greduce has three arguments : a multivariate
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.
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. 5.30.1),
greduce may have more than 3 arguments to specify ordering and
algorithm if they differ from the default (lexicographic ordering).