|  |  5.1.11 coef 
 
See
 coeffs.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_101.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 
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