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Fields with Analytic Structure.

Tue, 08/18/2009 - 11:53 — SomNambul
Authors: Raf Cluckers, Leonard Lipshitz
Subjects: Logic
link: http://arxiv.org/abs/0908.2376
Abstract

We present a unifying theory of fields with certain classes of analytic
functions, called fields with analytic structure. Both real closed fields and
Henselian valued fields are considered.

For real closed fields with analytic structure, o-minimality is shown. For
Henselian valued fields, both the model theory and the analytic theory are
developed. We give a list of examples that comprises, to our knowledge, all
principal, previously studied, analytic structures on Henselian valued fields,
as well as new ones. The b-minimality is shown, as well as other properties
useful for motivic integration on valued fields. The paper is reminiscent of
[Denef, van den Dries, "p-adic and real subanalytic sets" Ann. of Math. (2) 128
(1988) 79--138], of [Cohen, Paul J. "Decision procedures for real and p-adic
fields" Comm. Pure Appl. Math. 22 (1969) 131--151, and of [Fresnel, van der
Put, "Rigid analytic geometry and its applications" Progress in Mathematics,
218 Birkhauser (2004)], and unifies work by van den Dries, Haskell, Macintyre,
Macpherson, Marker, Robinson, and the authors.

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