Vol. 14, No. 6, 2020

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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
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 𝔽q[t]-points of bounded degrees of algebraic varieties, uniformly in the cardinality q of the finite field 𝔽q 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.

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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