# On the Hilbert Function of Four-Generated Numerical Semigroup Rings

• Anna Oneto
• Grazia Tamone
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 40)

## Abstract

In this article we study the Hilbert function HR of one-dimensional semigroup rings R = k[[S]], with embedding dimension four over an infinite field k. Let S =< e, n2, n3, n4 >  and let M = S ∖{0}. Consider the Apéry set of S with respect to the multiplicity e and its subsets Ah = {s ∈ Apéry(S) | s ∈ hM ∖ (h + 1)M}, h ≥ 2. Further let D2 ⊆{n3, n4} be the set of generators with torsion order 1. We prove that HR is non-decreasing at level ≤ 3 and that HR is non decreasing in each of the following cases: if A2 has cardinality ≤ 4, if A3 has cardinality ≤ 3, if A4 = ∅, if D2 has cardinality 2, if S has multiplicity ≤ 13.

## Keywords

Numerical semigroup Hilbert function Apéry set

## Mathematics Subject Classification

Primary: 13H10; Secondary: 14H20

## Notes

### Acknowledgements

This work is supported by the National Group of Algebraic and Geometric Structures and their Applications (GNSAGA)-INDAM—Italy.

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