degree(ProjectiveHilbertPolynomial)

Synopsis

Function: degree
Usage:

degree f
Inputs:
- f, a projective Hilbert polynomial, usually returned via hilbertPolynomial
Outputs:
- an integer, the degree of any graded module having this Hilbert polynomial

Description

This degree is obtained from the Hilbert polynomial f as follows: if f = d z^e/e! + lower terms in z, then d is returned. This is the lead coefficient of the highestP^e in the ProjectiveHilbertPolynomial display.

i1 : R = QQ[a..d];

i2 : I = ideal(a^3, b^2, a*b*c);

o2 : Ideal of R

i3 : F = hilbertPolynomial I

o3 = - 2*P  + 4*P
          0      1

o3 : ProjectiveHilbertPolynomial

i4 : degree F

o4 = 4

The degree of this polynomial may be recovered using dim:

i5 : dim F

o5 = 1

The dimension as a projective variety is also one less that the Krull dimension of R/I

i6 : (dim I - 1, degree I)

o6 = (1, 4)

o6 : Sequence

Ways to use this method: