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(y^{2}−2) modI where I is the ideal
generated by the Gröbner basis [x^{2}−y^{2},2xy−y^{2},y^{3}], because
y^{2}−2 is the euclidean division remainder of 2(xy−1) by G_{2}=2x y−y^{2}.

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`