Singular 2-0-4 Manual: 5.1.10 coeffs

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5.1.10 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, say J, as a univariate polynomial in the given ring_variable, say z, and returns the coefficients as a k x d matrix M, where:

    d-1 = maximum z-degree of all occurring polynomials
      k = 1 if J is a polynomial,
      k = number of generators  if J is an ideal.

If J is a vector or a module this procedure is repeated for each component and the resulting matrices are appended.
The third argument is used to return the matrix T of coefficients such that matrix(J) = T*M.

Note:

coeffs returns the coefficient 0 at the appropriate place if a monomial is not present, while coef considers only monomials which really occur in the given expression.
If $M=(m_{ij})$ then the j-th generator of an ideal J is equal to

$\begin{displaymath}J_j = z^0 \cdot m_{1j} + z^1 \cdot m_{2j} + ... + z^{d-1} \cdot m_{dj},\end{displaymath}$

while for a module J the i-th component of the j-th generator is equal to the entry [i,j] of matrix(J), and we get

$\begin{displaymath}J_{i,j} = z^0 \cdot m_{(i-1)d+1,j} + z^1 \cdot m_{(i-1)d+2,j} + ... + z^{d-1} \cdot m_{id,j}.\end{displaymath}$

Example:

  ring r;
  poly f=(x+y)^3;
  matrix M=coeffs(f,y);
  print(M);
→ x3, 
→ 3x2,
→ 3x, 
→ 1   
  ideal i=f,xyz+z10y4;
  print(coeffs(i,y));
→ x3, 0, 
→ 3x2,xz,
→ 3x, 0, 
→ 1,  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:

let the first argument be M, the second argument be K (a set of monomials, resp. vectors with monomial entries, in the variables appearing in P), the third argument be the product P of variables to consider (if this argument is not given, then the product of all ring variables is taken as default argument).
M is supposed to consist of elements of (resp. have entries in) a finitely generated module over a ring in the variables not appearing in P. K should contain the generators of M over this smaller ring. Then coeffs(M,K,P) returns a matrix A of coefficients with K*A=M such that the entries of A do not contain any variable from P.
If K does not contain all generators that are necessary to express M, then K*A=M’ where M’ is the part of M that can be expressed.

Example:

  ring r=32003,(x,y,z),dp;
  ideal M=x2z+y3,xy;
  print(coeffs(M,ideal(x2,xy,y2),xy));
→ z,0,
→ 0,1,
→ 0,0 
  print(coeffs(M,ideal(x2,xy,y2)));
→ 0,0,
→ 0,1,
→ 0,0

See coef; kbase.

User manual for Singular version 2-0-4, October 2002, generated by texi2html.