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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2404.02310 |
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| _version_ | 1866913296806838272 |
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| author | O'Neill, Christopher Ponomarenko, Vadim Ren, Eric |
| author_facet | O'Neill, Christopher Ponomarenko, Vadim Ren, Eric |
| contents | Fix $t\in [1,\infty]$. Let $S$ be an atomic commutative semigroup and, for all $x\in S$, let $\mathscr{L}_t(S):=\{\|f\|_t:f\in Z(x)\}$ be the "$t$-length set" of $x$ (using the standard $l_p$-space definition of $\|\cdot\|_t$). The $t$-Delta set of $x$ (denoted $Δ_t(S)$) is the set of gaps between consecutive elements of $\mathscr{L}_t(S)$; the Delta set of $S$ is then defined by $\bigcup\limits_{x\in S} Δ_t(S)$. Though all existing literature on this topic considers the $1$-Delta set, recent results on the $t$-elasticity of Numerical Semigroups (Behera et. al.) for $t\neq 1$ have brought attention to other invariants, such as the $t$-Delta set for $t\neq 1$, as well. Here we characterize $Δ_t(S)$ for all numerical semigroups $\langle a_1,a_2\rangle$ and all $t\in(1,\infty)$ outside a small family of extremal examples. We also determine the cardinality and describe the distribution of that aberrant family. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02310 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Perspicacious $l_p$ norm parameters O'Neill, Christopher Ponomarenko, Vadim Ren, Eric Commutative Algebra Fix $t\in [1,\infty]$. Let $S$ be an atomic commutative semigroup and, for all $x\in S$, let $\mathscr{L}_t(S):=\{\|f\|_t:f\in Z(x)\}$ be the "$t$-length set" of $x$ (using the standard $l_p$-space definition of $\|\cdot\|_t$). The $t$-Delta set of $x$ (denoted $Δ_t(S)$) is the set of gaps between consecutive elements of $\mathscr{L}_t(S)$; the Delta set of $S$ is then defined by $\bigcup\limits_{x\in S} Δ_t(S)$. Though all existing literature on this topic considers the $1$-Delta set, recent results on the $t$-elasticity of Numerical Semigroups (Behera et. al.) for $t\neq 1$ have brought attention to other invariants, such as the $t$-Delta set for $t\neq 1$, as well. Here we characterize $Δ_t(S)$ for all numerical semigroups $\langle a_1,a_2\rangle$ and all $t\in(1,\infty)$ outside a small family of extremal examples. We also determine the cardinality and describe the distribution of that aberrant family. |
| title | Perspicacious $l_p$ norm parameters |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2404.02310 |