Imaginaries, invariant types and pseudo p-adically closed fields

In this paper, we give a general criterion for elimination of imaginaries using an abstract independence relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these results to prove the elimination of imaginaries in bounded pseudop-adically closed fields.ValCoMo/[ANR-13-BS01-0006]//UCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de Investigaciones en Matemáticas Puras y Aplicadas (CIMPA

Hensel minimality

We present Hensel minimality, a new notion for non-archimedean tame geometry in Henselian valued fields. This notion resembles o-minimality for the field of reals, both in the way it is defined (though extra care for parameters of unary definable sets is needed) and in its consequences. In particular, it implies many geometric results that were previously known only under stronger assumptions like analyticity. As an application we show that Hensel minimality implies the existence of t-stratifications, as defined previously by the second author. Moreover, we obtain Taylor approximation results which lay the ground for analogues of point counting results by Pila and Wilkie, for analogues of Yomdin's $C^r$-parameterizations of definable sets, and for $p$-adic and motivic integration.Comment: 90 page

Hensel minimality II: Mixed characteristic and a diophantine application

In this paper, together with the preceding Part I [10], we develop a framework for tame geometry on Henselian valued fields of characteristic zero, called Hensel minimality. It adds to [10] the treatment of the mixed characteristic case. Hensel minimality is inspired by o-minimality and its role in real geometry and diophantine applications. We develop geometric results and applications for Hensel minimal structures that were previously known only under stronger or less axiomatic assumptions, and which often have counterparts in o-minimal structures. We prove a Jacobian property, a strong form of Taylor approximations of definable functions, resplendency results and cell decomposition, all under Hensel minimality – more precisely, $1$ -h-minimality. We obtain a diophantine application of counting rational points of bounded height on Hensel minimal curves

Imaginaries, invariant types and pseudo p-adically closed fields

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Enriching a predicate and tame expansions of the integers

Given a structure $\mathcal{M}$ and a stably embedded $\emptyset$-definable set $Q$, we prove tameness preservation results when enriching the induced structure on $Q$ by some further structure $\mathcal{Q}$. In particular, we show that if $T=\text{Th}(\mathcal{M})$ and $\text{Th}(\mathcal{Q})$ are stable (resp., superstable, $\omega$-stable), then so is the theory $T[\mathcal{Q}]$ of the enrichment of $\mathcal{M}$ by $\mathcal{Q}$. Assuming stability of $T$ and a further condition on $Q$ related to the behavior of algebraic closure, we also show that simplicity and NSOP$_1$ pass from $\text{Th}(\mathcal{Q})$ to $T[\mathcal{Q}]$. We then prove several applications for tame expansions of weakly minimal structures and, in particular, the group of integers. For example, we construct the first known examples of strictly stable expansions of $(\mathbb{Z},+)$. More generally, we show that any stable (resp., superstable, simple, NIP, NTP$_2$, NSOP$_1$) countable graph can be defined in a stable (resp., superstable, simple, NIP, NTP$_2$, NSOP$_1$) expansion of $(\mathbb{Z},+)$ by some unary predicate $A\subseteq \mathbb{N}$