Some Properties of Analytic Difference Valued Fields

Abstract

We prove field quantifier elimination for valued fields endowed with both an analytic structure and an automorphism that are σ-Henselian. From this result we can deduce various Ax-Kochen-Ersov type results with respect to completeness and the NIP property. The main example we are interested in is the field of Witt vectors on the algebraic closure of 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 is NIP.

Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero

“…The following result by S. Rideau in [29] (as a variation on results by Basarab in [2]) is obtained in loc. cit.…”

“…The elimination of valued field quantifiers only is more classical and can be proved in the line of [28], see e.g. [2] and the variants in [29] where this is done using model theoretic methods, and [19] where this is done in the line of Cohen's method of [12].…”

Imaginaries and invariant types in existentially closed valued differential fields

“…By [Sca00,Theorem 7.1] we have field quantifier elimination in the three sorted language. The stable embeddedness and purity results for k and Γ follow (see, for example, [Rid,Remark A.10.2]). Now, the theory induced on k and Γ are, respectively, differentially closed fields and divisible ordered Abelian groups.…”

“…A , by [Rid,Corollary 5.5] there exists an L G (M )-formula ψ(z, w, γ) and L K -terms r(x, λ) such that M ⊧ f λ (y, z) = γ if and only if M ⊧ ψ(z, r(y, λ), γ). Taking r to be the identity, the graph of f λ also has this form when T = ACVF G .…”

Non-Archimedean Yomdin–gromov Parametrizations and Points of Bounded Height

“…From now on in Section 3, definable sets and functions will be so for the language L. Note that the study of definable sets was initiated in the works of Macintyre [25] and Denef and van den Dries [19] in the p-adic case, and was generalized later to this and other settings in for example [1,12,14,36].…”