On Waring numbers of henselian rings

TL;DR

This paper investigates the Waring numbers of henselian rings, providing bounds, exact computations in many cases, and applications including explicit calculations for p-adic integers and fields.

Contribution

It introduces bounds and methods for computing the Waring number of henselian rings, extending to their fraction fields, with explicit results for certain p-adic cases.

Findings

Derived bounds for Waring numbers in henselian rings

Computed Waring numbers for $Z_p$ and $Q_p$ for n=3,4,5

Extended results to total rings of fractions

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 $p$.