Signature-based Gröbner basis computation

Transcription

1 Signature-based Gröbner basis computation Christian Eder POLSYS Team, UPMC, Paris, France March 08, / 22

2 1 The basic problem 2 Generic signature-based algorithms The basic idea Generic signature-based Gröbner basis algorithm Signature-based criteria 3 Implementations and recent work A good decade on signature-based algorithms Implementation in Singular 4 And what has really happened? Ongoing work 2 / 22

3 Example How to predict zero reductions? Let I = g 1, g 2 Q[x, y, z] be given where g 1 = xy z 2, g 2 = y 2 z 2, and let < be the graded reverse lexicographical ordering. 3 / 22

4 Example How to predict zero reductions? Let I = g 1, g 2 Q[x, y, z] be given where g 1 = xy z 2, g 2 = y 2 z 2, and let < be the graded reverse lexicographical ordering. spol(g 2, g 1 ) = xg 2 yg 1 = xy 2 xz 2 xy 2 + yz 2 = xz 2 + yz 2, so it reduces w.r.t. G to g 3 = xz 2 yz 2. 3 / 22

5 Example How to predict zero reductions? Let I = g 1, g 2 Q[x, y, z] be given where g 1 = xy z 2, g 2 = y 2 z 2, and let < be the graded reverse lexicographical ordering. spol(g 2, g 1 ) = xg 2 yg 1 = xy 2 xz 2 xy 2 + yz 2 = xz 2 + yz 2, so it reduces w.r.t. G to g 3 = xz 2 yz 2. spol(g 3, g 1 ) = xyz 2 y 2 z 2 xyz 2 + z 4 = y 2 z 2 + z 4. 3 / 22

6 Example How to predict zero reductions? Let I = g 1, g 2 Q[x, y, z] be given where g 1 = xy z 2, g 2 = y 2 z 2, and let < be the graded reverse lexicographical ordering. spol(g 2, g 1 ) = xg 2 yg 1 = xy 2 xz 2 xy 2 + yz 2 = xz 2 + yz 2, so it reduces w.r.t. G to g 3 = xz 2 yz 2. spol(g 3, g 1 ) = xyz 2 y 2 z 2 xyz 2 + z 4 = y 2 z 2 + z 4. We can reduce even further with z 2 g 2 : y 2 z 2 + z 4 + y 2 z 2 z 4 = 0. 3 / 22

7 Example How to predict zero reductions? Let I = g 1, g 2 Q[x, y, z] be given where g 1 = xy z 2, g 2 = y 2 z 2, and let < be the graded reverse lexicographical ordering. spol(g 2, g 1 ) = xg 2 yg 1 = xy 2 xz 2 xy 2 + yz 2 = xz 2 + yz 2, so it reduces w.r.t. G to g 3 = xz 2 yz 2. spol(g 3, g 1 ) = xyz 2 y 2 z 2 xyz 2 + z 4 = y 2 z 2 + z 4. We can reduce even further with z 2 g 2 : y 2 z 2 + z 4 + y 2 z 2 z 4 = 0. How can we discard such zero reductions in advance? 3 / 22

8 1 The basic problem 2 Generic signature-based algorithms The basic idea Generic signature-based Gröbner basis algorithm Signature-based criteria 3 Implementations and recent work A good decade on signature-based algorithms Implementation in Singular 4 And what has really happened? Ongoing work 4 / 22

9 Let I = f 1,..., f m. Idea: Give each f I a bit more structure: Signatures of polynomials 5 / 22

10 Let I = f 1,..., f m. Idea: Give each f I a bit more structure: Signatures of polynomials 1. Let R m be generated by e 1,..., e m, a well-ordering on the monomials of R m, and let π : R m R such that π(e i ) = f i for all i. 5 / 22

11 Let I = f 1,..., f m. Idea: Give each f I a bit more structure: Signatures of polynomials 1. Let R m be generated by e 1,..., e m, a well-ordering on the monomials of R m, and let π : R m R such that π(e i ) = f i for all i. 2. Each p I can be represented by an m s = h i e i R m such that p = π(s). i=1 5 / 22

12 Let I = f 1,..., f m. Idea: Give each f I a bit more structure: Signatures of polynomials 1. Let R m be generated by e 1,..., e m, a well-ordering on the monomials of R m, and let π : R m R such that π(e i ) = f i for all i. 2. Each p I can be represented by an m s = h i e i R m such that p = π(s). i=1 3. A signature of p is given by sig(p) = lm (s) with p = π(s). 5 / 22

