Bounds for rational points on algebraic curves, optimal in the degree, and dimension growth

Abstract

Bounding the number of rational points of height at most $H$ on irreducible algebraic plane curves of degree $d$ has been an intense topic of investigation since the work by Bombieri and Pila. In this paper we establish optimal dependence on $d$, by showing the upper bound $C d^2 H^{2/d} (\log H)^\kappa$ with some absolute constants $C$ and $\kappa$. This bound is optimal with respect to both $d$ and $H$, except for the constants $C$ and $\kappa$. This answers a question raised by Salberger, leading to a simplified proof of his results on the uniform dimension growth conjectures of Heath-Brown and Serre, and where at the same time we replace the $H^\epsilon$ factor by a power of $\log H$. The main strength of our approach comes from the combination of a new, efficient form of smooth parametrizations of algebraic curves with a century-old criterion of P\'olya, which allows us to save one extra power of $d$ compared with the standard approach using B\'ezout's theorem.