Signature Rewriting in Gröbner Basis Computation

Signature Rewriting in Gr¨obner Basis Computation Christian Eder joint work with Bjarke Hammersholt Roune POLSYS Team, UPMC, Paris, France

June 29, 2013

1 / 14

1 Signature-based algorithms

The basic idea Generic signature-based criteria

2 Rewritable signature criterion in detail

2 / 14

Signatures of polynomials Let I = hf1 , . . . , fm i. Idea: Give each f ∈ I a bit more structure:

3 / 14

Signatures of polynomials Let I = hf1 , . . . , fm i. Idea: Give each f ∈ I a bit more structure: 1. Let R m be generated by e1 , . . . , em , ≺ a well-ordering on the monomials of R m , and let α 7→ α : R m → R such that e i = fi for all i.

3 / 14

Signatures of polynomials Let I = hf1 , . . . , fm i. Idea: Give each f ∈ I a bit more structure: 1. Let R m be generated by e1 , . . . , em , ≺ a well-ordering on the monomials of R m , and let α 7→ α : R m → R such that e i = fi for all i. 2. Each p ∈ I can be represented by an α=

m X

hi ei ∈ R m such that p = α.

i=1

3 / 14

Signatures of polynomials Let I = hf1 , . . . , fm i. Idea: Give each f ∈ I a bit more structure: 1. Let R m be generated by e1 , . . . , em , ≺ a well-ordering on the monomials of R m , and let α 7→ α : R m → R such that e i = fi for all i. 2. Each p ∈ I can be represented by an α=

m X

hi ei ∈ R m such that p = α.

i=1

3. A signature of p is given by s(p) = lt≺ (α) with p = α.

3 / 14

Signatures of polynomials Let I = hf1 , . . . , fm i. Idea: Give each f ∈ I a bit more structure: 1. Let R m be generated by e1 , . . . , em , ≺ a well-ordering on the monomials of R m , and let α 7→ α : R m → R such that e i = fi for all i. 2. Each p ∈ I can be represented by an α=

m X

hi ei ∈ R m such that p = α.

i=1

3. A signature of p is given by s(p) = lt≺ (α) with p = α. 4. A minimal signature of p exists due to ≺. 3 / 14

Notations concerning signatures Let α ∈ R m , then α contains all data we need: I The polynomial data is α, its leading term denoted by lt (α). I The signature is s(α) = lt (α).

4 / 14

Notations concerning signatures Let α ∈ R m , then α contains all data we need: I The polynomial data is α, its leading term denoted by lt (α). I The signature is s(α) = lt (α). Moreover, we agree on the following: I For α, β ∈ R m , let α ' β if α = sβ for some s ∈ K . In the same sense we define α ' β if α = tβ for some t ∈ K . I G always denotes a finite subset of R m such that for all α, β ∈ G with s(α) ' s(β) it holds that α = β. I α ∈ R m is called a syzygy if α = 0.

4 / 14

Reductions concerning signatures Let α ∈ R m , and let t be a term of α. We can s-reduce t by β ∈ R m if
I ∃ a term b such that lt bβ = t and I s (bβ)  s (α).

5 / 14

Reductions concerning signatures Let α ∈ R m , and let t be a term of α. We can s-reduce t by β ∈ R m if
I ∃ a term b such that lt bβ = t and I s (bβ)  s (α).

Note We distinguish 2 types of s-reduction:

5 / 14

Reductions concerning signatures Let α ∈ R m , and let t be a term of α. We can s-reduce t by β ∈ R m if
I ∃ a term b such that lt bβ = t and I s (bβ)  s (α).

Note We distinguish 2 types of s-reduction:
1. If lt bβ ' lt (α) =⇒ top s-reduction step; otherwise =⇒ tail s-reduction step.

5 / 14

Reductions concerning signatures Let α ∈ R m , and let t be a term of α. We can s-reduce t by β ∈ R m if
I ∃ a term b such that lt bβ = t and I s (bβ)  s (α).

