# Singular

### 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 -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 k-dimensional 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 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:
  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.