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Solve much slower than Mathematica for simple system

asked 2022-02-18 19:57:18 +0100

ddd gravatar image

updated 2022-02-18 21:27:21 +0100

Mathematica solves the following system of multivariate polynomials almost instantaneously while sagemath takes around 10 seconds. Is there any way to speed up solve?

vars('a15, x13_14, a4, a11, a0, x14_15, x11_13, x5_5, x9_12, a5, a12, x3_4, a1, x7_7, x11_14, x12_15, x5_6, x4_4, x9_9, a6, a13, x3_5, a2, x10_10, x7_8, x6_6, x5_7, x12_12, x8_8, a7, x3_6, x6_7, x14_14, a9, x15_15, x4_6, x11_11, x8_9, a8, x10_12, a14, x13_13, a3, a10, x4_7, x12_14, x11_12, x13_15, x10_13')

polys = [-x3_4*x9_12, -x10_12*x3_4, -x10_13*x3_4, -x11_12*x3_4, -x11_13*x3_4, -x11_14*x3_4, x3_4^2*x7_7, x7_8*x8_9, -x11_12^2*x8_8, -x12_12*x3_4 + x3_4*x4_4, -x12_14*x3_4 + x3_4*x4_6, -x12_15*x3_4 + x3_4*x4_7, -x13_13*x3_4 + x3_4*x5_5, -x13_14*x3_4 + x3_4*x5_6, -x13_15*x3_4 + x3_4*x5_7, -x14_14*x3_4 + x3_4*x6_6, -x14_15*x3_4 + x3_4*x6_7, -x15_15*x3_4*x7_8 + x3_4*x7_8, x3_4*x3_5*x7_7, x3_4*x3_6*x7_7, a0 - 1, a1 - 1, a2 - 1, a3 - 1, a4*x4_4 - 1, a5*x5_5 - 1, a6*x6_6 - 1, a7*x7_7 - 1, a8*x8_8 - 1, a9*x9_9 - 1, a10*x10_10 - 1, a11*x11_11 - 1, a12*x12_12 - 1, a13*x13_13 - 1, a14*x14_14 - 1, a15*x15_15 - 1]

vs = [a15, x13_14, a4, a11, a0, x14_15, x11_13, x5_5, x9_12, a5, a12, x3_4, a1, x7_7, x11_14, x12_15, x5_6, x4_4, x9_9, a6, a13, x3_5, a2, x10_10, x7_8, x6_6, x5_7, x12_12, x8_8, a7, x3_6, x6_7, x14_14, a9, x15_15, x4_6, x11_11, x8_9, a8, x10_12, a14, x13_13, a3, a10, x4_7, x12_14, x11_12, x13_15, x10_13]

%time solve(polys, *vs)

This takes about 10 seconds

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Comments

Try solving via Grobner basis: https://ask.sagemath.org/question/584...

Max Alekseyev gravatar imageMax Alekseyev ( 2022-02-18 21:11:48 +0100 )edit
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2 Answers

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answered 2022-02-19 12:53:53 +0100

rburing gravatar image

updated 2022-02-19 12:54:34 +0100

You can calulate a Gröbner basis with respect to a lexicographic ordering to get an equivalent "triangular" system that can be solved back-to-front by successive solving and substitution (introducing parameters whenever there is a choice to be made).

