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 = | c3-bd2 ac2-b2d bc-ad b3-a2c |
     
                  1       4
     o3 : Matrix R  <--- R

     i4 = h = hilbertPolynomial S
     
               0      1
     o4 = - 3 P  + 4 P 
     
     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 = | 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).