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

TL;DR

The paper investigates factorization lengths in numerical semigroups using the $\ell_\infty$- and $\ell_0$-norms, providing a framework distinct from classical $\ell_1$-lengths. It proves that the infinity-length delta set $\Delta_\infty(x)$ is eventually periodic and develops a structure theorem showing that the infinity-length components $\mathcal{L}_\infty(x,i)$ are almost arithmetic progressions with gaps governed by modular data, leading to a periodic $\Delta_\infty(S)$. It classifies $\Delta_\infty(S)$ and $\Delta_0(S)$ for several well-studied families (e.g., supersymmetric, maximal embedding dimension, arithmetic and geometric sequences) and demonstrates that these delta sets can realize arbitrarily long intervals as well as long gaps, prompting a delta-realization conjecture. The work leverages trades, minimal presentations, and Betti elements to analyze infinity-length sets and advances understanding of non-unique factorization phenomena in numerical semigroups with potential implications for discrete optimization and knapsack-type problems.

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 $Δ_\infty(x)$ and $Δ_0(x)$ of the delta set $Δ(x)$ from classical factorization length. We prove that the $\infty$-delta set $Δ_\infty(x)$ is eventually periodic as a function of $x \in S$, classify $Δ_\infty(S)$ and the 0-delta set $Δ_0(S)$ for several well-studied families of numerical semigroups, and identify families of numerical semigroups demonstrating $Δ_\infty(S)$ and $Δ_0(S)$ can be arbitrarily long intervals and can avoid arbitrarily long subintervals.

Figures (1)

• Figure 1: Diagram of $\infty$-length set elements for large $x \in S$, where the $B_i$ are defined in Theorem \ref{['t:atlantis']}

Theorems & Definitions (29)

• Conjecture 1.1
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Proposition 2.5
• proof
• Theorem 2.6