sage: R.<a15, x13_14, a4, a11, a0, x14_15, x11_13, x5_5, x9_12, a5, a12, x3_4, a1, x7_7, x11_14, x12_15, x5_6, x4_4, x9_9, a6, a13, x3_5, a2, x10_10, x7_8, x6_6, x5_7, x12_12, x8_8, a7, x3_6, x6_7, x14_14, a9, x15_15, x4_6, x11_11, x8_9, a8, x10_12, a14, x13_13, a3, a10, x4_7, x12_14, x11_12, x13_15, x10_13> = PolynomialRing(QQ, order='invlex')
sage: polys = [-x3_4*x9_12, -x10_12*x3_4, -x10_13*x3_4, -x11_12*x3_4, -x11_13*x3_4, -x11_14*x3_4, x3_4^2*x7_7, x7_8*x8_9, -x11_12^2*x8_8, -x12_12*x3_4 + x3_4*x4_4, -x12_14*x3_4 + x3_4*x4_6, -x12_15*x3_4 + x3_4*x4_7, -x13_13*x3_4 + x3_4*x5_5, -x13_14*x3_4 + x3_4*x5_6, -x13_15*x3_4 + x3_4*x5_7, -x14_14*x3_4 + x3_4*x6_6, -x14_15*x3_4 + x3_4*x6_7, -x15_15*x3_4*x7_8 + x3_4*x7_8, x3_4*x3_5*x7_7, x3_4*x3_6*x7_7, a0 - 1, a1 - 1, a2 - 1, a3 - 1, a4*x4_4 - 1, a5*x5_5 - 1, a6*x6_6 - 1, a7*x7_7 - 1, a8*x8_8 - 1, a9*x9_9 - 1, a10*x10_10 - 1, a11*x11_11 - 1, a12*x12_12 - 1, a13*x13_13 - 1, a14*x14_14 - 1, a15*x15_15 - 1]
sage: I = R.ideal(polys)
sage: %time G = I.groebner_basis(algorithm='libsingular:std')
CPU times: user 4.45 ms, sys: 91 µs, total: 4.55 ms
Wall time: 4.57 ms
sage: list(G)
[a15*x3_4*x7_8 - x3_4*x7_8, a15*x15_15 - 1, x13_14*x3_4 - x3_4*x5_6, a4*x3_4 - a12*x3_4, a4*x4_4 - 1, a11*x11_11 - 1, a0 - 1, x14_15*x3_4 - x3_4*x6_7, x11_13*x3_4, x5_5*a5 - 1, x5_5*x3_4 - x3_4*x13_13, x9_12*x3_4, a5*x3_4 - x3_4*a13, a12*x12_12 - 1, x3_4^2, x3_4*x11_14, x3_4*x12_15 - x3_4*x4_7, x3_4*x4_4 - x3_4*x12_12, x3_4*a6 - x3_4*a14, x3_4*x3_5, x3_4*x7_8*x15_15 - x3_4*x7_8, x3_4*x6_6 - x3_4*x14_14, x3_4*x5_7 - x3_4*x13_15, x3_4*x3_6, x3_4*x4_6 - x3_4*x12_14, x3_4*x10_12, x3_4*x11_12, x3_4*x10_13, a1 - 1, x7_7*a7 - 1, x9_9*a9 - 1, a6*x6_6 - 1, a13*x13_13 - 1, a2 - 1, x10_10*a10 - 1, x7_8*x8_9, x8_8*a8 - 1, x14_14*a14 - 1, a3 - 1, x11_12^2]

This equivalent system can be passed to solve, which then does the job faster:

sage: %time solve(list(map(SR, G)), list(map(SR, R.gens())))
CPU times: user 3.04 s, sys: 170 ms, total: 3.21 s
Wall time: 2.53 s

Different variable orderings may lead to different speeds (the monomial ordering can be restricted to lex, I just used invlex as a shorthand for a variable ordering that happened to be faster). Another possible intermediate step is to calculate the radical of the ideal, though that is probably slow in general.

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Comments

1

This is significantly faster. Unfortunately it is still orders of magnitude behind Mathematica and I need to repeat this solving process millions of times.

ddd gravatar imageddd ( 2022-02-19 18:53:24 +0100 )edit
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0

answered 2022-02-19 19:20:44 +0100

tmonteil gravatar image

updated 2022-02-19 19:27:27 +0100

You can specify an algorithm to solve the system of equations:

