Approximation properties for some non-Noetherian local rings

Abstract

In this paper we study Artin approximation in power series rings in several variables over complete rank-one valuation rings. In particular we prove that the completion of the algebraic elements has the approximation property over the ring of algebraic power series. Moreover, for an important class of complete rank-one valuation rings, e.g. the ring of complex p-adic integers, we prove that the ring of algebraic power series is equal to the henselisation of the polynomial ring and that each algebraic power series has coefficients lying in a finitely generated /^-algebra, where R is discrete valuation rings.