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Bounding the number of rational points of height at most $H$ on irreduciblealgebraic plane curves of degree $d$ has been an intense topic of investigationsince the work by Bombieri and Pila. In this paper we establish optimaldependence 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 withrespect to both $d$ and $H$, except for the constants $C$ and $\kappa$. Thisanswers a question raised by Salberger, leading to a simplified proof of hisresults 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 anew, efficient form of smooth parametrizations of algebraic curves with acentury-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.

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