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 nth level s n (k). We give some upper and lower bounds for the nth 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 nth Waring number of the total ring of fractions of R are obtained. We then provide applications. In particular we compute w n (Z p ) and w n (Q p ) for n ∈ {3, 4, 5} and any prime p.