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5.1.45 hilb

Syntax:
hilb ( ideal_expression )
hilb ( module_expression )
hilb ( ideal_expression, int_expression )
hilb ( module_expression, int_expression )
hilb ( ideal_expression, int_expression , intvec_expression )
hilb ( module_expression, int_expression , intvec_expression )
Type:
none (if called with one argument)
intvec (if called with two or three arguments)
Purpose:
computes the (weighted) Hilbert series of the graded ring defined by the ideal generated by the leading terms of the input, respectively of the graded module coker(L(M)) where L(M) denotes the matrix with column entries the leading terms of the input.
If hilb is called with one argument, then the 1st and 2nd Hilbert series are computed and additional information (dimension and degree) are displayed.
If hilb is called with two arguments and if n=1 or n=2 is the second argument, then the nth Hilbert series is returned as an intvec.
If an integer vector w is given as 3rd argument, then the Hilbert series is computed with respect to the weights w (by default, all weights are set to 1).
Caution:
The last entry of the returned intvec is not part of the actual Hilbert series, but is used in the Hilbert driven standard basis computation (see stdhilb).
Note:
If the input is homogeneous w.r.t. the weights and a standard basis, the result is the (weighted) Hilbert series of the input ideal, resp. module.

The additional information displayed when calling hilb with one argument has to be interpreted as follows: If the input is homogeneous and a standard basis then the dimension and degree of the projetive variety defined by the input ideal, respectively module, are displayed. With one exception: if the projective dimension is -1, the affine dimension 0 and the degree of the projective cone are displayed.
If the input is not homogeneous but a standard basis with respect to a degree ordering, then the dimension of the affine variety and the degree of the projective variety defined by the homogenization of the ideal are displayed.
Example:
 
  ring R=32003,(x,y,z),dp;
  ideal i=x2,y2,z2;
  ideal s=std(i);
  hilb(s);
==> //         1 t^0
==> //        -3 t^2
==> //         3 t^4
==> //        -1 t^6
==> 
==> //         1 t^0
==> //         3 t^1
==> //         3 t^2
==> //         1 t^3
==> // dimension (affine)  = 0
==> // degree      = 8
  hilb(s,1);
==> 1,0,-3,0,3,0,-1,0
  hilb(s,2);
==> 1,3,3,1,0
  intvec w=2,2,2;
  hilb(s,1,w);
==> 1,0,0,0,-3,0,0,0,3,0,0,0,-1,0
See Hilbert function; ideal; intvec; module; std; stdhilb.

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