hilbertFunction(List,Ideal) -- compute the Hilbert function of the quotient of the ambient ring by an ideal

Synopsis

Usage:
hilbertFunction(d,I)
Function: hilbertFunction
Inputs:
- d, a list, a multidegree, or if I is singly graded, then one may write just ZZ
- I, an ideal
Outputs:
- an integer, the value of the Hilbert function of the quotient of the ambient ring by the ideal

Description

i1 : R = QQ[a..f, Degrees=>{6:{1,1}}];

i2 : I = ideal (a*b, c*d, e*f);

o2 : Ideal of R

i3 : hilbertFunction({2,2}, I)

o3 = 3

i4 : S = R/I;

i5 : basis({2,2},S)

o5 = | a2 ac ad ae af b2 bc bd be bf c2 ce cf d2 de df e2 f2 |

             1       18
o5 : Matrix S  <--- S

In the case where the ring is singly graded, then instead of having the input be a list of length 1 containing the degree, it is sufficient to write an integer.

i6 : R = QQ[a..f];

i7 : I = ideal (a*b, c*d, e*f);

o7 : Ideal of R

i8 : hilbertFunction(2, I)

o8 = 18

i9 : S = R/I;

i10 : basis(2,S)

o10 = | a2 ac ad ae af b2 bc bd be bf c2 ce cf d2 de df e2 f2 |

              1       18
o10 : Matrix S  <--- S

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.