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];
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i2 : I = monomialCurveIdeal(R, {1,3,4});
o2 : Ideal of R
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i3 : V = Proj(R/I)
o3 = V
o3 : ProjectiveVariety
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i4 : h = hilbertPolynomial V
o4 = - 3*P + 4*P
0 1
o4 : ProjectiveHilbertPolynomial
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i5 : hilbertPolynomial(V, Projective=>false)
o5 = 4i + 1
o5 : QQ[i]
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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
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i7 : apply(5, k-> hilbertFunction(k,ring V))
o7 = {1, 4, 9, 13, 17}
o7 : List
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