
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 zdegree 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
matrix and
,is the coefficient of
in f.

(ii) If J is an ideal with generators
then M is a
matrix and
,is the coefficient of
in
.

(iii) If J is a kdimensional vector with entries
then M is a
matrix and
,is the coefficient of
in
.

(iV) If J is a module generated by s vectors
of dimension k then M is a
matrix and
,is the coefficient of
in the jth 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:
 ring r;
poly f = (x+y)^3;
poly g = xyz+z10y4;
ideal i = f, g;
matrix M = coeffs(i, y);
print(M);
==> x3, 0,
==> 3x2,xz,
==> 3x, 0,
==> 1, 0,
==> 0, z10
vector v = [f, g];
M = coeffs(v, y);
print(M);
==> x3,
==> 3x2,
==> 3x,
==> 1,
==> 0,
==> 0,
==> xz,
==> 0,
==> 0,
==> z10

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:
 ring r=32003,(x,y,z),dp;
module M = [y3+x2z, xy], [xy, y2+x2z];
print(M);
==> y3+x2z,xy,
==> xy, x2z+y2
module K = [x2, xy], [y3, xy], [xy, x];
print(K);
==> x2,y3,xy,
==> xy,xy,x
matrix A = coeffs(M, K, xy); // leaving z as variable of interest
print(A); // attention: only the first row of M is reproduced by K*A
==> z,0,
==> 1,0,
==> 0,1

See
coef;
kbase.
