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Real algebraic curves with large finite number of real points

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Abstract

We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal curves of small degree. Our upper bound is sharp if the genus is small as compared to the degree. Some of the results are extended to other real algebraic surfaces, most notably ruled.

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Acknowledgements

Part of the work on this project was accomplished during the second and third authors’ stay at the Max-Planck-Institut für Mathematik, Bonn. We are grateful to the MPIM and its friendly staff for their hospitality and excellent working conditions. We extend our gratitude to Boris Shapiro, who brought the finite real curve problem to our attention and supported our work by numerous fruitful discussions. We would also like to thank Ilya Tyomkin for his help in specializing general statements from [20] to a few specific situations.

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Authors and Affiliations

  1. Université de Nantes, Laboratoire de Mathématiques Jean Leray, 2 rue de la Houssinière, 44322, Nantes Cedex 3, France

    Erwan Brugallé

  2. Department of Mathematics, Bilkent University, 06800, Ankara, Turkey

    Alex Degtyarev

  3. Institut de Mathématiques de Jussieu–Paris Rive Gauche, Sorbonne Université, 4 place Jussieu, 75252, Paris Cedex 5, France

    Ilia Itenberg

  4. Laboratoire angevin de recherche en mathématiques (LAREMA), Université d’Angers, CNRS, 49045, Angers Cedex 01, France

    Frédéric Mangolte

Authors
  1. Erwan Brugallé
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  2. Alex Degtyarev
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  3. Ilia Itenberg
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  4. Frédéric Mangolte
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Correspondence to Ilia Itenberg.

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The first author is partially supported by the Grant TROPICOUNT of Région Pays de la Loire. The first, third and fourth authors are partially supported by the ANR Grant ANR-18-CE40-0009 ENUMGEOM. The second author is partially supported by the TÜBİTAK Grant 116F211.

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Brugallé, E., Degtyarev, A., Itenberg, I. et al. Real algebraic curves with large finite number of real points. European Journal of Mathematics 5, 686–711 (2019). https://doi.org/10.1007/s40879-019-00324-9

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  • DOI: https://doi.org/10.1007/s40879-019-00324-9

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