Note We distinguish 2 types of s-reduction:
1. If lt bβ ' lt (α) =⇒ top s-reduction step; otherwise =⇒ tail s-reduction step. 2.

If s (bβ) ' s (α) otherwise

=⇒ singular s-reduction step; =⇒ regular s-reduction step.

5 / 14

Signature Gr¨obner bases I s-reductions are always performed w.r.t. a finite basis G ⊂ R m . I G is a signature Gr¨ obner basis in signature T if all α ∈ R m with s (α) = T s-reduce to zero w.r.t G. I G is a signature Gr¨ obner basis if it is a signature Gr¨obner basis in all signatures. I If G is a signature Gr¨ obner basis, then {α | α ∈ G} is a Gr¨obner basis for hf1 , . . . , fm i.

Note In the following we do not need much of the details of signature-based Gr¨obner basis algorithms, just one property: The pair set is ordered by increasing signatures. 6 / 14

Generic signature-based criteria

Non-minimal signature ( NM ) s(α) not minimal for α? ⇒ Remove α.

7 / 14

Generic signature-based criteria

Non-minimal signature ( NM ) s(α) not minimal for α? ⇒ Remove α.

Sketch of proof 1. There exists a syzygy β ∈ R m such that lt(β) = s(α). ⇒ We can represent α with a lower signature. 2. Pairs are handled by increasing signatures. ⇒ All relations of lower signature are already taken care of.

7 / 14

Generic signature-based criteria

Rewritable signature ( RW ) s(α) = s(β)? ⇒ Remove either α or β.

8 / 14

Generic signature-based criteria

Rewritable signature ( RW ) s(α) = s(β)? ⇒ Remove either α or β.

Sketch of proof 1. s(α − β) ≺ s(α), s(β). 2. Pairs are handled by increasing signatures. ⇒ All necessary computations of lower signature have already taken place. ⇒ We can represent β by β = α + elements of lower signature.

8 / 14

1 Signature-based algorithms

The basic idea Generic signature-based criteria

2 Rewritable signature criterion in detail

9 / 14

The concept of rewrite bases

Rewriter and rewritable elements

10 / 14

The concept of rewrite bases

Rewriter and rewritable elements I A rewrite order / is a total order on G such that s(α) | s(β) ⇒ α / β. (Exists due to s(α) ' s(β) ⇒ α = β.)

10 / 14

The concept of rewrite bases

Rewriter and rewritable elements I A rewrite order / is a total order on G such that s(α) | s(β) ⇒ α / β. (Exists due to s(α) ' s(β) ⇒ α = β.) I An element α ∈ G is a rewriter in signature T if s(α) | T .

10 / 14

The concept of rewrite bases

Rewriter and rewritable elements I A rewrite order / is a total order on G such that s(α) | s(β) ⇒ α / β. (Exists due to s(α) ' s(β) ⇒ α = β.) I An element α ∈ G is a rewriter in signature T if s(α) | T . I The /-maximal rewriter in signature T is the canonical rewriter in T .

10 / 14

The concept of rewrite bases

Rewriter and rewritable elements I A rewrite order / is a total order on G such that s(α) | s(β) ⇒ α / β. (Exists due to s(α) ' s(β) ⇒ α = β.) I An element α ∈ G is a rewriter in signature T if s(α) | T . I The /-maximal rewriter in signature T is the canonical rewriter in T . I A multiple of a basis element tα is called rewritable if α is not the canonical rewriter in s(tα).

10 / 14

The concept of rewrite bases

Rewrite Bases

11 / 14

The concept of rewrite bases

Rewrite Bases I G is a rewrite basis in signature T if the canonical rewriter in T is not regular s-reducible or if T is a syzygy signature.

