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

Synopsis

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]