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5.1.11 coef
Syntax:coef (poly_expression,product_of_ringvars)coef (ideal_expression,product_of_ringvars)Type:- matrix
Syntax:coef (vector_expression,product_of_ringvars,matrix_name,matrix_name)Type:- none
Purpose:- determines the monomials in f divisible by a ring variable of m
(where f is the first argument and m the second argument) and the coefficients
of these monomials as polynomials in the remaining variables. First case: returns a
matrix M, n being the number of the determined monomials.
The first row consists of these monomials, the second row
of the corresponding coefficients of the monomials in f.
Thus,
![$f = M[1,1]\cdot M[2,1]+\ldots+M[1,n]\cdot M[2,n].$](sing_102.png)
Second case: apply to all generators of the ideal and combine the results into one matrix.
Third case: the second matrix (i.e., the 4th argument) contains the monomials, the first matrix (i.e., the 3rd argument) the corresponding coefficients of the monomials in the vector.
Note:- coef considers only monomials which really occur in f (i.e., which are not 0),
while coeffs (see coeffs) returns the coefficient 0 at the
appropriate place if a monomial is not present.
Example:ring r=32003,(x,y,z),dp; poly f=x5+5x4y+10x2y3+y5; matrix m=coef(f,y); print(m); ==> y5,y3, y, 1, ==> 1, 10x2,5x4,x5 f=x20+xyz+xy+x2y+z3; print(coef(f,xy)); ==> x20,x2y,xy, 1, ==> 1, 1, z+1,z3 print(coef(maxideal(3),yz)); ==> y3,y2z,yz2,z3,y2,yz,z2,y, z, 1, ==> 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, ==> 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ==> 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ==> 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ==> 0, 0, 0, 0, 0, 0, x, 0, 0, 0, ==> 0, 0, 0, 0, 0, x, 0, 0, 0, 0, ==> 0, 0, 0, 0, x, 0, 0, 0, 0, 0, ==> 0, 0, 0, 0, 0, 0, 0, 0, x2,0, ==> 0, 0, 0, 0, 0, 0, 0, x2,0, 0, ==> 0, 0, 0, 0, 0, 0, 0, 0, 0, x3 vector v=[f,zy+77+xy]; print(v); ==> [x20+x2y+xyz+z3+xy,xy+yz+77] matrix mc; matrix mm; coef(v,y,mc,mm); print(mc); ==> x2+xz+x,x20+z3, ==> x+z, 77 print(mm); ==> y,1, ==> y,1