11 / 14

The concept of rewrite bases

Rewrite Bases I G is a rewrite basis in signature T if the canonical rewriter in T is not regular s-reducible or if T is a syzygy signature. I G is a rewrite basis if it is a rewrite basis in all signatures.

11 / 14

The concept of rewrite bases

Rewrite Bases I G is a rewrite basis in signature T if the canonical rewriter in T is not regular s-reducible or if T is a syzygy signature. I G is a rewrite basis if it is a rewrite basis in all signatures.

Lemma If G is a rewrite basis up to signature T then G is also a signature Gr¨obner basis up to T .

11 / 14

Improving the rewritable signature criterion RB (generic rewrite base algorithm) F5 (as presented in [Fa02])

AP/GVW/SB

Let α and β ∈ G such that s(α) = aei and s(β) = bej . α / β ⇐⇒ (i < j) or (i = j and a < b) Once an s-polynomial in a given signature T is computed all others are rewritable.

α / β ⇐⇒

s(α) lt(α)

≺

s(β) lt(β )

For any signature T define MT = {tα | α ∈ G, s(tα) = T } Choose tα such that lt(tα) is minimal.

Difference: There might be no such s-polynomial

12 / 14

Improving the rewritable signature criterion RB (generic rewrite base algorithm) F5 (as presented in [Fa02])

AP/GVW/SB

Let α and β ∈ G such that s(α) = aei and s(β) = bej . α / β ⇐⇒ (i < j) or (i = j and a < b) Once an s-polynomial in a given signature T is computed all others are rewritable.

α / β ⇐⇒

s(α) lt(α)

≺

s(β) lt(β )

For any signature T define MT = {tα | α ∈ G, s(tα) = T } Choose tα such that lt(tα) is minimal.

Difference: There might be no such s-polynomial

12 / 14

Improving the rewritable signature criterion RB (generic rewrite base algorithm) F5 (as presented in [Fa02])

AP/GVW/SB

Let α and β ∈ G such that s(α) = aei and s(β) = bej . α / β ⇐⇒ (i < j) or (i = j and a < b) Once an s-polynomial in a given signature T is computed all others are rewritable.

α / β ⇐⇒

s(α) lt(α)

≺

s(β) lt(β )

For any signature T define MT = {tα | α ∈ G, s(tα) = T } Choose tα such that lt(tα) is minimal.

Difference: There might be no such s-polynomial

12 / 14

Improving the rewritable signature criterion RB (generic rewrite base algorithm) F5 (as presented in [Fa02])

AP/GVW/SB

Let α and β ∈ G such that s(α) = aei and s(β) = bej . α / β ⇐⇒ (i < j) or (i = j and a < b) Once an s-polynomial in a given signature T is computed all others are rewritable.

α / β ⇐⇒

s(α) lt(α)

≺

s(β) lt(β )

For any signature T define MT = {tα | α ∈ G, s(tα) = T } Choose tα such that lt(tα) is minimal.

Difference: There might be no such s-polynomial

12 / 14

Improving the rewritable signature criterion RB (generic rewrite base algorithm) F5 (as presented in [Fa02])

AP/GVW/SB

Let α and β ∈ G such that s(α) = aei and s(β) = bej . α / β ⇐⇒ (i < j) or (i = j and a < b) Once an s-polynomial in a given signature T is computed all others are rewritable.

α / β ⇐⇒

s(α) lt(α)

≺

s(β) lt(β )

For any signature T define MT = {tα | α ∈ G, s(tα) = T } Choose tα such that lt(tα) is minimal.

Difference: There might be no such s-polynomial

12 / 14

Improving the rewritable signature criterion RB (generic rewrite base algorithm) F5 (as presented in [Fa02])

AP/GVW/SB

Let α and β ∈ G such that s(α) = aei and s(β) = bej . α / β ⇐⇒ (i < j) or (i = j and a < b) Once an s-polynomial in a given signature T is computed all others are rewritable.

