Saved in:
| Main Authors: | , , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.12241 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916653596409856 |
|---|---|
| author | Cyrusian, Sogol Domat, Alex O'Neill, Christopher Ponomarenko, Vadim Ren, Eric Ward, Mayla |
| author_facet | Cyrusian, Sogol Domat, Alex O'Neill, Christopher Ponomarenko, Vadim Ren, Eric Ward, Mayla |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12241 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On numerical semigroup elements and the $\ell_0$- and $\ell_\infty$-norms of their factorizations Cyrusian, Sogol Domat, Alex O'Neill, Christopher Ponomarenko, Vadim Ren, Eric Ward, Mayla Commutative Algebra 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. |
| title | On numerical semigroup elements and the $\ell_0$- and $\ell_\infty$-norms of their factorizations |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2503.12241 |