### 5.28.2 Gröbner reduction : greduce

greduce has three arguments : a multivariate polynomial, a vector made of polynomials which is supposed to be a Gröbner basis, and a vector of variable names.
greduce returns the reduction of the polynomial given as first argument with respect to the Gröbner basis given as the second argument. It is 0 if and only if the polynomial belongs to the ideal.

Input :

greduce(x*y-1,[x`^`2-y`^`2,2*x*y-y`^`2,y`^`3],[x,y])

Output :

y`^`2-2

that is to say xy−1=1/2(y2−2) modI where I is the ideal generated by the Gröbner basis [x2−y2,2xy−y2,y3], because y2−2 is the euclidean division remainder of 2(xy−1) by G2=2x y−y2.

Like gbasis (cf. 5.28.1), greduce may have more than 3 arguments to specify ordering and algorithm if they differ from the default (lexicographic ordering).
Input :

greduce(x1`^`2*x3`^`2,[x3`^`3-1,-x2`^`2-x2*x3-x3`^`2,x1+x2+x3], [x1,x2,x3],tdeg)

Output

`x2`