On differential operators of numerical semigroup rings

TL;DR

This paper investigates the structure of differential operators on numerical semigroup rings, especially for Arf semigroups, and explores derivations preserving monomial ideals within these rings.

Contribution

It provides a detailed description of the differential operators and their associated graded rings for numerical semigroup rings, with specific results for Arf semigroups and monomial ideals.

Findings

Differential operators are explicitly described for Arf semigroups.

The associated graded rings of these differential operators are characterized.

Results on derivations preserving monomial ideals are presented.

Abstract

If $S=<d_1,...,d_\nu>$ is a numerical semigroup, we call the ring $\C[S]=\C[t^{d_1},...,t^{d_\nu}]$ the semigroup ring of $S$. We study the ring of differential operators on $\C[S]$, and its associated graded in the filtration induced by the order of the differential operators. We find that these are easy to describe in case $S$ is a so called Arf semigroup. If $I$ is an ideal in $\C[S]$ that is generated by monomials, we also give some results on $\der(I,I)$ (the set of derivations which map $I$ into $I$).