Solution of polynomial systems using msolve#
msolve is a multivariate polynomial system solver based on Gröbner bases.
This module provide implementations of some operations on polynomial ideals based on msolve.
Note that the optional package msolve must be installed.
See also
sage.features.msolve
- sage.rings.polynomial.msolve.groebner_basis_degrevlex(ideal, proof=True)#
Compute a degrevlex Gröbner basis using msolve
EXAMPLES:
sage: from sage.rings.polynomial.msolve import groebner_basis_degrevlex sage: R.<a,b,c> = PolynomialRing(GF(101), 3, order='lex') sage: I = sage.rings.ideal.Katsura(R,3) sage: gb = groebner_basis_degrevlex(I); gb # optional - msolve [a + 2*b + 2*c - 1, b*c - 19*c^2 + 10*b + 40*c, b^2 - 41*c^2 + 20*b - 20*c, c^3 + 28*c^2 - 37*b + 13*c] sage: gb.universe() is R # optional - msolve False sage: gb.universe().term_order() # optional - msolve Degree reverse lexicographic term order sage: ideal(gb).transformed_basis(other_ring=R) # optional - msolve [c^4 + 38*c^3 - 6*c^2 - 6*c, 30*c^3 + 32*c^2 + b - 14*c, a + 2*b + 2*c - 1]
Gröbner bases over the rationals require \(proof=False\):
sage: R.<x, y> = PolynomialRing(QQ, 2) sage: I = Ideal([ x*y - 1, (x-2)^2 + (y-1)^2 - 1]) sage: I.groebner_basis(algorithm='msolve') # optional - msolve Traceback (most recent call last): ... ValueError: msolve relies on heuristics; please use proof=False sage: I.groebner_basis(algorithm='msolve', proof=False) # optional - msolve [x*y - 1, x^2 + y^2 - 4*x - 2*y + 4, y^3 - 2*y^2 + x + 4*y - 4]
- sage.rings.polynomial.msolve.variety(ideal, ring, proof)#
Compute the variety of a zero-dimensional ideal using msolve.
Part of the initial implementation was loosely based on the example interfaces available as part of msolve, with the authors’ permission.
EXAMPLES:
sage: from sage.rings.polynomial.msolve import variety sage: p = 536870909 sage: R.<x, y> = PolynomialRing(GF(p), 2, order='lex') sage: I = Ideal([ x*y - 1, (x-2)^2 + (y-1)^2 - 1]) sage: sorted(variety(I, GF(p^2), proof=False), key=str) # optional - msolve [{x: 1, y: 1}, {x: 254228855*z2 + 114981228, y: 232449571*z2 + 402714189}, {x: 267525699, y: 473946006}, {x: 282642054*z2 + 154363985, y: 304421338*z2 + 197081624}]