Real algebraic curves with large finite number of real points

Erwan Brugallé (1) , Alex Degtyarev (2) , Ilia Itenberg (3) , Frédéric Mangolte (4)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 Department of Mathematics [Ankara]
3 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
4 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

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.

Domaines: Géométrie algébrique [math.AG]

Erwan Brugallé, Alex Degtyarev, Ilia Itenberg, Frédéric Mangolte. Real algebraic curves with large finite number of real points. 2018. ⟨hal-01832739v1⟩