13 Let I = f 1,..., f m. Idea: Give each f I a bit more structure: Signatures of polynomials 1. Let R m be generated by e 1,..., e m, a well-ordering on the monomials of R m, and let π : R m R such that π(e i ) = f i for all i. 2. Each p I can be represented by an m s = h i e i R m such that p = π(s). i=1 3. A signature of p is given by sig(p) = lm (s) with p = π(s). 4. A minimal signature of p exists due to. 5 / 22

14 Our example now with signatures and pot We have already computed the following data: g 1 = xy z 2, sig(g 1 ) = e 1, g 2 = y 2 z 2, sig(g 2 ) = e 2, g 3 = spol(g 2, g 1 ) = xg 2 yg 1 sig(g 3 ) = x sig(g 2 ) = xe 2. 6 / 22

15 Our example now with signatures and pot We have already computed the following data: g 1 = xy z 2, sig(g 1 ) = e 1, g 2 = y 2 z 2, sig(g 2 ) = e 2, g 3 = spol(g 2, g 1 ) = xg 2 yg 1 sig(g 3 ) = x sig(g 2 ) = xe 2. spol(g 3, g 1 ) = yg 3 z 2 g 1 : sig (spol(g 3, g 1 )) = y sig(g 3 ) = xye 2. 6 / 22

16 Our example now with signatures and pot We have already computed the following data: g 1 = xy z 2, sig(g 1 ) = e 1, g 2 = y 2 z 2, sig(g 2 ) = e 2, g 3 = spol(g 2, g 1 ) = xg 2 yg 1 sig(g 3 ) = x sig(g 2 ) = xe 2. spol(g 3, g 1 ) = yg 3 z 2 g 1 : sig (spol(g 3, g 1 )) = y sig(g 3 ) = xye 2. Note that sig (spol(g 3, g 1 )) = xye 2 and lm(g 1 ) = xy. 6 / 22

17 Our example now with signatures and pot We have already computed the following data: g 1 = xy z 2, sig(g 1 ) = e 1, g 2 = y 2 z 2, sig(g 2 ) = e 2, g 3 = spol(g 2, g 1 ) = xg 2 yg 1 sig(g 3 ) = x sig(g 2 ) = xe 2. spol(g 3, g 1 ) = yg 3 z 2 g 1 : sig (spol(g 3, g 1 )) = y sig(g 3 ) = xye 2. Note that sig (spol(g 3, g 1 )) = xye 2 and lm(g 1 ) = xy. We know that spol(g 3, g 1 ) will reduce to zero w.r.t. G. 6 / 22

18 Why do we know this? The general idea is to check the signatures of the generated s-polynomials. 7 / 22

19 Why do we know this? The general idea is to check the signatures of the generated s-polynomials. If sig ( spol(f, g) ) is not minimal for spol(f, g) then spol(f, g) is discarded. 7 / 22

20 Why do we know this? The general idea is to check the signatures of the generated s-polynomials. If sig ( spol(f, g) ) is not minimal for spol(f, g) then spol(f, g) is discarded. Our goal Find and discard as many s-polynomials as possible for which the algorithm computes a non-minimal signature. 7 / 22

21 Why do we know this? The general idea is to check the signatures of the generated s-polynomials. If sig ( spol(f, g) ) is not minimal for spol(f, g) then spol(f, g) is discarded. Our goal Find and discard as many s-polynomials as possible for which the algorithm computes a non-minimal signature. Our task We need to take care of the correctness of the signatures throughout the computations. 7 / 22

22 Generic signature-based Gröbner basis algorithm Input: Ideal I = f 1,..., f m Output: Gröbner Basis poly(g) for I 1. G 2. G G {(e i, f i )} for all i {1,..., m} 3. P {(g i, g j ) g i, g j G, i > j} 8 / 22

23 Generic signature-based Gröbner basis algorithm Input: Ideal I = f 1,..., f m Output: Gröbner Basis poly(g) for I 1. G 2. G G {(e i, f i )} for all i {1,..., m} 3. P {(g i, g j ) g i, g j G, i > j} 4. While P (a) Choose (f, g) P such that sig (spol(f, g)) minimal, P P \ {(f, g)} (b) If sig (spol(f, g)) minimal for spol(f, g): 8 / 22

