On Waring numbers of henselian rings

Abstract

Let $$ be a positive integer. Let $$ be a henselian local ring with residue field $$ of $$th level $$. We give some upper and lower bounds for the $$th Waring number $$ in terms of $$ and $$. In large number of cases, we are able to compute $$. Similar results for the $$th Waring number of the total ring of fractions of $$ are obtained. We then provide applications. In particular, we compute $$ and $$ for $$ and any prime $$.