On the number of ordinary conics

TL;DR

This paper establishes a lower bound on the number of ordinary conics determined by a finite point set in the plane, improving understanding of geometric configurations and their combinatorial properties.

Contribution

It provides a simpler proof of a Sylvester-Gallai-type theorem for conics and establishes a tight lower bound on the number of ordinary conics for certain point sets.

Findings

Proved a lower bound of |S|^4 for ordinary conics

Provided a construction showing the bound's optimality

Simplified the proof of a key Sylvester-Gallai-type result

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 $\Omega_c(|S|^4)$ 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.