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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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