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 :
^2-y^2,2*x*y-y^2,y^3],[x,y])Output :
^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 euclidian division rmainder of 2(xy−1) by G2=2x y−y2.
Like gbasis (cf. 2.27.1), greduce may have more than 3 arguments to specify ordering and algorithm if they differ from the default (lexicographic ordering). Input :
^2*x3^2,[x3^3-1,-x2^2-x2*x3-x3^2,x1+x2+x3], [x1,x2,x3],tdeg)
Output
x2