Bounds for rational points on algebraic curves and dimension growth

Gal Binyamini; Raf Cluckers; Dmitry Novikov

doi:10.1093/imrn/rnae034

Bounds for rational points on algebraic curves and dimension growth

Gal Binyamini, Raf Cluckers, Dmitry Novikov

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

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 cd2H2/d(logH)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 d2. 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.

Original language	English
Article number	rnae034
Number of pages	10
Journal	International Mathematics Research Notices
DOIs	https://doi.org/10.1093/imrn/rnae034
Publication status	Published - 4 Mar 2024

Bibliographical note

G.B. and R.C. would like to thank the Royal Swedish Academy for their hospitality during the Schock Prize Symposium 2022 for Jonathan Pila, where this work began. It is also our pleasure to thank Per Salberger for fruitful discussions during this event, and in particular for pointing out the importance of the estimate (1), which is the main topic of this paper. R.C. would like to thank Tim Browning, Wouter Castryck, Philip Dittmann, Marcelo Paredes, Jonathan Pila, Per Salberger, and Roman Sasyk for interesting discussions on the topics of the paper.
Funding - G.B. was supported by funding from the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 802107), by the ISRAEL SCIENCE FOUNDATION (grant No. 2067/23) and by the Shimon and Golde Picker - Weizmann Annual Grant. R.C. was partially supported by KU Leuven IF C16/23/010 and the Labex CEMPI (ANR-11-LABX-0007-01). D.N. was supported by the ISRAEL SCIENCE FOUNDATION (grant no. 1167/17) and by funding received from the MINERVA Stiftung with the funds from the BMBF of the Federal Republic of Germany.

Communicated by Prof. Pila

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title = "Bounds for rational points on algebraic curves and dimension growth",

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 cd2H2/d(logH)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 d2. 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 P{\'o}lya to save one extra power of d; this is applied instead of B{\'e}zout after obtaining efficient forms of smooth parametrizations for curves of degree d.",

author = "Gal Binyamini and Raf Cluckers and Dmitry Novikov",

note = "G.B. and R.C. would like to thank the Royal Swedish Academy for their hospitality during the Schock Prize Symposium 2022 for Jonathan Pila, where this work began. It is also our pleasure to thank Per Salberger for fruitful discussions during this event, and in particular for pointing out the importance of the estimate (1), which is the main topic of this paper. R.C. would like to thank Tim Browning, Wouter Castryck, Philip Dittmann, Marcelo Paredes, Jonathan Pila, Per Salberger, and Roman Sasyk for interesting discussions on the topics of the paper. Funding - G.B. was supported by funding from the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 802107), by the ISRAEL SCIENCE FOUNDATION (grant No. 2067/23) and by the Shimon and Golde Picker - Weizmann Annual Grant. R.C. was partially supported by KU Leuven IF C16/23/010 and the Labex CEMPI (ANR-11-LABX-0007-01). D.N. was supported by the ISRAEL SCIENCE FOUNDATION (grant no. 1167/17) and by funding received from the MINERVA Stiftung with the funds from the BMBF of the Federal Republic of Germany. Communicated by Prof. Pila",

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Bounds for rational points on algebraic curves and dimension growth

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