For one reason on another I found myself computing Gröbner bases over fields which are not . Thus, my interest in Sage’s interface to Singular and Magma increased quite a bit again recently (before I was mainly using PolyBoRi). Hence I wrote two patches which need review (hint, hint):

A natural question when it comes to Gröbner basis computations is of course how much resources a particular input system is going to require. For many systems (in fact it is conjectured that it holds for *most* systems) it turns out one can estimate the degree which will be reached during the computation by computing the degree of semi-regularity of the homogenisation of the system, i.e. one computes the first non-zero coefficient of the following power series:

where are the respective degrees of the input polynomials. While there is a Magma script readily available for computing the degree of semi-regularity (by Luk Bettale) we didn’t have it for Sage.

sage: sr = mq.SR(1,2,1,4) sage: F,s = sr.polynomial_system() sage: I = F.ideal() sage: I.degree_of_semi_regularity() 3

Thus, we expect to only reach degree three to compute a Gröbner basis for :

sage: I = F.ideal() sage: _ = I.groebner_basis('singular:slimgb',prot='sage') Leading term degree: 1. Leading term degree: 1. Critical pairs: 56. Leading term degree: 2. Leading term degree: 2. Critical pairs: 69. Leading term degree: 3. Leading term degree: 3. Critical pairs: 461. Leading term degree: 3. Leading term degree: 3. Critical pairs: 535. Leading term degree: 2. Leading term degree: 2. Critical pairs: 423. Leading term degree: 2. Leading term degree: 2. Critical pairs: 367. Leading term degree: 3. Leading term degree: 3. Critical pairs: 0. Leading term degree: 1. Leading term degree: 1. Critical pairs: 39. Leading term degree: 1. Critical pairs: 38. Leading term degree: 1. Critical pairs: 37. Leading term degree: 1. Critical pairs: 36. Leading term degree: 1. Critical pairs: 35. Leading term degree: 1. Critical pairs: 34. Leading term degree: 1. Critical pairs: 33. Leading term degree: 1. Critical pairs: 32. Leading term degree: 1. Critical pairs: 31. Leading term degree: 1. Critical pairs: 30. Leading term degree: 1. Critical pairs: 29. Leading term degree: 1. Critical pairs: 28. Leading term degree: 1. Critical pairs: 27. Leading term degree: 1. Critical pairs: 26. Leading term degree: 1. Critical pairs: 25. Leading term degree: 1. Critical pairs: 24. Leading term degree: 1. Critical pairs: 23. Leading term degree: 1. Critical pairs: 22. Leading term degree: 1. Critical pairs: 21. Leading term degree: 1. Critical pairs: 20. Leading term degree: 1. Critical pairs: 19. Leading term degree: 1. Critical pairs: 18. Leading term degree: 1. Critical pairs: 17. Leading term degree: 1. Critical pairs: 16. Leading term degree: 1. Critical pairs: 15. Leading term degree: 1. Critical pairs: 14. Leading term degree: 1. Critical pairs: 13. Leading term degree: 1. Critical pairs: 12. Leading term degree: 1. Critical pairs: 11. Leading term degree: 1. Critical pairs: 10. Leading term degree: 1. Critical pairs: 9. Leading term degree: 1. Critical pairs: 8. Leading term degree: 1. Critical pairs: 7. Leading term degree: 1. Critical pairs: 6. Leading term degree: 1. Critical pairs: 5. Leading term degree: 1. Critical pairs: 4. Leading term degree: 1. Critical pairs: 3. Leading term degree: 1. Critical pairs: 2. Highest degree reached during computation: 3.

Which leads me to my next point.

It’s nice to see how far a particular computation is progressed and to get some summary information about the computation in the end. Hence, a new patch which enables live logs from Singular and Magma in Sage.

