Computation of Delta sets of numerical monoids

Abstract

Let $\{a_1,\dots ,a_p\}$ be the minimal generating set of a numerical monoid $S$. For any $s\in S$, its Delta set is defined by $\Delta (s)=\{l_i-l_{i-1}|i=2,\dots ,k\}$ where $\{l_1<\cdots <l_k\}$ is the set $\{{\sum }_{i=1}^p{x}_i|s={\sum }_{i=1}^p{x}_i{a}_i\text{ and }{x}_i\in \mathbb{N}\text{ for all }i\}.$ The Delta set of $S$, denoted by $\Delta (S)$, is the union of all the sets $\Delta (s)$ with $s\in S.$ As proved in [S.T. Chapman, R. Hoyer, and N. Kaplan. Delta sets of numerical monoids are eventually periodic. Aequationes Math. 77 (2009), no. 3, 273--279], there exists a bound $N$ such that $\Delta (S)$ is the union of the sets $\Delta (s)$ with $s\in S$ and $s<N$. In this work, by using geometrical tools, we obtain a sharpened bound and we give an algorithm to compute $\Delta (S)$ from the factorizations of only $a_1$ elements.