Top
Back: coef
Forward: contract
FastBack: Functions and system variables
FastForward: Control structures
Up: Functions
Top: Singular Manual
Contents: Table of Contents
Index: Index
About: About this document

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.

Top Back: coef Forward: contract FastBack: Functions and system variables FastForward: Control structures Up: Functions Top: Singular Manual Contents: Table of Contents Index: Index About: About this document
            User manual for Singular version 3-1-6, Dec 2012, generated by texi2html.