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European Journal of Mathematics

, Volume 5, Issue 3, pp 686–711 | Cite as

Real algebraic curves with large finite number of real points

  • Erwan Brugallé
  • Alex Degtyarev
  • Ilia ItenbergEmail author
  • Frédéric Mangolte
Research Article
  • 33 Downloads

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.

Keywords

Positive polynomials Real algebraic curves Real algebraic surfaces Patchworking 

Mathematics Subject Classification

14P25 14H50 14M25 

Notes

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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Erwan Brugallé
    • 1
  • Alex Degtyarev
    • 2
  • Ilia Itenberg
    • 3
    Email author
  • Frédéric Mangolte
    • 4
  1. 1.Université de Nantes, Laboratoire de Mathématiques Jean LerayNantes Cedex 3France
  2. 2.Department of MathematicsBilkent UniversityAnkaraTurkey
  3. 3.Institut de Mathématiques de Jussieu–Paris Rive GaucheSorbonne UniversitéParis Cedex 5France
  4. 4.Laboratoire angevin de recherche en mathématiques (LAREMA)Université d’Angers, CNRSAngers Cedex 01France

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