Real algebraic curves with large finite number of real points
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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.
KeywordsPositive polynomials Real algebraic curves Real algebraic surfaces Patchworking
Mathematics Subject Classification14P25 14H50 14M25
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  to a few specific situations.
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