The greduce command will find a polynomial modulo I, where
I is an ideal as in Section 6.31.1.
greduce takes three arguments mandatory arguments and
three optional arguments:
P, a multivariate polynomial.
gbasis, a vector made of polynomials which is
supposed to be a Gröbner basis.
vars, and a vector of variable names.
Optionally, the same ordering options and CoCaA options
as gbasis (see Section 6.31.1).
greduce(P,gbasis,vars ⟨
,options⟩)
returns the reduction of P with respect to the Gröbner basis
gbasis. It is 0 if and only if the polynomial belongs to the ideal.
Examples.
Input:
greduce(x*y-1,[x^2-y^2,2*x*y-y^2,y^3],[x,y])
Output:
1
2
y2−1
that is to say xy−1=1/2y2−1 modI where I is the ideal
generated by the Gröbner basis [x2−y2,2xy−y2,y3], because
1/2y2−1 is the Euclidean division remainder of xy−1 by G2=2xy−y2.