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SOME PROPERTIES OF ANALYTIC DIFFERENCE VALUED FIELDS

Part of: Non-Archimedean analysis Model theory Differential and difference algebra

Published online by Cambridge University Press:  29 May 2015

Silvain Rideau
Silvain Rideau*
Affiliation:
Département des Mathématiques et Applications, École normale supérieure, 45 rue d’Ulm, 75005 Paris, France (silvain.rideau@ens.fr)
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Abstract

We prove field quantifier elimination for valued fields endowed with both an analytic structure that is $\unicode[STIX]{x1D70E}$-Henselian and an automorphism that is $\unicode[STIX]{x1D70E}$-Henselian. From this result we can deduce various Ax–Kochen–Eršov type results with respect to completeness and the independence property. The main example we are interested in is the field of Witt vectors on the algebraic closure of $\mathbb{F}_{p}$ endowed with its natural analytic structure and the lifting of the Frobenius. It turns out we can give a (reasonable) axiomatization of its first-order theory and that this theory does not have the independence property.

Keywords

valued fields with an automorphismanalytic structureseparated power series𝜎-HenselianityWitt vectorsNIP

MSC classification

Primary: 03C10: Quantifier elimination, model completeness and related topics 12H10: Difference algebra
Secondary: 03C45: Classification theory, stability and related concepts 32P05: Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem) 32B05: Analytic algebras and generalizations, preparation theorems
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 3 , June 2017 , pp. 447 - 499
Copyright
© Cambridge University Press 2016 

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