sage: %time solve(polys, vs, algorithm='giac', solution_dict=True)
// Giac share root-directory:/opt/sagemath/sage-source/local/share/giac/
// Giac share root-directory:/opt/sagemath/sage-source/local/share/giac/
Help file /opt/sagemath/sage-source/local/share/giac/doc/fr/aide_cas not found
Added 0 synonyms
CPU times: user 531 ms, sys: 32.8 ms, total: 564 ms
Wall time: 563 ms
[{a15: 1/x15_15,
  x13_14: x13_14,
  a4: 1/x4_4,
  a11: 1/x11_11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: 1/a5,
  x9_12: x9_12,
  a5: a5,
  a12: 1/x12_12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: x11_14,
  x12_15: x12_15,
  x5_6: x5_6,
  x4_4: x4_4,
  x9_9: x9_9,
  a6: a6,
  a13: a13,
  x3_5: x3_5,
  a2: 1,
  x10_10: x10_10,
  x7_8: x7_8,
  x6_6: 1/a6,
  x5_7: x5_7,
  x12_12: x12_12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: x3_6,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: x15_15,
  x4_6: x4_6,
  x11_11: x11_11,
  x8_9: 0,
  a8: 1/x8_8,
  x10_12: x10_12,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: x10_13},
 {a15: 1/x15_15,
  x13_14: x13_14,
  a4: 1/x4_4,
  a11: 1/x11_11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: 1/a5,
  x9_12: x9_12,
  a5: a5,
  a12: 1/x12_12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: x11_14,
  x12_15: x12_15,
  x5_6: x5_6,
  x4_4: x4_4,
  x9_9: x9_9,
  a6: a6,
  a13: a13,
  x3_5: x3_5,
  a2: 1,
  x10_10: x10_10,
  x7_8: x7_8,
  x6_6: 1/a6,
  x5_7: x5_7,
  x12_12: x12_12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: x3_6,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: x15_15,
  x4_6: x4_6,
  x11_11: x11_11,
  x8_9: 0,
  a8: 1/x8_8,
  x10_12: x10_12,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: 0},
 {a15: 1/x15_15,
  x13_14: x13_14,
  a4: 1/x4_4,
  a11: 1/x11_11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: 1/a5,
  x9_12: x9_12,
  a5: a5,
  a12: 1/x12_12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: x11_14,
  x12_15: x12_15,
  x5_6: x5_6,
  x4_4: x4_4,
  x9_9: x9_9,
  a6: a6,
  a13: a13,
  x3_5: x3_5,
  a2: 1,
  x10_10: x10_10,
  x7_8: x7_8,
  x6_6: 1/a6,
  x5_7: x5_7,
  x12_12: x12_12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: x3_6,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: x15_15,
  x4_6: x4_6,
  x11_11: x11_11,
  x8_9: 0,
  a8: 1/x8_8,
  x10_12: 0,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: 0},
 {a15: 1/x15_15,
  x13_14: x13_14,
  a4: 1/x4_4,
  a11: 1/x11_11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: 1/a5,
  x9_12: x9_12,
  a5: a5,
  a12: 1/x12_12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: x11_14,
  x12_15: x12_15,
  x5_6: x5_6,
  x4_4: x4_4,
  x9_9: x9_9,
  a6: a6,
  a13: a13,
  x3_5: x3_5,
  a2: 1,
  x10_10: x10_10,
  x7_8: x7_8,
  x6_6: 1/a6,
  x5_7: x5_7,
  x12_12: x12_12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: 0,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: x15_15,
  x4_6: x12_14,
  x11_11: x11_11,
  x8_9: 0,
  a8: 1/x8_8,
  x10_12: 0,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: 0},
 {a15: 1/x15_15,
  x13_14: x13_14,
  a4: 1/x4_4,
  a11: 1/x11_11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: 1/a5,
  x9_12: x9_12,
  a5: a5,
  a12: 1/x12_12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: x11_14,
  x12_15: x12_15,
  x5_6: x5_6,
  x4_4: x4_4,
  x9_9: x9_9,
  a6: a6,
  a13: a13,
  x3_5: x3_5,
  a2: 1,
  x10_10: x10_10,
  x7_8: x7_8,
  x6_6: 1/a6,
  x5_7: x13_15,
  x12_12: x12_12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: 0,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: x15_15,
  x4_6: x12_14,
  x11_11: x11_11,
  x8_9: 0,
  a8: 1/x8_8,
  x10_12: 0,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: 0},
 {a15: 1/x15_15,
  x13_14: x13_14,
  a4: 1/x4_4,
  a11: 1/x11_11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: 1/a5,
  x9_12: x9_12,
  a5: a5,
  a12: 1/x12_12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: x11_14,
  x12_15: x12_15,
  x5_6: x5_6,
  x4_4: x4_4,
  x9_9: x9_9,
  a6: 1/x14_14,
  a13: a13,
  x3_5: 0,
  a2: 1,
  x10_10: x10_10,
  x7_8: 0,
  x6_6: x14_14,
  x5_7: x13_15,
  x12_12: x12_12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: 0,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: x15_15,
  x4_6: x12_14,
  x11_11: x11_11,
  x8_9: 0,
  a8: 1/x8_8,
  x10_12: 0,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: 0},
 {a15: 1/x15_15,
  x13_14: x13_14,
  a4: 1/x12_12,
  a11: 1/x11_11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: 1/a5,
  x9_12: x9_12,
  a5: a5,
  a12: 1/x12_12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: x11_14,
  x12_15: x12_15,
  x5_6: x5_6,
  x4_4: x12_12,
  x9_9: x9_9,
  a6: 1/x14_14,
  a13: a13,
  x3_5: 0,
  a2: 1,
  x10_10: x10_10,
  x7_8: 0,
  x6_6: x14_14,
  x5_7: x13_15,
  x12_12: x12_12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: 0,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: x15_15,
