Hilbert function of a proper ideal

Definition

Let \(S\) be a numerical semigroup and \(E\) a proper ideal of \(S\). It is defined the Hilbert function of \(E\) as the application

\[ HF_E(n) = |nE \setminus (n+1)E|, \] where

\[ nE = \{e_1 + \cdots + e_n ~ | ~ e_1, \ldots, e_n \in E\}. \]

This definition comes from the concept of Hilbert function of an ideal in \(\mathbb{K}[x_1, \ldots, x_n]\) for some fixed field \(\mathbb{K}\).

Examples

\(\circ\) Let \(S = \langle 4, 7, 9 \rangle = \{0, 4, 7, 8, 9, 11, \rightarrow \}\), \(I = \{-3, -1, 0\} + S\) and \(n = 2\). Doing the computations, if \(L = \{0, 4, 7, 8, 9\}\) and \(L - x = \{l - x ~ | ~ l \in L\}\),

\[ I = (L-3) \cup (L - 1) \cup L \cup \{8, \rightarrow\} = \{-3, -1, 0, 1, 3, \rightarrow\} \]

where it is deduced that

\[ 2I = \{-3, -1, 0, 1, 3, \rightarrow\} + \{-3, -1, 0, 1, 3, \rightarrow\} = (\{-3, -1, 0, 1\} + \{-3, -1, 0, 1 \}) \cup \{0, \rightarrow\} = \{-6, -4, \rightarrow\} \]

and

\[ 3I = 2I + I = \{-6, -4, \rightarrow \} + \{-3, -1, 0, 1, 3, \rightarrow\} = \{-9, -7, \rightarrow\}. \]

Therefore, the Hilbert function of \(I\) in \(n = 1\) and \(N = 2\) is

\[ HF_I(1) = |I \setminus 2I| = 0, ~~ HF_I(2) = |2I \setminus 3I| = 0. \]

Examples with GAP

The following example is made with the package NumericalSgps in GAP.

\(\diamond\) Let \(S = \langle 39, 56, 93, 96, 104, 105 \rangle\) and \(I = \{47, 57, 70, 92, 100\} + S\), in GAP:

gap> S := NumericalSemigroup(39, 56, 93, 96, 104, 105);
<Numerical semigroup with 6 generators>
gap> I := [47, 57, 70, 92, 100] + S;
<Ideal of numerical semigroup>

Given a proper ideal \(I\) of a numerical semigroup and a non-negative integer \(n\), the function HilbertFunctionOfIdealOfNumericalSemigroup returns \(HF_I(n)\).

gap> List([1..20], l -> HilbertFunctionOfIdealOfNumericalSemigroup(l, I));
[ 37, 44, 52, 54, 54, 54, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53 ]

Moreover, given a proper ideal \(I\) of a numerical semigroup, the function HilbertFunction returns a functions that maps to each \(n\) the value \(HF_I(n)\).

gap> f := HilbertFunction(I);
function( n ) ... end
gap> f(50);
53

References

Delgado, M., P. A. Garcia-Sanchez, and J. Morais. 2024. “NumericalSgps, a Package for Numerical Semigroups, Version 1.4.0.” https://gap-packages.github.io/ numericalsgps.