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5.1.11 coeffs
Syntax:
coeffs ( poly_expression , ring_variable )
coeffs ( ideal_expression, ring_variable )
coeffs ( vector_expression, ring_variable )
coeffs ( module_expression, ring_variable )
coeffs ( poly_expression, ring_variable, matrix_name )
coeffs ( ideal_expression, ring_variable, matrix_name )
coeffs ( vector_expression, ring_variable, matrix_name )
coeffs ( module_expression, ring_variable, matrix_name )
Type:
- matrix
Purpose:
- develops each polynomial of the first argument J
as a univariate polynomial in the given ring_variable
z, and returns the coefficients as a matrix M.
With e denoting the maximal z-degree occuring in the polynomials of J, and d:=e+1, M =
satisfies the following conditions:
-
(i) If J is a single polynomial f, then M is a
is the coefficient of
in f.
-
(ii) If J is an ideal with generators
then M is a
is the coefficient of
in
-
(iii) If J is a k-dimensional vector with entries
then M is a
is the coefficient of
in
-
(iV) If J is a module generated by s vectors
of dimension k then M is a
is the coefficient of
in the j-th entry of
The optional third argument T can be used to return the matrix of powers of z
such that matrix(J) = T*M holds in each of the previous four cases.
Note:
coeffs returns the coefficient 0 at the appropriate matrix entry if a monomial
is not present, while coef considers only monomials which actually occur
in the given expression.
Example:
Syntax:
coeffs ( ideal_expression, ideal_expression )
coeffs ( module_expression, module_expression )
coeffs ( ideal_expression, ideal_expression, product_of_ringvars )
coeffs ( module_expression, module_expression, product_of_ringvars )
Type:
- matrix
Purpose:
- expresses each polynomial of the first argument M as a sum
where the
come from a specified set of monomials, the
are from the underlying
coefficient ring (or field), and the
are powers of a specified ring variable x.
The second parameter K provides the set of monomials which should be sufficient to generate all entries of M.
Both M and K can be thought of as the matrices obtained by matrix(M) and matrix(K), respectively. (If M and K
are given by ideals, then this matrix has just one row.)
The optional parameter product_of_ringvars determines the variable x: It is expected to be either the product of
all ring variables (then x is 1, and each polynomial will be expressed as
or product_of_ringvars is the product of all ring variables except one variable (which then determines x).
If product_of_ringvars is omitted then x = 1 as default.
If K contains all monomials that are necessary to express the entries of
M, then the returned matrix A satisfies
Otherwise only a subset of entries of
and M will coincide.
In this case, the valid entries start at M[1,1] and run from left to right, top to bottom.
Note:
- Note that in general not all entries of K*A and M will coincide, depending on the set of monomials
provided by K.
Example:
See
coef;
kbase.
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