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

Abstract

In this article, we prove several new uniform upper bounds on the number of points of bounded height on varieties over F q [ t ] $\mathbb {F}_q[t]$ . For projective curves, we prove the analogue of Walsh' result with polynomial dependence on q $q$ and the degree d $d$ 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 ⩾ 64 $d\geqslant 64$ , building on work by Castryck, Cluckers, Dittmann and Nguyen in characteristic zero. These bounds depend polynomially on q $q$ and d $d$ , and it is this dependence which simplifies the treatment of the dimension growth conjecture.