On numerical semigroup elements and the $\ell_0$- and $\ell_\infty$-norms of their factorizations

Abstract

A numerical semigroup $S$ is a cofinite, additively-closed subset of $\mathbb Z_{\ge 0}$ that contains 0, and a factorization of $x \in S$ is a $k$-tuple $z = (z_1, \ldots, z_k)$ where $x = z_1a_1 + \cdots + z_ka_k$ expresses $x$ as a sum of generators of $S = \langle a_1, \ldots, a_k \rangle$. Much~of the study of non-unique factorization centers on factorization length $z_1 + \cdots + z_k$, which coincies with the $\ell_1$-norm of $z$ as the $k$-tuple. In this paper, we study the $\ell_\infty$-norm and $\ell_0$-norm of factorizations, viewed as alternative notions of length, with particular focus on the generalizations $\Delta_\infty(x)$ and $\Delta_0(x)$ of the delta set $\Delta(x)$ from classical factorization length. We prove that the $\infty$-delta set $\Delta_\infty(x)$ is eventually periodic as a function of $x \in S$, classify $\Delta_\infty(S)$ and the 0-delta set $\Delta_0(S)$ for several well-studied families of numerical semigroups, and identify families of numerical semigroups demonstrating $\Delta_\infty(S)$ and $\Delta_0(S)$ can be arbitrarily long intervals and can avoid arbitrarily long subintervals.