# hilbertSeries(ProjectiveHilbertPolynomial) -- compute the Hilbert series of a projective Hilbert polynomial

## Synopsis

• Function: hilbertSeries
• Usage:
hilbertSeries P
• Inputs:
• Optional inputs:
• Order => ..., default value infinity, display the truncated power series expansion
• Reduce => ..., default value false, reduce the Hilbert series
• Outputs:
• , the Hilbert series

## Description

We compute the Hilbert series of a projective Hilbert polynomial.
 i1 : P = projectiveHilbertPolynomial 3 o1 = P 3 o1 : ProjectiveHilbertPolynomial i2 : s = hilbertSeries P 1 o2 = -------- 4 (1 - T) o2 : Expression of class Divide i3 : numerator s o3 = 1 o3 : ZZ[T]
Computing the Hilbert series of a projective variety can be useful for finding the h-vector of a simplicial complex from its f-vector. For example, consider the octahedron. The ideal below is its Stanley-Reisner ideal. We can see its f-vector (1, 6, 12, 8) in the Hilbert polynomial, and then we get the h-vector (1,3,3,1) from the coefficients of the Hilbert series projective Hilbert polynomial.
 i4 : R = QQ[a..h]; i5 : I = ideal (a*b, c*d, e*f); o5 : Ideal of R i6 : P=hilbertPolynomial(I) o6 = - P + 6*P - 12*P + 8*P 1 2 3 4 o6 : ProjectiveHilbertPolynomial i7 : s = hilbertSeries P 2 3 1 + 3T + 3T + T o7 = ----------------- 5 (1 - T) o7 : Expression of class Divide i8 : numerator s 2 3 o8 = 1 + 3T + 3T + T o8 : ZZ[T]