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 $z=(z_1,\dots ,z_k)$ where $x=z_1{a}_1+\cdots +z_k{a}_k$ expresses $x$ as a sum of generators of $S=⟨a_1,\dots ,a_k⟩$. 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.