# hilbertPolynomial(Ring) -- compute the Hilbert polynomial of the ring

## Description

We compute the Hilbert polynomial of a coordinate ring of the rational quartic curve in P^3.
 i1 : R = ZZ/101[a..d]; i2 : S = coimage map(R, R, {a^4, a^3*b, a*b^3, b^4}); i3 : presentation S o3 = | bc-ad c3-bd2 ac2-b2d b3-a2c | 1 4 o3 : Matrix R <--- R i4 : h = hilbertPolynomial S o4 = - 3*P + 4*P 0 1 o4 : ProjectiveHilbertPolynomial i5 : hilbertPolynomial(S, Projective=>false) o5 = 4i + 1 o5 : QQ[i]
The rational quartic curve in P^3 is therefore 'like' 4 copies of P^1, with three points missing. One can see this by noticing that there is a deformation of the rational quartic to the union of 4 lines, or 'sticks', which intersect in three successive points.

These Hilbert polynomials can serve as Hilbert functions too since the values of the Hilbert polynomial eventually are the same as the Hilbert function.

 i6 : apply(5, k-> h(k)) o6 = {1, 5, 9, 13, 17} o6 : List i7 : apply(5, k-> hilbertFunction(k,S)) o7 = {1, 4, 9, 13, 17} o7 : List