Submission date: 8 February 2023

Abstract:

We prove that the number of rational points of height at most H lying on an irreducible algebraic curve of degree d is bounded by c d^2 H^{2/d}(log H)^κ where c, κ are universal constants. This bound is optimal except for the constants c and κ; the new aspect of the bound is the factor d^2. This result provides a positive answer to a question raized by Salberger, and allows to reprove and sharpen his result on uniform dimension growth in a short way. The main novelty in our proof is the application of a century-old theorem of Pólya to save one extra power of d; this is applied instead of Bézout after obtaining efficient forms of smooth parametrizations for curves of degree d.

Mathematics Subject Classification: Primary 11D45, Secondary 14G05, 34C10, 11G35

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Full text arXiv 2302.04209: pdf, ps.