24 Generic signature-based Gröbner basis algorithm Input: Ideal I = f 1,..., f m Output: Gröbner Basis poly(g) for I 1. G 2. G G {(e i, f i )} for all i {1,..., m} 3. P {(g i, g j ) g i, g j G, i > j} 4. While P (a) Choose (f, g) P such that sig (spol(f, g)) minimal, P P \ {(f, g)} (b) If sig (spol(f, g)) minimal for spol(f, g): (i) h spol(f, g) (ii) If poly(h) G 0 8 / 22

25 Generic signature-based Gröbner basis algorithm Input: Ideal I = f 1,..., f m Output: Gröbner Basis poly(g) for I 1. G 2. G G {(e i, f i )} for all i {1,..., m} 3. P {(g i, g j ) g i, g j G, i > j} 4. While P (a) Choose (f, g) P such that sig (spol(f, g)) minimal, P P \ {(f, g)} (b) If sig (spol(f, g)) minimal for spol(f, g): (i) h spol(f, g) (ii) If poly(h) G 0 (iii) If poly(h) G poly(r) 0 5. Return poly(g). P P {(r, g) g G} G G {r} 8 / 22

26 Generic signature-based Gröbner basis algorithm Input: Ideal I = f 1,..., f m Output: Gröbner Basis poly(g) for I 1. G 2. G G {(e i, f i )} for all i {1,..., m} 3. P {(g i, g j ) g i, g j G, i > j} 4. While P (a) Choose (f, g) P such that sig (spol(f, g)) minimal, P P \ {(f, g)} (b) If sig (spol(f, g)) minimal for spol(f, g): (i) h spol(f, g) (ii) If poly(h) G 0 signature-safe (iii) If poly(h) G poly(r) 0 signature-safe & g G such that m sig(g) = sig(r) and m lm(poly(g)) = lm(poly(r)) P P {(r, g) g G} G G {r} 5. Return poly(g). 8 / 22

27 Signature-safe reductions Let p and q in R be given such that m lm(q) = lm(p), c = lc(p) lc(q). Assume p cmq. 9 / 22

28 Signature-safe reductions Let p and q in R be given such that m lm(q) = lm(p), c = lc(p) lc(q). Assume p cmq. signature-safe: sig(p cmq) = sig(p) 9 / 22

29 Signature-safe reductions Let p and q in R be given such that m lm(q) = lm(p), c = lc(p) lc(q). Assume p cmq. signature-safe: sig(p cmq) = sig(p) signature-safe: signature-increasing: sig(p cmq) = m sig(q) 9 / 22

30 Signature-safe reductions Let p and q in R be given such that m lm(q) = lm(p), c = lc(p) lc(q). Assume p cmq. signature-safe: sig(p cmq) = sig(p) signature-safe: signature-increasing: sig(p cmq) = m sig(q) signature-decreasing: sig(p cmq) sig(p), m sig(q) 9 / 22

31 How does this work? Termination If sig(r) = m sig(g) and lm (poly(r)) = m lm (poly(g)) is not added to G. Each new element in G enlarges (sig(r), lm(poly(r))). 10 / 22

32 How does this work? Termination If sig(r) = m sig(g) and lm (poly(r)) = m lm (poly(g)) is not added to G. Each new element in G enlarges (sig(r), lm(poly(r))). Correctness All possible s-polynomials are taken care of: signature-increasing reduction new pair in the next step. All elements r with poly(r) 0 are added to G besides those fulfilling sig(r) = m sig(g) and lm (poly(r)) = m lm (poly(g)). 10 / 22

33 Signature-based criteria Non-minimal signature ( NM ) sig(h) not minimal for h? Remove h. 11 / 22

34 Signature-based criteria Non-minimal signature ( NM ) sig(h) not minimal for h? Remove h. Sketch of proof 1. There exists a syzygy s R m such that lm(s) = sig(h). We can represent h with a lower signature. 2. Pairs are handled by increasing signatures. All relations of lower signature are already taken care of. 11 / 22

35 Signature-based criteria Non-minimal signature ( NM ) sig(h) not minimal for h? Remove h. Sketch of proof 1. There exists a syzygy s R m such that lm(s) = sig(h). We can represent h with a lower signature. 2. Pairs are handled by increasing signatures. All relations of lower signature are already taken care of. Our example with pot revisited sig (spol(g 3, g 1 )) = xye 2 g 1 = xy z 2 } g 2 = y 2 z 2 psyz(g 2, g 1 ) = g 1 e 2 g 2 e 1 = xye / 22

