On semigroup rings with decreasing Hilbert function

TL;DR

This paper investigates conditions under which the Hilbert function of one-dimensional semigroup rings decreases, providing classifications and explicit descriptions for specific embedding dimensions and multiplicities, especially in Gorenstein cases.

Contribution

It offers new criteria for decreasing Hilbert functions in semigroup rings and classifies semigroups with specific parameters, extending understanding of their algebraic structure.

Findings

Decreasing Hilbert functions occur under specific generator conditions.

Explicit description of Apery sets for certain embedding dimensions.

Hilbert function is non-decreasing for Gorenstein semigroup rings with e ≤ v+4.

Abstract

In this paper we study the Hilbert function HR of one-dimensional semigroup rings R = k[[S]]. For some classes of semigroups, by means of the notion of support of the elements in S, we give conditions on the generators of S in order to have decreasing HR. When the embedding dimension v and the multiplicity e verify v + 3 ? e ? v + 4, the decrease of HR gives explicit description of the Apery set of S. In particular for e = v+3, we classify the semigroups with e = 13 and HR decreasing, further we show that HR is non-decreasing if e < 12. Finally we deduce that HR is non-decreasing for every Gorenstein semigroup ring with e ? v + 4.