# degree(ProjectiveHilbertPolynomial)

## Synopsis

• Function: degree
• Usage:
degree f
• Inputs:
• 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