On semigroup rings with decreasing Hilbert function

Given a one-dimensional semigroup ring R=k[[S]], in this article we study the behaviour of the Hilbert function HR. By means of the notion of support of the elements in S, for some classes of semigroup rings 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 an explicit description of the Apéry set of S. In particular for e=v+3, we prove that HR is non-decreasing if e≤12 and we classify the semigroups with e=13 and HR decreasing. Finally we deduce that HR is non-decreasing for every Gorenstein semigroup ring with e≤v+4. This fact is not true in general: through numerical duplication and some of the above results another recent paper shows the existence of infinitely many one-dimensional Gorenstein rings with decreasing Hilbert function.