36 Signature-based criteria Rewritable signature ( RW ) sig(g) = sig(h)? Remove either g or h. 12 / 22

37 Signature-based criteria Rewritable signature ( RW ) sig(g) = sig(h)? Remove either g or h. Sketch of proof 1. sig(g h) sig(g), sig(h). 2. Pairs are handled by increasing signatures. All necessary computations of lower signature have already taken place. We can represent h by h = g + elements of lower signature. 12 / 22

38 1 The basic problem 2 Generic signature-based algorithms The basic idea Generic signature-based Gröbner basis algorithm Signature-based criteria 3 Implementations and recent work A good decade on signature-based algorithms Implementation in Singular 4 And what has really happened? Ongoing work 13 / 22

39 A good decade on signature-based algorithms SB Roune, Stillmann Quasihomog Safey El-Din, Verron (2013) using sym Svartz (2013) AP1 Arri,Perry, AP Arri,Perry Involutive Gerdt, Hashemi, Alizadeh (2013) F4/5 Albrecht, Perry (2010) SAGBI Rahmany AP2 Arri,Perry, n Matrix Bardet (2002) Faugère (2002) ic with BC Ars (2005) C Perry, Bihomog Safey El-Din, Spaenlehauer G2V Gao,Guan,Volny (2010) ig2v A Perry, Extended Criteria Ars, Hashemi GVW Gao,Volny,Wang ia 14 / 22

40 A good decade on signature-based algorithms SB Roune, Stillmann Quasihomog Safey El-Din, Verron (2013) using sym Svartz (2013) AP1 Arri,Perry, AP Arri,Perry Involutive Gerdt, Hashemi, Alizadeh (2013) F4/5 Albrecht, Perry (2010) SAGBI Rahmany AP2 Arri,Perry, n Matrix Bardet (2002) Faugère (2002) ic with BC Ars (2005) C Perry, Bihomog Safey El-Din, Spaenlehauer G2V Gao,Guan,Volny (2010) ig2v A Perry, Extended Criteria Ars, Hashemi GVW Gao,Volny,Wang ia 14 / 22

41 A good decade on signature-based algorithms SB Roune, Stillmann Quasihomog Safey El-Din, Verron (2013) using sym Svartz (2013) AP1 Arri,Perry, AP Arri,Perry Involutive Gerdt, Hashemi, Alizadeh (2013) F4/5 Albrecht, Perry (2010) SAGBI Rahmany AP2 Arri,Perry, n Matrix Bardet (2002) Faugère (2002) ic with BC Ars (2005) C Perry, Bihomog Safey El-Din, Spaenlehauer G2V Gao,Guan,Volny (2010) ig2v A Perry, Extended Criteria Ars, Hashemi GVW Gao,Volny,Wang ia 14 / 22

42 A good decade on signature-based algorithms SB Roune, Stillmann Quasihomog Safey El-Din, Verron (2013) using sym Svartz (2013) AP1 Arri,Perry, AP Arri,Perry Involutive Gerdt, Hashemi, Alizadeh (2013) F4/5 Albrecht, Perry (2010) SAGBI Rahmany AP2 Arri,Perry, n Matrix Bardet (2002) Faugère (2002) ic with BC Ars (2005) C Perry, Bihomog Safey El-Din, Spaenlehauer G2V Gao,Guan,Volny (2010) ig2v A Perry, Extended Criteria Ars, Hashemi GVW Gao,Volny,Wang ia 14 / 22

43 A good decade on signature-based algorithms SB Roune, Stillmann Quasihomog Safey El-Din, Verron (2013) using sym Svartz (2013) AP1 Arri,Perry, AP Arri,Perry Involutive Gerdt, Hashemi, Alizadeh (2013) F4/5 Albrecht, Perry (2010) SAGBI Rahmany AP2 Arri,Perry, n Matrix Bardet (2002) Faugère (2002) ic with BC Ars (2005) C Perry, Bihomog Safey El-Din, Spaenlehauer G2V Gao,Guan,Volny (2010) A Perry, Extended Criteria Ars, Hashemi GVW Gao,Volny,Wang ia 14 / 22

