Dimension theory and parameterized normalization for D-semianalytic sets
  over non-Archimedean fields

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Abstract

We develop a dimension theory for D-semianalytic sets over an arbitrary non-Archimedean complete field. Our main results are the equivalence of several notions of dimension and a theorem on additivity of dimensions of projections and fibers in characteristic 0. We also prove a parameterized version of normalization for D-semianalytic sets.

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p-adic and Real Subanalytic Sets

L van den Dries, J. Denef (1988)

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On definable subsets of p-adic fields

Angus Macintyre (1976)

The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p-adically closed fields, from the model-theoretical point of view. Cohen [5], from a standpoint less model-theoretic, also contributed much to this analogy.