Singular Manual: reduce (letterplace)

7.8.9 reduce (letterplace)

Syntax:

reduce ( poly_expression, ideal_expression )
reduce ( poly_expression, ideal_expression, int_expression )
reduce ( vector_expression, ideal_expression )
reduce ( vector_expression, ideal_expression, int_expression )
reduce ( vector_expression, module_expression, int_expression )
reduce ( ideal_expression, ideal_expression )
reduce ( ideal_expression, ideal_expression, int_expression )

Type:

the type of the first argument

Purpose:

reduces a polynomial, vector, or ideal (the first argument) to its two-sided normal form with respect to the second argument, meant to be an ideal, represented by its two-sided Groebner basis (otherwise, the result may have no meaning).
returns 0 if and only if the polynomial (resp. vector, ideal) is an element (resp. subideal) of the ideal.
The third (optional) argument of type int modifies the behavior:

0 default
1 consider only the leading term and do no tail reduction.
2 tail reduction: in the local/mixed ordering case: reduce also with bad ecart
4 reduce without division, return possibly a non-zero constant multiple of the remainder

Note:

The commands reduce and NF are synonymous.

Note:

A two-sided Groebner presentation of a polynomial with respect to a two-sided ideal can be computed by the procedure lpDivision from freegb_lib.

Example:

LIB "freegb.lib";
ring r = 0,(x,y),dp;
ring R = freeAlgebra(r,5);
ideal I = x*x + y*y - 1; // 2D sphere
ideal J = twostd(I); // computes a two-sided Groebner basis
J; // it is finite and nice
==> J[1]=x*x+y*y-1
==> J[2]=y*y*x-x*y*y
poly g = x*y*y - y*y*x;
reduce(g,J); // 0, hence g belongs to J
==> 0
poly h = x*y*y*x - y*x*x;
reduce(h,J); // the rest of two-sided division of h by J
==> -y*y*y*y+y*y*y+y*y-y
qring Q = J; // switch to K<x,y>/J
reduce(x*y*y - y*y*x,twostd(0)); //image of g above
==> 0
reduce(x*y*y*x - y*x*x,std(0)); //image of h above
==> -y*y*y*y+y*y*y+y*y-y