Flattening and subanalytic sets in rigid analytic geometry

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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)⊂ X; such an image is a typical example of a subanalytic set. Using Embedded Resolution of Singularities, we derive in the zero characteristic case a Uniformization Theorem for subanalytic sets: after finitely many local blowing ups with smooth centres, a subanalytic set becomes semi-analytic. To prove this we establish a Flattening Theorem for affinoid varieties in the style of Hironaka, which allows a reduction to the study of subanalytic sets arising from flat maps. Specifically we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups. The case of an image under a flat map is then dealt with by a small extension of a result of Raynaud. Our result can be conveniently stated as a Quantifier Elimination theorem for the valuation ring R in an analytic expansion of the language of valued fields. This formulation is in the style of Denef and van den Dries.