Real algebraic curves with large finite number of real points

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.