On Waring numbers of henselian rings
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866911999704694784 |
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| author | Kowalczyk, Tomasz Miska, Piotr |
| author_facet | Kowalczyk, Tomasz Miska, Piotr |
| contents | Let $n>1$ be a positive integer. Let $R$ be a henselian local ring with residue field $k$ of $n$th level $s_n(k)$. We give some upper and lower bounds for the $n$th Waring number $w_n(R)$ in terms of $w_n(k)$ and $s_n(k)$. In large number of cases we are able to compute $w_n(R)$. Similar results for the $n$th Waring number of the total ring of fractions of $R$ are obtained. We then provide applications. In particular we compute $w_n(\mathbb{Z}_p)$ and $w_n(\mathbb{Q}_p)$ for $n\in\{3,4,5\}$ and any prime $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_09904 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On Waring numbers of henselian rings Kowalczyk, Tomasz Miska, Piotr Commutative Algebra Number Theory 12D15, 11P05 Let $n>1$ be a positive integer. Let $R$ be a henselian local ring with residue field $k$ of $n$th level $s_n(k)$. We give some upper and lower bounds for the $n$th Waring number $w_n(R)$ in terms of $w_n(k)$ and $s_n(k)$. In large number of cases we are able to compute $w_n(R)$. Similar results for the $n$th Waring number of the total ring of fractions of $R$ are obtained. We then provide applications. In particular we compute $w_n(\mathbb{Z}_p)$ and $w_n(\mathbb{Q}_p)$ for $n\in\{3,4,5\}$ and any prime $p$. |
| title | On Waring numbers of henselian rings |
| topic | Commutative Algebra Number Theory 12D15, 11P05 |
| url | https://arxiv.org/abs/2206.09904 |