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5.30.2  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 :


Output :


that is to say xy−1=1/2(y2−2) modI where I is the ideal generated by the Gröbner basis [x2y2,2xyy2,y3], because y2−2 is the euclidean division remainder of 2(xy−1) by G2=2x yy2.

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).
Input :

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



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