44 A good decade on signature-based algorithms SB Roune, Stillmann Quasihomog Safey El-Din, Verron (2013) using sym Svartz (2013) AP1 Arri,Perry, AP Arri,Perry Involutive Gerdt, Hashemi, Alizadeh (2013) F4/5 Albrecht, Perry (2010) SAGBI Rahmany AP2 Arri,Perry, n Matrix Bardet (2002) Faugère (2002) ic with BC Ars (2005) C Perry, Bihomog Safey El-Din, Spaenlehauer G2V Gao,Guan,Volny (2010) ig2v A Perry, Extended Criteria Ars, Hashemi GVW Gao,Volny,Wang ia 14 / 22

45 A good decade on signature-based algorithms SB Roune, Stillmann Quasihomog Safey El-Din, Verron (2013) using sym Svartz (2013) AP1 Arri,Perry, AP Arri,Perry Involutive Gerdt, Hashemi, Alizadeh (2013) F4/5 Albrecht, Perry (2010) SAGBI Rahmany AP2 Arri,Perry, n Matrix Bardet (2002) Faugère (2002) ic with BC Ars (2005) C Perry, Bihomog Safey El-Din, Spaenlehauer G2V Gao,Guan,Volny (2010) ig2v A Perry, Extended Criteria Ars, Hashemi GVW Gao,Volny,Wang ia 14 / 22

46 A good decade on signature-based algorithms SB Roune, Stillmann Quasihomog Safey El-Din, Verron (2013) using sym Svartz (2013) AP1 Arri,Perry, AP Arri,Perry Involutive Gerdt, Hashemi, Alizadeh (2013) F4/5 Albrecht, Perry (2010) SAGBI Rahmany AP2 Arri,Perry, n Matrix Bardet (2002) Faugère (2002) ic with BC Ars (2005) C Perry, Bihomog Safey El-Din, Spaenlehauer G2V Gao,Guan,Volny (2010) ig2v A Perry, Extended Criteria Ars, Hashemi GVW Gao,Volny,Wang ia 14 / 22

47 A good decade on signature-based algorithms SB Roune, Stillmann Quasihomog Safey El-Din, Verron (2013) using sym Svartz (2013) AP1 Arri,Perry, AP Arri,Perry Involutive Gerdt, Hashemi, Alizadeh (2013) F4/5 Albrecht, Perry (2010) SAGBI Rahmany AP2 Arri,Perry, n Matrix Bardet (2002) Faugère (2002) ic with BC Ars (2005) C Perry, Bihomog Safey El-Din, Spaenlehauer G2V Gao,Guan,Volny (2010) ig2v A Perry, Extended Criteria Ars, Hashemi GVW Gao,Volny,Wang ia 14 / 22

48 Implemented in Singular SB Roune, Stillmann Quasihomog Safey El-Din, Verron (2013) using sym Svartz (2013) AP Arri,Perry AP1 Arri,Perry, Involutive Gerdt, Hashemi, Alizadeh (2013) F4/5 Albrecht, Perry (2010) SAGBI Rahmany AP2 Arri,Perry, n Matrix Bardet (2002) Faugère (2002) ic with BC Ars (2005) C Perry, Bihomog Extended Safey El-Din, Criteria Spaenlehauer Ars, Hashemi G2V Gao,Guan,Volny (2010) ig2v A Perry, GVW Gao,Volny,Wang ia 14 / 22

49 Implementation in Singular 100% 60% 20% Virasoro-8 Red-Cyc-7 Fabrice-24 Noon-9 Rose Red-Eco-11 Sendra Wang / 22

50 Implementation in Singular 100% 60% 20% Virasoro-8 Red-Cyc-7 Fabrice-24 Noon-9 Rose Red-Eco-11 Sendra Wang % 60% 20% Cyclic-8 Ext-Cyc-6 Katsura-13 Berth Eco-11 F-855 Schrans-Troost RPBL 15 / 22

51 1 The basic problem 2 Generic signature-based algorithms The basic idea Generic signature-based Gröbner basis algorithm Signature-based criteria 3 Implementations and recent work A good decade on signature-based algorithms Implementation in Singular 4 And what has really happened? Ongoing work 16 / 22

52 And what has really happened? Rather boring We have (hopefully) understood the criteria. We have proven termination of et al. We have implemented signature-based Buchberger-style Gröbner basis algorithms quite a lot. 17 / 22

