2023/02/08 by Gal Binyamini, Binyamini, Gal, Raf Cluckers +3 · 2 citations
Mathematics · #11G35 #34C10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Differential Geometry (math.DG) #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #Primary 11D45 #Secondary 14G05
paper · pdf · doi:10.48550/arxiv.2302.04209
openalex publication_date 2023/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 d2 H2/d (log H)κ with some absolute constants C and κ. This bound is optimal with respect to both d and H, except for the constants C and κ. 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 Hε 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 Pólya, which allows us to save one extra power of d compared with the standard approach using Bézout's theorem.