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Macaulay2Doc :: hilbertPolynomial(ProjectiveVariety)

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

Synopsis

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