1971], [Satake, 1973], [Kempf, Knudsen, Mumford & Saint-Donat, 1973], [Miyake & Oda, 1975], [Ehlers, 1975], [Bernstein, 1975], [Kusnirenko, 1976] [Khovanskii, 1977], . . .

. . . [Oda, 1988] . . . [Fulton, 1993] . . . [Cox, Little & Schenck, 2011]

Sparse resultant

[Gelfand, Kapranov & Zelevinsky, 1990], [Kapranov, Sturmfels & Zelevinsky, 1992], [Sturmfels, 1993], [Pedersen & Sturmfels, 1993], [Gelfand, Kapranov

& Zelevinsky, 1994], [Canny & Emiris, 1995], [D´Andrea, 2002], [D´Andrea

& Sombra, 2013]

Gr¨obner basis over semigroup algebras

[Sturmfels, 1991], [Faug`ere, Spaenlehauer & Svartz, 2014], [B., Faug`ere & Tsigaridas, 2018]

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Semigroup algebra K[S

^h

] : Unmixed case

K[S^h]

−→

Graded algebra [Faug`ere, Spaenlehauer & Svartz, 2014]

GB over K[S^h] → exists and finite. Algor. to compute GB over K[S^h]

→ Lazard’s algorithm + F5. If the homogenization of f1, . . . , fm

over K[S^h] is a regular sequence, No redundant computations

(F5 criterion).

? 0-dim systems → Complexity bounds (CM regularity).

Generic unmixed systems → homogenization regular.

Generic mixed systems → homogenization NOTregular.

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Semigroup algebra K[S

^h

] : Unmixed case

K[S^h]

−→

GB over K[S^h] → exists and finite. Algor. to compute GB over K[S^h]

(F5 criterion).

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Semigroup algebra K[S

^h

] : Unmixed case

K[S^h]

−→

−→

GB over K[S^h] → exists and finite.

Algor. to compute GB over K[S^h]

→ Lazard’s algorithm + F5.

If the homogenization of f1, . . . , fm

(F5 criterion).

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Semigroup algebra K[S

^h

] : Unmixed case

K[S^h]

−→

GB over K[S^h] → exists and finite.

Algor. to compute GB over K[S^h]

→ Lazard’s algorithm + F5.

If the homogenization of f1, . . . , fm

(F5 criterion).

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Relaxing the regular sequence condition

We need a new algorithm

Generic mixed systems are NOTregular sequences...

... but they behave as them for “big degrees”.

Idea

Check what happens at specific multidegrees.

Consider multigrading for K[S^h] related to the different polytopes.

Characterize the optimality of F5 at a multideg. → Koszul complex.

Do not compute in every multidegree,

only consider the ones where we can predict the syzygies.

Warning

This approach does not make the systems regular sequences.

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Semigroup algebra K[S

^h

] : Mixed case

Minkowski sum

∆1

+

∆₂

=

∆₁+ ∆₂

Semigroup algebra

K[S^h]

K[S^h] multigraded by N^m X^α ∈ K[S^h]_(d₁_,...,d_m₎

x

 y

α ∈ (d₁∆₁+ · · · + d_m∆_m) ∩ Zⁿ

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Semigroup algebra K[S

^h

] : Mixed case

Minkowski sum

∆1

+

∆2

=

∆1+ ∆2

Semigroup algebra

K[S^h]

x

 y

α ∈ (d1∆1+ · · · + dm∆m) ∩ Zⁿ

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Semigroup algebra K[S

^h

] : Mixed case

Minkowski sum

∆1

+

∆2

=

∆1+ ∆2

Semigroup algebra

K[S^h]

x

 y

α ∈ (d1∆1+ · · · + dm∆m) ∩ Zⁿ

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Semigroup algebra K[S

^h

] : Mixed case

Minkowski sum

∆1

+

∆2

=

∆1+ ∆2

Semigroup algebra

K[S^h]

x

 y

α ∈ (d1∆1+ · · · + dm∆m) ∩ Zⁿ

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Semigroup algebra K[S

^h

] : Mixed case

] : Mixed case

X^α ∈ K[S^h]_(d₁_,d₂₎←→ α ∈ (d₁∆2+ d2∆2) ∩ Zⁿ

+ ∆₁ 2∆₁

(2,0)

∆2

2∆2

Solving sparse polynomial systems using Gröbner basis

Solving sparse polynomial systems using Gr¨ obner basis

Resum´ e of the talk

Gr¨ obner basics

Gr¨ obner basics

Computing Gr¨ obner basis : Lazard’s approach

Computing Gr¨ obner basis : Lazard’s approach

Computing Gr¨ obner basis : Lazard’s approach

Computing Gr¨ obner basis : Lazard’s approach

Computing Gr¨ obner basis : Lazard’s approach

Computing Gr¨ obner basis : Lazard’s approach for Grevlex

Computing Gr¨ obner basis : Lazard’s approach for Grevlex

Computing Gr¨ obner basis : Lazard’s approach for Grevlex

Computing Gr¨ obner basis : Lazard’s approach for Grevlex

F5 criterion to detect trivial syzygies

F5 criterion to detect trivial syzygies

F5 criterion to detect trivial syzygies

F5 criterion to detect trivial syzygies

Computing Gr¨ obner bases

(J k ) d

Computing Gr¨ obner bases

(J k ) d

Computing Gr¨ obner bases

(J k ) d

Computing Gr¨ obner bases

(J k ) d

Computing Gr¨ obner bases

(J k ) d

Computing Gr¨ obner bases

(J k ) d

Computing Gr¨ obner bases

(J k ) d

Summing up

Summing up

Summing up

Summing up

Summing up

Summing up

Sparse systems

Sparse systems

Sparse systems

Previous work (Non-exhaustive!)

Semigroup algebra K[S

] : Unmixed case

−→

Semigroup algebra K[S

] : Unmixed case

−→

Semigroup algebra K[S

] : Unmixed case

−→

Semigroup algebra K[S

] : Unmixed case

−→

Semigroup algebra K[S

] : Unmixed case

−→

Semigroup algebra K[S

] : Unmixed case

−→

Semigroup algebra K[S

] : Unmixed case

−→

Semigroup algebra K[S

] : Unmixed case

−→

Semigroup algebra K[S

] : Unmixed case

−→

Relaxing the regular sequence condition

Semigroup algebra K[S

] : Mixed case

Semigroup algebra K[S

] : Mixed case

Semigroup algebra K[S

] : Mixed case

Semigroup algebra K[S

] : Mixed case

Semigroup algebra K[S

] : Mixed case

(J _k ) _d

(J _k ) _d

(J _k ) _d

(J _k ) _d

(J _k ) _d

(J _k ) _d

(J _k ) _d