Non-archimedean Yomdin-Gromov parametrizations and points of bounded height

TL;DR

This paper develops a non-archimedean analogue of the Yomdin-Gromov Lemma for $p$-adic definable sets, enabling bounds on rational points and polynomial dimensions in non-archimedean geometry.

Contribution

It introduces a non-archimedean Yomdin-Gromov parametrization that accounts for Taylor approximation in totally disconnected settings, extending key results in $p$-adic and algebraic geometry.

Findings

Bound on rational points of bounded height on $p$-adic sets

Dimension bounds for polynomials on algebraic varieties over $ ext{C}((t))$

Local Lipschitz implies piecewise global Lipschitz in non-archimedean geometry

Abstract

We prove an analogue of the Yomdin-Gromov Lemma for $p$-adic definable sets and more broadly in a non-archimedean, definable context. This analogue keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of $p$-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over $\mathbb{C} ((t))$, in analogy to results by Pila and Wilkie, resp. by Bombieri and Pila. Along the way we prove, for definable functions in a general context of non-archimedean geometry, that local Lipschitz continuity implies piecewise global Lipschitz continuity.