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 =  bcad c3bd2 ac2b2d b3a2c 
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
