Point-curve incidences in the complex plane

Adam Sheffer, Joshua Zahl

Published 2015-02-24Version 1

We prove an incidence theorem for points and curves in the complex plane. Given a set of $m$ points in $\mathBB{R}^2$ and a set of $n$ curves with $k$ degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is $O\big(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n\big)$. We establish the slightly weaker bound $O_\varepsilon\big(m^{\frac{k}{2k-1}+\varepsilon}n^{\frac{2k-2}{2k-1}}+m+n\big)$ on the number of incidences between $m$ points and $n$ (complex) algebraic curves in $\mathbb{C}^2$ with $k$ degrees of freedom. We introduce a new tool to the study of geometric incidences by combining polynomial partitioning with foliations. Specifically, we rely on Frobenius' theorem on integrable distributions. We also apply various algebraic geometry machinery such as Chevalley's upper semi-continuity theorem.