On the ultrametric Stone-Weierstrass theorem and Mahler's expansion

Abstract

We describe an ultrametric version of the Stone-Weierstrass theorem, without any assumption on the residue field. If E is a subset of a rank-one valuation domain V, we show that the ring of polynomial functions is dense in the ring of continuous functions from E to V if and only if the topological closure E ^ of E in the completion V ^ of V is compact. We then show how to expand continuous functions in sums of polynomials.