D.6.2.4 reduction

Procedure from library ainvar.lib (see ainvar_lib).

Usage:

reduction(p,I[,q,n]); p poly, I ideal, [q monomial, n int (optional)]

Return:

a polynomial equal to p-H(f1,...,fr), in case the leading term LT(p) of p is of the form H(LT(f1),...,LT(fr)) for some polynomial H in r variables over the base field, I=f1,...,fr; if q is given, a maximal power a is computed such that q^a divides p-H(f1,...,fr), and then (p-H(f1,...,fr))/q^a is returned; return p if no H is found
if n=1, a different algorithm is chosen which is sometimes faster (default: n=0; q and n can be given (or not) in any order)

Note:

this is a kind of SAGBI reduction in the subalgebra K[f1,...,fr] of the basering

Example:

LIB "ainvar.lib";
ring q=0,(x,y,z,u,v,w),dp;
poly p=x2yz-x2v;
ideal dom =x-w,u2w+1,yz-v;
reduction(p,dom);
→ 2xyzw-yzw2-2xvw+vw2
reduction(p,dom,w);
→ 2xyz-yzw-2xv+vw