On the number of ordinary conics

2015/11/11 by Thomas Boys, Boys, Thomas, Claudiu Valculescu +3
Computer Science · Mathematics · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications #Robotic Path Planning Algorithms

paper · pdf · doi:10.48550/arxiv.1511.03588

openalex publication_date 2015/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a lower bound on the number of ordinary conics determined by a finite point set in ℝ². An ordinary conic for a subset S of ℝ² is a conic that is determined by five points of S, and contains no other points of S. Wiseman and Wilson proved the Sylvester-Gallai-type statement that if a finite point set is not contained in a conic, then it determines at least one ordinary conic. We give a simpler proof of their result and then combine it with a result of Green and Tao to prove our main result: If S is not contained in a conic and has at most c|S| points on a line, then S determines Ω_c(|S|⁴) ordinary conics. We also give a construction, based on the group structure of elliptic curves, that shows that the exponent in our bound is best possible.

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