  x4_6: x12_14,
  x11_11: x11_11,
  x8_9: 0,
  a8: 1/x8_8,
  x10_12: 0,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: 0},
 {a15: 1/x15_15,
  x13_14: x13_14,
  a4: 1/x12_12,
  a11: 1/x11_11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: 1/a5,
  x9_12: x9_12,
  a5: a5,
  a12: 1/x12_12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: x11_14,
  x12_15: x4_7,
  x5_6: x5_6,
  x4_4: x12_12,
  x9_9: x9_9,
  a6: 1/x14_14,
  a13: a13,
  x3_5: 0,
  a2: 1,
  x10_10: x10_10,
  x7_8: 0,
  x6_6: x14_14,
  x5_7: x13_15,
  x12_12: x12_12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: 0,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: x15_15,
  x4_6: x12_14,
  x11_11: x11_11,
  x8_9: 0,
  a8: 1/x8_8,
  x10_12: 0,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: 0},
 {a15: 1/x15_15,
  x13_14: x13_14,
  a4: 1/x12_12,
  a11: 1/x11_11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: 1/a5,
  x9_12: x9_12,
  a5: a5,
  a12: 1/x12_12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: 0,
  x12_15: x4_7,
  x5_6: x5_6,
  x4_4: x12_12,
  x9_9: x9_9,
  a6: 1/x14_14,
  a13: a13,
  x3_5: 0,
  a2: 1,
  x10_10: x10_10,
  x7_8: 0,
  x6_6: x14_14,
  x5_7: x13_15,
  x12_12: x12_12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: 0,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: x15_15,
  x4_6: x12_14,
  x11_11: x11_11,
  x8_9: 0,
  a8: 1/x8_8,
  x10_12: 0,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
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  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: 0},
 {a15: 1/x15_15,
  x13_14: x13_14,
  a4: 1/x12_12,
  a11: 1/x11_11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: 1/a5,
  x9_12: x9_12,
  a5: a5,
  a12: 1/x12_12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: x11_14,
  x12_15: x12_15,
  x5_6: x5_6,
  x4_4: x12_12,
  x9_9: x9_9,
  a6: 1/x14_14,
  a13: a13,
  x3_5: 0,
  a2: 1,
  x10_10: x10_10,
  x7_8: 0,
  x6_6: x14_14,
  x5_7: x13_15,
  x12_12: x12_12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: 0,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: x15_15,
  x4_6: x12_14,
  x11_11: x11_11,
  x8_9: x8_9,
  a8: 1/x8_8,
  x10_12: 0,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: 0},
 {a15: 1/x15_15,
  x13_14: x13_14,
  a4: 1/x12_12,
  a11: 1/x11_11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: 1/a5,
  x9_12: x9_12,
  a5: a5,
  a12: 1/x12_12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: x11_14,
  x12_15: x4_7,
  x5_6: x5_6,
  x4_4: x12_12,
  x9_9: x9_9,
  a6: 1/x14_14,
  a13: a13,
  x3_5: 0,
  a2: 1,
  x10_10: x10_10,
  x7_8: 0,
  x6_6: x14_14,
  x5_7: x13_15,
  x12_12: x12_12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: 0,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: x15_15,
  x4_6: x12_14,
  x11_11: x11_11,
  x8_9: x8_9,
  a8: 1/x8_8,
  x10_12: 0,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: 0},
 {a15: 1/x15_15,
  x13_14: x13_14,
  a4: 1/x12_12,
  a11: 1/x11_11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: 1/a5,
  x9_12: x9_12,
  a5: a5,
  a12: 1/x12_12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: 0,
  x12_15: x4_7,
  x5_6: x5_6,
  x4_4: x12_12,
  x9_9: x9_9,
  a6: 1/x14_14,
  a13: a13,
  x3_5: 0,
  a2: 1,
  x10_10: x10_10,
  x7_8: 0,
  x6_6: x14_14,
  x5_7: x13_15,
  x12_12: x12_12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: 0,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: x15_15,
  x4_6: x12_14,
  x11_11: x11_11,
  x8_9: x8_9,
  a8: 1/x8_8,
  x10_12: 0,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: 0},
 {a15: 1/x15_15,
  x13_14: x13_14,
  a4: 1/x4_4,
  a11: 1/x11_11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: 1/a5,
  x9_12: x9_12,
  a5: a5,
  a12: 1/x12_12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: x11_14,
  x12_15: x12_15,
  x5_6: x5_6,
  x4_4: x4_4,
  x9_9: x9_9,
  a6: 1/x14_14,
  a13: a13,
  x3_5: x3_5,
  a2: 1,
  x10_10: x10_10,
  x7_8: 0,
  x6_6: x14_14,
  x5_7: x13_15,
  x12_12: x12_12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: 0,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: x15_15,
  x4_6: x12_14,
  x11_11: x11_11,
  x8_9: x8_9,
  a8: 1/x8_8,
  x10_12: 0,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: 0},
 {a15: 1/x15_15,
  x13_14: x13_14,
  a4: 1/x4_4,
  a11: 1/x11_11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: 1/a5,
  x9_12: x9_12,
  a5: a5,
  a12: 1/x12_12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: x11_14,
  x12_15: x12_15,
  x5_6: x5_6,
  x4_4: x4_4,
  x9_9: x9_9,
  a6: a6,
  a13: a13,
  x3_5: x3_5,
  a2: 1,
  x10_10: x10_10,
  x7_8: 0,
  x6_6: 1/a6,
  x5_7: x5_7,
  x12_12: x12_12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: x3_6,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: x15_15,
  x4_6: x12_14,
  x11_11: x11_11,
  x8_9: x8_9,
  a8: 1/x8_8,
  x10_12: 0,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: 0}]

