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 $\mathrm{\Delta }(s)=\{l_i-l_{i-1}\mathrm{\mid }i=2,\dots ,k\}$ where $\{l_1<\cdots <l_k\}$ is the set ${\sum_{i=1}^px_i,|, s=\sum_{i=1}^px_ia_i $\text{ and }$ x_i\in \N \textrm{ for all }i}.$TheDeltasetof$S$,denotedby$\Delta(S)$, is the union of all the sets $\mathrm{\Delta }(s)$ with $s\in S\mathrm{.}$ 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 $\mathrm{\Delta }(S)$ is the union of the sets $\mathrm{\Delta }(s)$ 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 $\mathrm{\Delta }(S)$ from the factorizations of only $a_1$ elements.