Flattening and subanalytic sets in rigid analytic geometry

Abstract

Let K be an algebraically closed field endowed with a complete non-archimedean norm with valuation ring R. Let f:Y -> X be a map of K-affinoid varieties. In this paper we study the analytic structure of the image f(Y) in X; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are precisely the D-semianalytic sets, where D is the truncated division function first introduced by Denef and van den Dries. This result is most conveniently stated as a Quantifier Elimination result for the valuation ring R in an analytic expansion of the language of valued fields.