Points of bounded height on curves and the dimension growth conjecture over $\mathbb{F}_q[t]$

TL;DR

This paper establishes new uniform upper bounds on the number of points of bounded height on varieties over $\\mathbb{F}_q[t]$, advancing understanding of the dimension growth conjecture with polynomial dependencies on key parameters.

Contribution

It provides the first polynomial-dependent bounds for points on curves and hypersurfaces over function fields, extending previous results and simplifying the dimension growth conjecture.

Findings

Proved polynomial bounds for points on projective curves over $\\mathbb{F}_q[t]$.

Improved bounds for affine curves compared to prior work.

Extended dimension growth results to higher-dimensional hypersurfaces with degree $d\geq 64$.

Abstract

In this article we prove several new uniform upper bounds on the number of points of bounded height on varieties over $\mathbb{F}_q[t]$. For projective curves, we prove the analogue of Walsh' result with polynomial dependence on $q$ and the degree $d$ of the curve. For affine curves, this yields an improvement to bounds by Sedunova, and Cluckers, Forey and Loeser. In higher dimensions, we prove a version of dimension growth for hypersurfaces of degree $d\geq 64$, building on work by Castryck, Cluckers, Dittmann and Nguyen in characteristic zero. These bounds depend polynomially on $q$ and $d$, and it is this dependence which simplifies the treatment of the dimension growth conjecture.