α / β ⇐⇒

s(α) lt(α)

≺

s(β) lt(β )

For any signature T define MT = {tα | α ∈ G, s(tα) = T } Choose tα such that lt(tα) is minimal.

Difference: There might be no such s-polynomial

12 / 14

Improving the rewritable signature criterion RB (generic rewrite base algorithm) F5 (as presented in [Fa02])

AP/GVW/SB

Let α and β ∈ G such that s(α) = aei and s(β) = bej . α / β ⇐⇒ (i < j) or (i = j and a < b) Once an s-polynomial in a given signature T is computed all others are rewritable.

α / β ⇐⇒

s(α) lt(α)

≺

s(β) lt(β )

For any signature T define MT = {tα | α ∈ G, s(tα) = T } Choose tα such that lt(tα) is minimal.

Difference: There might be no such s-polynomial

12 / 14

Example for differences in the rewritable signature criterion Let K be the finite field with 13 elements and let R ..= K [x, y , z, t]. Let < be the graded reverse lexicographic monomial ordering. Consider the three input elements g1 ..= −2y 3 − x 2 z − 2x 2 t − 3y 2 t,

g2 ..= 3xyz + 2xyt,

g3 ..= 2xyz − 2yz 2 + 2z 3 + 4yzt.

13 / 14

Example for differences in the rewritable signature criterion Let K be the finite field with 13 elements and let R ..= K [x, y , z, t]. Let < be the graded reverse lexicographic monomial ordering. Consider the three input elements g1 ..= −2y 3 − x 2 z − 2x 2 t − 3y 2 t,

g2 ..= 3xyz + 2xyt,

g3 ..= 2xyz − 2yz 2 + 2z 3 + 4yzt. αi ∈ G α1 α2 α3 α4 α5 α6 α7 α8 α9 α10 α11 α12 α13 α14

reduced from e1 e2 y 2 α2 − xzα1 = S (α2 , α1 ) e3 xα4 − zα2 = S (α4 , α2 ) y 2 α4 − z 2 α1 = S (α4 , α1 ) y α5 − z 2 α2 = S (α5 , α2 ) xα5 − α6 = S (α5 , α6 ) xα6 − zα3 = S (α6 , α3 ) y α8 − z 3 α4 = S (α8 , α4 ) x 3 α4 − y α3 = S (α4 , α3 ) zα11 − x 3 α2 = S (α11 , α2 ) y α10 − x 3 α1 = S (α10 , α1 ) xα12 − α9 = S (α12 , α9 )

lt (αi ) y3 xyz x 3z 2 yz 2 xz 3 x 2z 3 x 2y 2t z5 4 x zt x 3y 2t x 4 yt x 3 zt 3 x 5 zt x 4t4

s(αi ) e1 e2 y 2 e2 e3 xe3 y 2 e3 xy e3 x 2 e3 xy 2 e3 x 2 y e3 x 3 e3 x 3 ze3 x 2 y 2 e3 x 4 ze3

←−

13 / 14

Example for differences in the rewritable signature criterion Let K be the finite field with 13 elements and let R ..= K [x, y , z, t]. Let < be the graded reverse lexicographic monomial ordering. Consider the three input elements g1 ..= −2y 3 − x 2 z − 2x 2 t − 3y 2 t,

g2 ..= 3xyz + 2xyt,

g3 ..= 2xyz − 2yz 2 + 2z 3 + 4yzt. αi ∈ G α1 α2 α3 α4 α5 α6 α7 α8 α9 α10 α11 α12 α13 α14

reduced from e1 e2 y 2 α2 − xzα1 = S (α2 , α1 ) e3 xα4 − zα2 = S (α4 , α2 ) y 2 α4 − z 2 α1 = S (α4 , α1 ) y α5 − z 2 α2 = S (α5 , α2 ) xα5 − α6 = S (α5 , α6 ) xα6 − zα3 = S (α6 , α3 ) y α8 − z 3 α4 = S (α8 , α4 ) x 3 α4 − y α3 = S (α4 , α3 ) zα11 − x 3 α2 = S (α11 , α2 ) y α10 − x 3 α1 = S (α10 , α1 ) xα12 − α9 = S (α12 , α9 )

