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5.1.22 division

Syntax:
division ( ideal_expression, ideal_expression )
division ( module_expression, module_expression )
division ( ideal_expression, ideal_expression, int_expression )
division ( module_expression, module_expression, int_expression )
division ( ideal_expression, ideal_expression, int_expression, intvec_expression )
division ( module_expression, module_expression, int_expression,
intvec_expression )
Type:
list
Purpose:
division computes a division with remainder. For two ideals resp. modules M (first argument) and N (second argument), it returns a list T,R,U where T is a matrix, R is an ideal resp. a module, and U is a diagonal matrix of units such that matrix(M)*U=matrix(N)*T+matrix(R) is a standard representation for the normal form R of M with respect to a standard basis of N. division uses different algorithms depending on whether N is represented by a standard basis. For a polynomial basering, the matrix U is the identity matrix. A matrix T as above is also computed by lift.
For additional arguments n (third argument) and w (fourth argument), division returns a list T,R as above such that matrix(M)=matrix(N)*T+matrix(R) is a standard representation for the normal form R of M with respect to N up to weighted degree n with respect to the weight vector w. The weighted degree of T and R respect to w is at most n. If the weight vector w is not given, division uses the standard weight vector w=1,...,1.
Example:
 
See ideal; lift; module.

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