Uniform properties of rigid subanalytic sets

Lipshitz, Leonard; Robinson, Zachary

doi:10.1090/S0002-9947-05-04003-1

Uniform properties of rigid subanalytic sets
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Trans. Amer. Math. Soc. 357 (2005), 4349-4377 Request permission

Abstract:

In the context of rigid analytic spaces over a non-Archimedean valued field, a rigid subanalytic set is a Boolean combination of images of rigid analytic maps. We give an analytic quantifier elimination theorem for (complete) algebraically closed valued fields that is independent of the field; in particular, the analytic quantifier elimination is independent of the valued field’s characteristic, residue field and value group, in close analogy to the algebraic case. This provides uniformity results about rigid subanalytic sets. We obtain uniform versions of smooth stratification for subanalytic sets and the Łojasiewicz inequalities, as well as a unfiorm description of the closure of a rigid semianalytic set.

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