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arXiv · 2206.09904

On Waring numbers of henselian rings

Tomasz Kowalczyk·Piotr Miska
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Abstract

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 $p$.

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Tomasz Kowalczyk, Piotr Miska. 2022-06-20. On Waring numbers of henselian rings. https://doi.org/10.1112/mtk.12276

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