euler(ProjectiveHilbertPolynomial) -- constant term of the Hilbert polynomial

Synopsis

Function: euler
Usage:

euler P
Inputs:
- P, a projective Hilbert polynomial
Outputs:
- an integer, the constant term of the Hilbert polynomial

Description

The command returns P(0) the constant term of P. This is also the Euler characteristic of the sheaf of rings of a projective variety with Hilbert polynomial P.

i1 : R = QQ[x_0..x_3]

o1 = R

o1 : PolynomialRing

i2 : C = Proj(R/monomialCurveIdeal(R, {1,3,4}));

i3 : P = hilbertPolynomial C

o3 = - 3*P  + 4*P
          0      1

o3 : ProjectiveHilbertPolynomial

i4 : euler P

o4 = 1

Ways to use this method: