# 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```