On Waring numbers of henselian rings

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 $p$.

34 pages, final version, to appear in Mathematika

References in corpus (4)

• Positive semidefinite analytic functions on real analytic surfaces
• Computing the Length of Sum of Squares and Pythagoras Element in a Global Field
• Sums of even powers of k-regulous functions
• Sums of squares on hypersurfaces