CODEWITHSUNDEEP
sympysymbolic-mathpolynomialssagegroebner-basis
KludgeShySir
KludgeShySir

Reputation: 51

In Sympy is there a way to get a representation of Groebner basis in terms of defining polynomials?

In Sympy one can obtain the representation of a polynomial's reduction by a Grobner basis:

from sympy import groebner, expand
from sympy.abc import x, y
f = 2*x**4 - x**2 + y**3 + y**2
G = groebner([x**3 - x, y**3 - y])
Q, r = G.reduce(f)

assert f == expand(sum(q*g for q, g in zip(Q, G)) + r)

But what I'm looking for is a way to get the expression of the elements of a Groebner basis in terms of the polynomials defining it, essentially just storing the computations performed in the Buchberger algorithm used in producing the basis.

For example

groebner([2*x+3*y+5, 3*x+5*y+2, 5*x+2*y+3])   # GroebnerBasis([1], x, y, domain='ZZ', order='lex')

It indicates that these three polynomials generate the unit ideal, but I would like an explicit combination of these polynomials that equals 1. In the example given it is a linear combination, but I would like a method that works with nonlinear as well.

In the first example I obtained Q,r. In the second example I obtained the analog of the remainder r but I would like the polynomials Q realizing it.

Similarly, the method G.contains() will indicate if the ideal contains the polynomial, but it won't tell you how to produce it. Is there a way to do this too?

Upvotes: 5

Views: 696

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