5.1.106 reduce

Syntax:

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

Type:

the type of the first argument

Purpose:

reduces a polynomial, vector, ideal or module to its normal form with respect to an ideal or module represented by a standard basis. Returns 0 if and only if the polynomial (resp. vector, ideal, module) is an element (resp. subideal, submodule) of the ideal (resp. module). The result may have no meaning if the second argument is not a standard basis.
The third (optional) argument 1 of type int forces a reduction which considers only the leading term and does no tail reduction.
If a third argument u of type poly or matrix is given, the first argument p is replaced by p/u. This works only for zero dimensional ideals (resp. modules) in the second argument and gives, even in a local ring, a reduced normal form which is the projection to the quotient by the ideal (resp. module).

Note:

The commands reduce and NF are synonymous.

Example:

ring r1 = 0,(z,y,x),ds; poly s1=2x5y+7x2y4+3x2yz3; poly s2=1x2y2z2+3z8; poly s3=4xy5+2x2y2z3+11x10; ideal i=s1,s2,s3; ideal j=std(i); reduce(3z3yx2+7y4x2+yx5+z12y2x2,j); → -yx5+2401/81y14x2+2744/81y11x5+392/27y8x8+224/81y5x11+16/81y2x14 reduce(3z3yx2+7y4x2+yx5+z12y2x2,j,1); → -yx5+z12y2x2

See ideal; module; std; vector.