Uniform Yomdin–Gromov parametrizations and points of bounded height in valued fields

Cluckers, Raf; Forey, Arthur; Loeser, François

doi:10.2140/ant.2020.14.1423

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Uniform Yomdin–Gromov parametrizations and points of bounded height in valued fields

Raf Cluckers, Arthur Forey and François Loeser

Vol. 14 (2020), No. 6, 1423–1456

DOI: 10.2140/ant.2020.14.1423

Abstract

We prove a uniform version of non-Archimedean Yomdin–Gromov parametrizations in a definable context with algebraic Skolem functions in the residue field. The parametrization result allows us to bound the number of $𝔽_{q} [t]$ -points of bounded degrees of algebraic varieties, uniformly in the cardinality $q$ of the finite field $𝔽_{q}$ and the degree, generalizing work by Sedunova for fixed $q$ . We also deduce a uniform non-Archimedean Pila–Wilkie theorem, generalizing work by Cluckers–Comte–Loeser.

Keywords

rational points, points of bounded height, parametrizations

Mathematical Subject Classification 2010

Primary: 14G05

Secondary: 03C98, 11D88, 11G50

Milestones

Received: 18 February 2019

Revised: 9 January 2020

Accepted: 3 March 2020

Published: 30 July 2020

Authors

Raf Cluckers
University of Lille CNRS, UMR 8524 – Laboratoire Paul Painlevé F-59000 Lille France	KU Leuven Department of Mathematics Leuven Belgium

Arthur Forey
D-Math ETH Zürich Zürich Switzerland

François Loeser
Institut Universitaire de France Sorbonne Université UMR 7586 CNRS Institut Mathématique de Jussieu Paris France

Vol. 14, No. 6, 2020