Gröbner reduction: greduce

### 5.31.2 Gröbner reduction: greduce

The greduce command will find a polynomial modulo I, where I is an ideal as in Section 5.31.1.

- 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 5.31.1).
- 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=2x y−y2.
- Input:
  greduce(x1^2*x3^2,[x3^3-1,-x2^2-x2*x3-x3^2,x1+x2+x3], [x1,x2,x3],tdeg)
  Output:

  x2