sage: _ = I.groebner_basis('singular:slimgb',prot=True) # Singular native 1461888602 > def sage584=slimgb(sage581); def sage584=slimgb(sage581); 1M[23,23](56)2M[56,eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeebbbbb32](69)3M[69,bbb69](461)3M[100,bbbbbb74](535)2M[100,beebbbbbb3](423)2M[23,eeeebbbbbb0](367)3M[28,eeeeebb0](0) NF:399 product criterion:16403, ext_product criterion:0 [7:3]1(39)s(38)s(37)s(36)s(35)s(34)s(33)s(32)s(31)s(30)s(29)s(28)s(27)s(26)s(25)s(24)s(23)s(22)s(21)s(20)s(19)s(18)s(17)s(16)s(15)s(14)s(13)s(12)s(11)s(10)s(9)s(8)s(7)s(6)s(5)s(4)s(3)s(2)sss (S:39)--------------------------------------- > sage: _ = I.groebner_basis('magma', prot=True) # Magma native Append(~_sage_, 0); Append(~_sage_, 0); >>>_sage_[126]:=_sage_[128]; _sage_[126]:=_sage_[128]; >>>Append(~_sage_, 0); Append(~_sage_, 0); >>>_sage_[86]:=GroebnerBasis(_sage_[126]); _sage_[86]:=GroebnerBasis(_sage_[126]); Homogeneous weights search Number of variables: 40, nullity: 0 Exact search time: 0.000 ******************** FAUGERE F4 ALGORITHM ******************** Coefficient ring: GF(2^4) Rank: 40 Order: Graded Reverse Lexicographical NEW hash table Matrix kind: Zech short Datum size: 2 No queue sort Initial length: 80 Inhomogeneous Initial queue setup time: 0.009 Initial queue length: 48 ******* STEP 1 Basis length: 80, queue length: 48, step degree: 1, num pairs: 8 Basis total mons: 264, average length: 3.300 Number of pair polynomials: 8, at 25 column(s), 0.000 Average length for reductees: 6.00 [8], reductors: 10.00 [8] Symbolic reduction time: 0.000, column sort time: 0.000 8 + 8 = 16 rows / 25 columns, 32% / 37.641% (8/r) Before ech memory: 7.9MB Row sort time: 0.000 0.000 + 0.000 = 0.000 [8] Delete 1 memory chunk(s); time: 0.000 After ech memory: 7.9MB Queue insertion time: 0.000 Step 1 time: 0.000, [0.001], mat/total: 0.000/0.009 [0.005], mem: 7.9MB ******* STEP 2 Basis length: 88, queue length: 48, step degree: 2, num pairs: 32 Basis total mons: 340, average length: 3.864 Number of pair polynomials: 32, at 169 column(s), 0.000 Average length for reductees: 3.88 [32], reductors: 7.12 [192] Symbolic reduction time: 0.000, column sort time: 0.000 32 + 192 = 224 rows / 293 columns, 2.2733% / 5.8884% (6.6607/r) Before ech memory: 7.9MB Row sort time: 0.000 0.000 + 0.000 = 0.000 [8] Delete 1 memory chunk(s); time: 0.000 After ech memory: 7.9MB Queue insertion time: 0.000 Step 2 time: 0.000, [0.003], mat/total: 0.000/0.009 [0.008], mem: 7.9MB ******* STEP 3 Basis length: 96, queue length: 69, step degree: 3, num pairs: 69 Basis total mons: 472, average length: 4.917 Number of pair polynomials: 69, at 540 column(s), 0.000 Average length for reductees: 13.20 [69], reductors: 5.50 [343] Symbolic reduction time: 0.000, column sort time: 0.000 69 + 343 = 412 rows / 719 columns, 0.94387% / 2.4223% (6.7864/r), bv Before ech memory: 7.9MB Row sort time: 0.000 0.000 + 0.000 = 0.000 [69] Delete 1 memory chunk(s); time: 0.000 After ech memory: 7.9MB Queue insertion time: 0.010 Step 3 time: 0.010, [0.015], mat/total: 0.000/0.019 [0.023], mem: 7.9MB ******* STEP 4 Basis length: 165, queue length: 736, step degree: 3, num pairs: 461 Basis total mons: 1368, average length: 8.291 Number of pair polynomials: 461, at 802 column(s), 0.000 Average length for reductees: 10.35 [461], reductors: 7.49 [654] Symbolic reduction time: 0.000, column sort time: 0.000 461 + 654 = 1115 rows / 804 columns, 1.0789% / 2.7301% (8.6744/r) Before ech memory: 7.9MB Row sort time: 0.000 0.010 + 0.000 = 0.010 [129] Delete 1 memory chunk(s); time: 0.000 Number of unused reductors: 2 After ech memory: 7.9MB Queue insertion time: 0.020 Step 4 time: 0.030, [0.022], mat/total: 0.010/0.049 [0.045], mem: 7.9MB ******* STEP 5 Basis length: 294, queue length: 2094, step degree: 2, num pairs: 132 Basis total mons: 1669, average length: 5.677 Number of pair polynomials: 132, at 149 column(s), 0.000 Average length for reductees: 2.00 [132], reductors: 5.22 [148] Symbolic reduction time: 0.000, column sort time: 0.000 132 + 148 = 280 rows / 149 columns, 2.4856% / 7.1242% (3.7036/r) Before ech memory: 7.9MB Row sort time: 0.000 0.000 + 0.000 = 0.000 [0] Delete 1 memory chunk(s); time: 0.000 After ech memory: 7.9MB Queue insertion time: 0.000 Step 5 time: 0.000, [0.001], mat/total: 0.010/0.049 [0.046], mem: 7.9MB ******* STEP 6 Basis length: 294, queue length: 1962, step degree: 3, num pairs: 848 Basis total mons: 1669, average length: 5.677 Number of pair polynomials: 57, at 85 column(s), 0.000 Average length for reductees: 2.00 [57], reductors: 2.51 [84] Symbolic