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7.3.4 division (plural)

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 left division with remainder. For two left ideals resp. modules M (first argument) and N (second argument), it returns a list T,R,U where T is a matrix, R is a left ideal resp. a module, and U is a diagonal matrix of units such that transpose(U)*transpose(matrix(M))=transpose(T)*transpose(matrix(N)) + transpose(matrix(R)). From this data one gets a left standard representation for the left normal form R of M with respect to a left Groebner basis of N. division uses different algorithms depending on whether N is represented by a Groebner basis. For a GR-algebra, 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 transpose(matrix(M))=transpose(T)*transpose(matrix(N)) + transpose(matrix(R)) is a left standard representation for the left 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; poly; vector.


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