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

## Synopsis

• Function: hilbertPolynomial
• Usage:
hilbertPolynomial V
• Inputs:
• Optional inputs:
• Projective => ..., default value true, choose how to display the Hilbert polynomial
• Outputs:
• , unless the option Projective is false

## 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 of the sheaf of rings or of the underlying ring.

 i6 : apply(5, k-> h(k)) o6 = {1, 5, 9, 13, 17} o6 : List i7 : apply(5, k-> hilbertFunction(k,ring V)) o7 = {1, 4, 9, 13, 17} o7 : List