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

TL;DR

This paper establishes an optimal upper bound on the number of rational points of bounded height on algebraic plane curves, improving previous results by refining dependence on degree and height, and simplifying proofs of related conjectures.

Contribution

It provides a new, optimal bound for rational points on algebraic curves with respect to degree and height, using innovative parametrization and classical criteria.

Findings

Bound $C d^2 H^{2/d} ( ext{log } H)^ ext{kappa}$ is optimal in degree and height.

Simplifies proofs of uniform dimension growth conjectures.

Replaces $H^ extepsilon$ with a power of $ ext{log } H$.

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.