reduction time: 0.000, column sort time: 0.000 57 + 84 = 141 rows / 85 columns, 2.7117% / 8.4127% (2.305/r) Before ech memory: 7.9MB Row sort time: 0.000 0.000 + 0.000 = 0.000 [0] Delete 1 memory chunk(s); time: 0.000 After ech memory: 7.9MB Queue insertion time: 0.000 Step 6 time: 0.000, [0.000], mat/total: 0.010/0.049 [0.046], mem: 7.9MB ******* STEP 7 Basis length: 294, queue length: 1114, step degree: 4, num pairs: 1098 Basis total mons: 1669, average length: 5.677 Number of pair polynomials: 0, at 0 column(s), 0.000 Symbolic reduction time: 0.000, column sort time: 0.000 0 + 0 = 0 rows / 0 columns, 0% / 0% (0/r), bv Before ech memory: 7.9MB Row sort time: 0.000 0.000 + 0.000 = 0.000 [0] After ech memory: 7.9MB Queue insertion time: 0.000 Step 7 time: 0.000, [0.000], mat/total: 0.010/0.049 [0.046], mem: 7.9MB ******* STEP 8 Basis length: 294, queue length: 16, step degree: 5, num pairs: 16 Basis total mons: 1669, average length: 5.677 Number of pair polynomials: 0, at 0 column(s), 0.000 Symbolic reduction time: 0.000, column sort time: 0.000 0 + 0 = 0 rows / 0 columns, 0% / 0% (0/r), bv Before ech memory: 7.9MB Row sort time: 0.000 0.000 + 0.000 = 0.000 [0] After ech memory: 7.9MB Queue insertion time: 0.000 Step 8 time: 0.000, [0.000], mat/total: 0.010/0.049 [0.046], mem: 7.9MB Reduce 294 final polynomial(s) by 294 16 redundant polynomial(s) removed; time: 0.000 Interreduce 40 (out of 294) polynomial(s) Symbolic reduction time: 0.000 Column sort time: 0.000 40 + 0 = 40 rows / 41 columns, 10.976% / 24.736% (4.5/r) Row sort time: 0.000 0.000 + 0.000 = 0.000 [40] Delete 1 memory chunk(s); time: 0.000 Total reduction time: 0.000 Reduction time: 0.000 Final number of polynomials: 278 Number of pairs: 759 Total pair setup time: 0.000 Max num entries matrix: 1115 by 804 Max num rows matrix: 1115 by 804 Total symbolic reduction time: 0.000 Total column sort time: 0.000 Total row sort time: 0.000 Total matrix time: 0.010 Total new polys time: 0.000 Total queue update time: 0.030 Total Faugere F4 time: 0.049, real time: 0.046 >>>_sage_[126]:=0; _sage_[126]:=0; >>> sage: _ = I.groebner_basis('singular:slimgb',prot='sage') # Singular parsed Leading term degree: 1. Leading term degree: 1. Critical pairs: 56. Leading term degree: 2. Leading term degree: 2. Critical pairs: 69. Leading term degree: 3. Leading term degree: 3. Critical pairs: 461. Leading term degree: 3. Leading term degree: 3. Critical pairs: 535. Leading term degree: 2. Leading term degree: 2. Critical pairs: 423. Leading term degree: 2. Leading term degree: 2. Critical pairs: 367. Leading term degree: 3. Leading term degree: 3. Critical pairs: 0. Leading term degree: 1. Leading term degree: 1. Critical pairs: 39. Leading term degree: 1. Critical pairs: 38. Leading term degree: 1. Critical pairs: 37. Leading term degree: 1. Critical pairs: 36. Leading term degree: 1. Critical pairs: 35. Leading term degree: 1. Critical pairs: 34. Leading term degree: 1. Critical pairs: 33. Leading term degree: 1. Critical pairs: 32. Leading term degree: 1. Critical pairs: 31. Leading term degree: 1. Critical pairs: 30. Leading term degree: 1. Critical pairs: 29. Leading term degree: 1. Critical pairs: 28. Leading term degree: 1. Critical pairs: 27. Leading term degree: 1. Critical pairs: 26. Leading term degree: 1. Critical pairs: 25. Leading term degree: 1. Critical pairs: 24. Leading term degree: 1. Critical pairs: 23. Leading term degree: 1. Critical pairs: 22. Leading term degree: 1. Critical pairs: 21. Leading term degree: 1. Critical pairs: 20. Leading term degree: 1. Critical pairs: 19. Leading term degree: 1. Critical pairs: 18. Leading term degree: 1. Critical pairs: 17. Leading term degree: 1. Critical pairs: 16. Leading term degree: 1. Critical pairs: 15. Leading term degree: 1. Critical pairs: 14. Leading term degree: 1. Critical pairs: 13. Leading term degree: 1. Critical pairs: 12. Leading term degree: 1. Critical pairs: 11. Leading term degree: 1. Critical pairs: 10. Leading term degree: 1. Critical pairs: 9. Leading term degree: 1. Critical pairs: 8. Leading term degree: 1. Critical pairs: 7. Leading term degree: 1. Critical pairs: 6. Leading term degree: 1. Critical pairs: 5. Leading term degree: 1. Critical pairs: 4. Leading term degree: 1. Critical pairs: 3. Leading term degree: 1. Critical pairs: 2. Highest degree reached during computation: 3.

Note, that these logs are all live, i.e. the are displayed during the computation as soon as the respective system makes them available.

Both patches are up for review on Sage’s trac server.