53 And what has really happened? Rather boring We have (hopefully) understood the criteria. We have proven termination of et al. We have implemented signature-based Buchberger-style Gröbner basis algorithms quite a lot. At least some new ideas We use different module monomial orderings on the signatures to allow non-incremental computations. We have improved the incremental variants a bit (reduced intermediate bases) There are some slight improvements on the signature-based criteria. 17 / 22

54 Improving the non-minimal signature criterion (as presented in [Fa02]) G2V/AP/GVW/SB/A use principal syzygies use signatures of zero reductions 18 / 22

55 Improving the non-minimal signature criterion (as presented in [Fa02]) G2V/AP/GVW/SB/A use principal syzygies use signatures of zero reductions 18 / 22

56 Improving the non-minimal signature criterion (as presented in [Fa02]) G2V/AP/GVW/SB/A use principal syzygies use signatures of zero reductions 18 / 22

57 Improving the non-minimal signature criterion (as presented in [Fa02]) G2V/AP/GVW/SB/A use principal syzygies use signatures of zero reductions Remark This helps only if the input sequence is not regular. 18 / 22

58 Improving the rewritable signature criterion (as presented in [Fa02]) Fix a total ordering on G. A basis element g G is a rewriter in signature T if sig(g) T. The -maximal rewriter in T is the canonical rewriter. An element mg is rewritable if g is not the canonical rewriter in sig(mg). AP/GVW/SB For any signature T define M T = {mg g G, sig(mg) = T } Choose mg such that m lm (poly(g)) is minimal. Compute the corresponding s-polynomial with mg. Difference: There might be no such s-polynomial 19 / 22

59 Improving the rewritable signature criterion (as presented in [Fa02]) Fix a total ordering on G. A basis element g G is a rewriter in signature T if sig(g) T. The -maximal rewriter in T is the canonical rewriter. An element mg is rewritable if g is not the canonical rewriter in sig(mg). AP/GVW/SB For any signature T define M T = {mg g G, sig(mg) = T } Choose mg such that m lm (poly(g)) is minimal. Compute the corresponding s-polynomial with mg. Difference: There might be no such s-polynomial 19 / 22

60 Improving the rewritable signature criterion (as presented in [Fa02]) Fix a total ordering on G. A basis element g G is a rewriter in signature T if sig(g) T. The -maximal rewriter in T is the canonical rewriter. An element mg is rewritable if g is not the canonical rewriter in sig(mg). AP/GVW/SB For any signature T define M T = {mg g G, sig(mg) = T } Choose mg such that m lm (poly(g)) is minimal. Compute the corresponding s-polynomial with mg. Difference: There might be no such s-polynomial 19 / 22

61 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 g 1.= 2y 3 x 2 z 2x 2 t 3y 2 t, g 3.= 2xyz 2yz 2 + 2z 3 + 4yzt. g 2.= 3xyz + 2xyt, 20 / 22

62 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 g 1.= 2y 3 x 2 z 2x 2 t 3y 2 t, g 3.= 2xyz 2yz 2 + 2z 3 + 4yzt. g 2.= 3xyz + 2xyt, g i G reduced from lm (poly(g i )) sig(g i ) g 1 e 1 y 3 e 1 g 2 e 2 xyz e 2 g 3 y 2 g 2 xzg 1 = spol (g 2, g 1) x 3 z 2 y 2 e 2 g 4 e 3 yz 2 e 3 g 5 xg 3 zg 2 = spol (g 3, g 2) xz 3 xe 3 g 6 y 2 g 3 z 2 g 1 = spol (g 3, g 1) x 2 z 3 y 2 e 3 g 7 yg 5 z 2 g 2 = spol (g 5, g 2) x 2 y 2 t xye 3 g 8 xg 5 g 6 = spol (g 5, g 6) z 5 x 2 e 3 g 9 xg 6 zg 3 = spol (g 6, g 3) x 4 zt xy 2 e 3 g 10 yg 8 z 3 g 4 = spol (g 8, g 4) x 3 y 2 t x 2 ye 3 g 11 x 3 g 4 yg 3 = spol (g 4, g 3) x 4 yt x 3 e 3 g 12 zg 11 x 3 g 2 = spol (g 11, g 2) x 3 zt 3 x 3 ze 3 g 13 yg 10 x 3 g 1 = spol (g 10, g 1) x 5 zt x 2 y 2 e 3 g 14 xg 12 g 9 = spol (g 12, g 9) x 4 t 4 x 4 ze 3 20 / 22

