Description
We compute the
Hilbert series of a projective Hilbert polynomial.
i1 : P = projectiveHilbertPolynomial 3
o1 = P
3
o1 : ProjectiveHilbertPolynomial
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i2 : s = hilbertSeries P
1
o2 = --------
4
(1 - T)
o2 : Expression of class Divide
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i3 : numerator s
o3 = 1
o3 : ZZ[T]
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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];
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i5 : I = ideal (a*b, c*d, e*f);
o5 : Ideal of R
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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]
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