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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 (H) ^ with some absolute constants C and. This bound is optimal with respect to both d and H, except for the constants C and. 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^ factor by a power of 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 Pólya, which allows us to save one extra power of d compared with the standard approach using Bézout’s theorem.
Binyamini et al. (Mon,) studied this question.