Existential uniform pp-adic integration and descent for integrability and largest poles

Since the work by Denef, $p$-adic cell decomposition provides a well-established method to study $p$-adic and motivic integrals. In this paper, we present a variant of this method that keeps track of existential quantifiers. This enables us to deduce descent properties for $p$-adic integrals. In particular, we show that integrability for `existential' functions descends from any $p$-adic field to any $p$-adic subfield. As an application, we obtain that the largest pole of the Serre-Poincar\'e series can only increase when passing to field extensions. As a side result, we prove a relative quantifier elimination statement for Henselian valued fields of characteristic zero that preserves existential formulas.Comment: 38 page

Motivic integration and the Grothendieck group of pseudo-finite fields

We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how p-adic integrals of a very general type depend on p.Comment: 11 page

Analytic cell decomposition and analytic motivic integration

The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language, and its application to analytic motivic integrals and analytic integrals over \FF_q((t)) of big enough characteristic. To accomplish this, we introduce a general framework for Henselian valued fields $K$ with analytic structure, and we investigate the structure of analytic functions in one variable, defined on annuli over $K$. We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper pave the way for a theory of \emph{analytic} motivic integration and \emph{analytic} motivic constructible functions in the line of R. Cluckers and F. Loeser [\emph{Fonctions constructible et int\'egration motivic I}, Comptes rendus de l'Acad\'emie des Sciences, {\bf 339} (2004) 411 - 416]

A closedness theorem and applications in geometry of rational points over Henselian valued fields

We develop geometry of algebraic subvarieties of $K^{n}$ over arbitrary Henselian valued fields $K$. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach is again the closedness theorem that the projections $K^{n} \times \mathbb{P}^{m}(K) \to K^{n}$ are definably closed maps. It enables application of resolution of singularities in much the same way as over locally compact ground fields. As before, the proof of that theorem uses i.a. the local behavior of definable functions of one variable and fiber shrinking, being a relaxed version of curve selection. But now, to achieve the former result, we first examine functions given by algebraic power series. All our previous results will be established here in the general settings: several versions of curve selection (via resolution of singularities) and of the \L{}ojasiewicz inequality (via two instances of quantifier elimination indicated below), extending continuous hereditarily rational functions as well as the theory of regulous functions, sets and sheaves, including Nullstellensatz and Cartan's theorems A and B. Two basic tools applied in this paper are quantifier elimination for Henselian valued fields due to Pas and relative quantifier elimination for ordered abelian groups (in a many-sorted language with imaginary auxiliary sorts) due to Cluckers--Halupczok. Other, new applications of the closedness theorem are piecewise continuity of definable functions, H\"{o}lder continuity of definable functions on closed bounded subsets of $K^{n}$, the existence of definable retractions onto closed definable subsets of $K^{n}$, and a definable, non-Archimedean version of the Tietze--Urysohn extension theorem. In a recent preprint, we established a version of the closedness theorem over Henselian valued fields with analytic structure along with some applications.Comment: This paper has been published in Journal of Singularities 21 (2020), 233-254. arXiv admin note: substantial text overlap with arXiv:1704.01093, arXiv:1703.08203, arXiv:1702.0784

Cell Decomposition for semibounded p-adic sets

We study a reduct L\ast of the ring language where multiplication is restricted to a neighbourhood of zero. The language is chosen such that for p-adically closed fields K, the L\ast-definable subsets of K coincide with the semi-algebraic subsets of K. Hence structures (K,L\ast) can be seen as the p-adic counterpart of the o-minimal structure of semibounded sets. We show that in this language, p-adically closed fields admit cell decomposition, using cells similar to p-adic semi-algebraic cells. From this we can derive quantifier-elimination, and give a characterization of definable functions. In particular, we conclude that multi- plication can only be defined on bounded sets, and we consider the existence of definable Skolem functions.Comment: 20 page