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arXiv · 1511.03588

On the number of ordinary conics

Thomas Boys·Claudiu Valculescu·Frank de Zeeuw
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Abstract

We prove a lower bound on the number of ordinary conics determined by a finite point set in $\mathbb{R}^2$. An ordinary conic for a subset $S$ of $\mathbb{R}^2$ 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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Thomas Boys, Claudiu Valculescu, Frank de Zeeuw. 2016-05-21. On the number of ordinary conics. https://arxiv.org/abs/1511.03588

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