11.5.2  Gröbner reduction

The greduce command finds a polynomial modulo I, where I is an ideal as in Section 11.5.1.

• 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 CoCoA options as gbasis (see Section 11.5.1).
• 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

that is to say xy−1≡ 1/2y²−1(mod I ) 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=2x y−y².