hilbertPolynomial(ProjectiveVariety) -- compute the Hilbert polynomial of the projective variety

Synopsis

Usage:
hilbertPolynomial V
Function: hilbertPolynomial
Inputs:
- V, a projective variety
Outputs:
- a projective Hilbert polynomial, unless the option Projective is false
Optional inputs:
- Projective => ..., -- choose how to display the Hilbert polynomial

Description

We compute an example of the Hilbert polynomial of a projective Hilbert variety. This is the same as the Hilbert polynomial of its coordinate ring.

i1 : R = QQ[a..d];

i2 : I = monomialCurveIdeal(R, {1,3,4});

o2 : Ideal of R

i3 : V = Proj(R/I)

o3 = V

o3 : ProjectiveVariety

i4 : h = hilbertPolynomial V

o4 = - 3*P  + 4*P
          0      1

o4 : ProjectiveHilbertPolynomial

i5 : hilbertPolynomial(V, Projective=>false)

o5 = 4i + 1

o5 : QQ [i]

These Hilbert polynomials can serve as Hilbert functions too since the values of the Hilbert polynomial eventually are the same as the Hilbert function.

i6 : apply(5, k-> h(k))

o6 = {1, 5, 9, 13, 17}

o6 : List

i7 : apply(5, k-> hilbertFunction(k,V))

o7 = {1, 4, 9, 13, 17}

o7 : List