Real algebraic curves with large finite number of real points
Erwan Brugall\'e (LMJL), Alex Degtyarev, Ilia Itenberg (IMJ-PRG),, Fr\'ed\'eric Mangolte (LAREMA)

TL;DR
This paper investigates the maximum number of real points on algebraic curves in the projective plane, improving bounds and constructing near-optimal examples, especially for small degrees and genus.
Contribution
It provides improved bounds and explicit constructions for real algebraic curves with many real points, extending results to other surfaces.
Findings
Upper bounds are sharp for small genus
Constructed curves close to optimal for small degree
Extended results to ruled surfaces
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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