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

TL;DR

This paper proves the uniform dimension growth conjecture for varieties of degree at least 4 over global fields, generalizing bounds for curves and improving existing estimates.

Contribution

It extends previous work to establish uniform bounds for rational points on higher-degree varieties over global fields, including a generalization of Bombieri and Pila's bounds.

Findings

Proved the uniform dimension growth conjecture for degree ≥ 4 varieties.

Generalized Bombieri and Pila bounds to curves over global fields.

Improved the logarithmic factor in bounds from B^ε to log(B).

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 $B^{\varepsilon}$ factor by a $\log(B)$ factor.