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Uniform bounds for the number of rational points on varieties over global fields

Marcelo Paredes and Román Sasyk

Vol. 16 (2022), No. 8, 1941–2000
Abstract

We extend the work of Salberger; Walsh; Castryck, Cluckers, Dittmann and Nguyen; and Vermeulen to prove the uniform dimension growth conjecture of Heath-Brown and Serre for varieties of degree at least 4 over global fields. As an intermediate step, we generalize the bounds of Bombieri and Pila to curves over global fields and in doing so we improve the B𝜀 factor by a log (B) factor.

Keywords
varieties over global fields, heights in global fields, number of rational solutions of diophantine equations, determinant method
Mathematical Subject Classification
Primary: 11D45, 11G35, 11G50, 14G05
Milestones
Received: 28 February 2021
Revised: 14 August 2021
Accepted: 12 November 2021
Published: 29 November 2022
Authors
Marcelo Paredes
Department of Mathematics
Facultad de Ciencias Exactas y Naturales
Universidad de Buenos Aires
Ciudad Universitaria – Pabellón I
1428
Ciudad Autónoma de Buenos Aires
Argentina
Román Sasyk
Department of Mathematics
Facultad de Ciencias Exactas y Naturales
Universidad de Buenos Aires
Ciudad Universitaria – Pabellón I
1428
Ciudad Autónoma de Buenos Aires
Argentina
Instituto Argentino de Matemáticas Alberto P. Calderón-CONICET
Saavedra 15, Piso 3
1083
Ciudad Autónoma de Buenos Aires
Argentina