hilbertPolynomial(Ideal) -- compute the Hilbert polynomial of the quotient of the ambient ring by the ideal

Function: hilbertPolynomial
Usage:

hilbertPolynomial I
Inputs:
- I, an ideal
Optional inputs:
- Projective => ..., default value true, choose how to display the Hilbert polynomial
Outputs:
- a projective Hilbert polynomial, unless the option Projective is false

Description

We compute the Hilbert polynomial of the quotient of the ambient ring by an ideal.

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

i2 : I = monomialCurveIdeal(R, {1,3,4});

o2 : Ideal of R

i3 : h = hilbertPolynomial I

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

o3 : ProjectiveHilbertPolynomial

i4 : hilbertPolynomial (R/I)

o4 = - 3*P  + 4*P
          0      1

o4 : ProjectiveHilbertPolynomial

i5 : hilbertPolynomial(I, Projective=>false)

o5 = 4i + 1

o5 : QQ[i]

These Hilbert polynomials can serve as Hilbert functions too since the values of the Hilbert polynomial eventually are the same as the Hilbert function.

i6 : apply(10, k-> h(k))

o6 = {1, 5, 9, 13, 17, 21, 25, 29, 33, 37}

o6 : List

i7 : apply(10, k-> hilbertFunction(k,I))

o7 = {1, 4, 9, 13, 17, 21, 25, 29, 33, 37}

o7 : List

Caveat

As is often the case, calling this function on an ideal I actually computes it for R/I where R is the ring of I.

Ways to use this method:

hilbertPolynomial(Ideal) -- compute the Hilbert polynomial of the quotient of the ambient ring by the ideal

The source of this document is in Macaulay2Doc/functions/hilbertPolynomial-doc.m2:124:0.