63 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 g 1.= 2y 3 x 2 z 2x 2 t 3y 2 t, g 3.= 2xyz 2yz 2 + 2z 3 + 4yzt. g 2.= 3xyz + 2xyt, g i G reduced from lm (poly(g i )) sig(g i ) g 1 e 1 y 3 e 1 g 2 e 2 xyz e 2 g 3 y 2 g 2 xzg 1 = spol (g 2, g 1) x 3 z 2 y 2 e 2 g 4 e 3 yz 2 e 3 g 5 xg 3 zg 2 = spol (g 3, g 2) xz 3 xe 3 g 6 y 2 g 3 z 2 g 1 = spol (g 3, g 1) x 2 z 3 y 2 e 3 g 7 yg 5 z 2 g 2 = spol (g 5, g 2) x 2 y 2 t xye 3 g 8 xg 5 g 6 = spol (g 5, g 6) z 5 x 2 e 3 g 9 xg 6 zg 3 = spol (g 6, g 3) x 4 zt xy 2 e 3 g 10 yg 8 z 3 g 4 = spol (g 8, g 4) x 3 y 2 t x 2 ye 3 g 11 x 3 g 4 yg 3 = spol (g 4, g 3) x 4 yt x 3 e 3 g 12 zg 11 x 3 g 2 = spol (g 11, g 2) x 3 zt 3 x 3 ze 3 g 13 yg 10 x 3 g 1 = spol (g 10, g 1) x 5 zt x 2 y 2 e 3 g 14 xg 12 g 9 = spol (g 12, g 9) x 4 t 4 x 4 ze 3 20 / 22

64 Why am I here? F4: linear algebra for reduction purposes Heuristics: orderings on signatures; orderings for critical pairs (sugar degree), reducers Parallelisation: modular methods, parallel criteria checks Computation of syzygies: implementation Generalization of signature-based criteria: more terms per signature, relaxing criteria for combination with Buchberger s criteria 21 / 22

65 [AP10] M. Albrecht und J. Perry. F4/5 [A05] G. Ars. Applications des bases de Groeobner a la cryptographie [AH09] G. Ars und A. Hashemi. Extended Criteria [AP11] A. Arri und J. Perry. The Criterion revised [E12a] C. Eder. Improving incremental signature-based Gröbner bases algorithms [E12b] C. Eder. Sweetening the sour taste of inhomogeneous signature-based Gröbner basis computations [EGP11] C. Eder, J. Gash and J. Perry. Modifying Faugère s Algorithm to ensure termination [EP10] C. Eder and J. Perry. C: A variant of Faugère s Algorithm with reduced Gröbner bases [EP11] C. Eder and J. Perry. Signature-based algorithms to compute Gröbner bases [ER13] C. Eder and B. H. Roune. Signature Rewriting in Gröbner Basis Computation Bibliography [Fa02] J.-C. Faugère. A new efficient algorithm for computing Gröbner bases without reduction to zero F 5 [FR09] [FSS11] [FSV13] [FS12] J.-C. Faugère and S. Rahmany. Solving systems of polynomial equations with symmetries using SAGBI-Gröbner bases J.-C. M. Safey El-Din and P.-J. Spaenlehauer. Gröbner Bases of Bihomogeneous Ideals Generated by Polynomials of Bidegree (1,1) J.-C. M. Safey El-Din and T. Verron. Computing Gröbner bases for quasi-homogeneous systems J.-C. Faugère and J. Svartz. Solving Polynomial Systems Globally Invariant Under an Action of the Symmetric Group and Application to the Equilibria of N vortices in the Plane [Ga12a] V. Galkin. Termination of original F 5 [Ga12b] V. Galkin. Simple signature-based Groebner basis algorithm [GGV10] S. Gao, Y. Guan and F. Volny IV. A New Incremental Algorithm for Computing Gröbner Bases [GHA13] V. Gerdt, A. Hashemi and B. M.-Alizadeh. Involutive Bases Algorithm Incorporating Criterion [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 [PHW13] S. Pan, Y. Hu and B. Wang. The Termination of Algorithms for Computing Gröbner Bases [RS12] B. H. Roune and M. Stillman. Practical Gröbner Basis Computation [SW10] Y. Sun und D. Wang. A new proof of the Algorithm [SW11] Y. Sun and D. Wang. A Generalized Criterion for Signature Related Gröbner Basis Algorithms 22 / 22