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## 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 = (y2 -2)  mod I 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 = 2xy - y2.

Like gbasis (cf. 1.27.1), 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

giac documentation written by Renée De Graeve and Bernard Parisse