Computation of Delta sets of numerical monoids

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 $Δ(s)=\{l_{i}-l_{i-1}|i=2,\dots,k\}$ where $\{l_1<\dots<l_k\}$ is the set $\{\sum_{i=1}^px_i\,|\, s=\sum_{i=1}^px_ia_i \textrm{ and } x_i\in \N \textrm{ for all }i\}.$ The Delta set of $S$, denoted by $Δ(S)$, is the union of all the sets $Δ(s)$ 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 $Δ(S)$ is the union of the sets $Δ(s)$ 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 $Δ(S)$ from the factorizations of only $a_1$ elements.

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