lt (αi ) y3 xyz x 3z 2 yz 2 xz 3 x 2z 3 x 2y 2t z5 4 x zt x 3y 2t x 4 yt x 3 zt 3 x 5 zt x 4t4

s(αi ) e1 e2 y 2 e2 e3 xe3 y 2 e3 xy e3 x 2 e3 xy 2 e3 x 2 y e3 x 3 e3 x 3 ze3 x 2 y 2 e3 x 4 ze3

←−

←−

←− ←−

13 / 14

Example for differences in the rewritable signature criterion Let K be the finite field with 13 elements and let R ..= K [x, y , z, t]. Let < be the graded reverse lexicographic monomial ordering. Consider the three input elements g1 ..= −2y 3 − x 2 z − 2x 2 t − 3y 2 t,

g2 ..= 3xyz + 2xyt,

g3 ..= 2xyz − 2yz 2 + 2z 3 + 4yzt. αi ∈ G α1 α2 α3 α4 α5 α6 α7 α8 α9 α10 α11 α12 α13 α14

reduced from e1 e2 y 2 α2 − xzα1 = S (α2 , α1 ) e3 xα4 − zα2 = S (α4 , α2 ) y 2 α4 − z 2 α1 = S (α4 , α1 ) y α5 − z 2 α2 = S (α5 , α2 ) xα5 − α6 = S (α5 , α6 ) xα6 − zα3 = S (α6 , α3 ) y α8 − z 3 α4 = S (α8 , α4 ) x 3 α4 − y α3 = S (α4 , α3 ) zα11 − x 3 α2 = S (α11 , α2 ) y α10 − x 3 α1 = S (α10 , α1 ) xα12 − α9 = S (α12 , α9 )

lt (αi ) y3 xyz x 3z 2 yz 2 xz 3 x 2z 3 x 2y 2t z5 4 x zt x 3y 2t x 4 yt x 3 zt 3 x 5 zt x 4t4

s(αi ) e1 e2 y 2 e2 e3 xe3 y 2 e3 xy e3 x 2 e3 xy 2 e3 x 2 y e3 x 3 e3 x 3 ze3 x 2 y 2 e3 x 4 ze3

←−

←−

←− ←−

13 / 14

Bibliography [AP11]

A. Arri und J. Perry. The F5 Criterion revised

[EP10]

C. Eder and J. Perry. F5C: A variant of Faug` ere’s F5 Algorithm with reduced Gr¨ obner bases

[EP11]

C. Eder and J. Perry. Signature-based algorithms to compute Gr¨ obner bases

[ER13]

C. Eder and B. H. Roune. Signature Rewriting in Gr¨ obner Basis Computation

[Fa02]

J.-C. Faug` ere. A new efficient algorithm for computing Gr¨ obner bases without reduction to zero F5

[Ga12b]

V. Galkin. Simple signature-based Groebner basis algorithm

[GGV10]

S. Gao, Y. Guan and F. Volny IV. A New Incremental Algorithm for Computing Gr¨ obner Bases

[GVW11]

S. Gao, F. Volny IV and M. Wang. A New Algorithm For Computing Grobner Bases

[MSWZ12]

X. Ma, Y. Sun, D. Wang, and Y. Zhang. A Signature-Based Algorithm for Computing Groebner Bases in Solvable Polynomial Algebras

[RS12]

B. H. Roune and M. Stillman. Practical Gr¨ obner Basis Computation

[SW11]

Y. Sun and D. Wang. A Generalized Criterion for Signature Related Gr¨ obner Basis Algorithms

[V11]

F. Volny IV. New Algorithms for Computing Gr¨ obner Bases

14 / 14