Hilbert

Hilbert

the Hilbert-Poincare' function

Syntax


Hilbert(R:RING or TAGGED("Quotient")):TAGGED("$hp.Hilbert")
Hilbert(R:RING or TAGGED("Quotient"), N:INT):INT

Description

The first form of this function computes the Hilbert function for R. The second form computes the N-th value of the Hilbert function. The weights of the indeterminates of R must all be 1. If the input is not homogeneous, the Hilbert function of the corresponding leading term (initial) ideal or module is calculated. For repeated evaluations of the Hilbert function, use EvalHilbertFn instead of Hilbert(R, N) in order to speed up execution.

This function is the same as HilbertFn .

The coefficient ring must be a field.

Example

  Use R ::= QQ[t,x,y,z];
  Hilbert(R/Ideal(z^2-xy,xz^2+t^3));
H(0) = 1
H(1) = 4
H(t) = 6t-3   for t >= 2
-------------------------------
  M := R^2/Module([x^2-t,xy-z^3],[zy,tz-x^3y+3]);
  Hilbert(M);
H(0) = 2
H(1) = 8
H(2) = 20
H(3) = 39
H(t) = 3t^2 + 6t-7   for t >= 4
-------------------------------
  Hilbert(M,3)
39
-------------------------------
  Hilbert(M,5);
98
-------------------------------

See Also