Note that, in contrast with maxima, giac does not seem to introduce free parameter like r19 to describe the solutions, instead the values of the dict are unchanged if they are free.

You can also use this to work with @rburing answer (giac knows about Groebner basis, but the solution is shorted here, perhaps because of a different monomial ordering):

sage: %time solve(list(map(SR, G)), list(map(SR, R.gens())), algorithm='giac', solution_dict=True)
CPU times: user 128 ms, sys: 0 ns, total: 128 ms
Wall time: 127 ms
[{a15: a15,
  x13_14: x13_14,
  a4: a4,
  a11: a11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: x5_5,
  x9_12: x9_12,
  a5: 1/x5_5,
  a12: a12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: x11_14,
  x12_15: x12_15,
  x5_6: x5_6,
  x4_4: 1/a4,
  x9_9: x9_9,
  a6: a6,
  a13: a13,
  x3_5: x3_5,
  a2: 1,
  x10_10: x10_10,
  x7_8: x7_8,
  x6_6: 1/a6,
  x5_7: x5_7,
  x12_12: 1/a12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: x3_6,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: 1/a15,
  x4_6: x4_6,
  x11_11: 1/a11,
  x8_9: 0,
  a8: 1/x8_8,
  x10_12: x10_12,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: x10_13},
 {a15: a15,
  x13_14: x13_14,
  a4: a4,
  a11: a11,
  a0: 1,
  x14_15: x14_15,
  x11_13: x11_13,
  x5_5: x5_5,
  x9_12: x9_12,
  a5: 1/x5_5,
  a12: a12,
  x3_4: 0,
  a1: 1,
  x7_7: x7_7,
  x11_14: x11_14,
  x12_15: x12_15,
  x5_6: x5_6,
  x4_4: 1/a4,
  x9_9: x9_9,
  a6: a6,
  a13: a13,
  x3_5: x3_5,
  a2: 1,
  x10_10: x10_10,
  x7_8: 0,
  x6_6: 1/a6,
  x5_7: x5_7,
  x12_12: 1/a12,
  x8_8: x8_8,
  a7: 1/x7_7,
  x3_6: x3_6,
  x6_7: x6_7,
  x14_14: x14_14,
  a9: 1/x9_9,
  x15_15: 1/a15,
  x4_6: x4_6,
  x11_11: 1/a11,
  x8_9: x8_9,
  a8: 1/x8_8,
  x10_12: x10_12,
  a14: 1/x14_14,
  x13_13: 1/a13,
  a3: 1,
  a10: 1/x10_10,
  x4_7: x4_7,
  x12_14: x12_14,
  x11_12: 0,
  x13_15: x13_15,
  x10_13: x10_13}]
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Asked: 2022-02-18 19:57:18 +0100

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Last updated: Feb 19 '22

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