Projective

Projective => true -- an option to hilbertPolynomial which specifies that the Hilbert polynomial produced should be expressed in terms of the Hilbert polynomials of projective spaces. This is the default.
Projective => false -- an option to hilbertPolynomial which specifies that the Hilbert polynomial produced should be expressed as a polynomial in the degree.

i1 : R = ZZ/101[a..d]



o1 = R



o1 : PolynomialRing

i2 : S = image map(R, R, {a^4, a^3*b, a*b^3, b^4})



o2 = S



o2 : QuotientRing

i3 : presentation S



o3 = {0} | c3-bd2 ac2-b2d bc-ad b3-a2c |



             1       4

o3 : Matrix R  <--- R

i4 : h = hilbertPolynomial S



o4 = - 3*P  + 4*P 

          0      1



o4 : ProjectiveHilbertPolynomial

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.

i5 : h 3



o5 = 13

i6 : basis(3,S)



o6 = {0} | a3 a2b a2c a2d ab2 abd acd ad2 b2d bd2 c2d cd2 d3 |



             1        ZZ 13

o6 : Matrix S  <--- (---)

                     101

i7 : rank source basis(3,S)



o7 = 13

Note that the Hilbert polynomial of P^i is z |--> binomial(z + i, i).