Point-Curve Incidences in the Complex Plane

• DOI: 10.1007/s00493-016-3441-7
• Publication Date: 2017-02-13T01:44:10Z
• Authors: Adam Sheffer; Endre Szabó; Joshua Zahl
• Keywords: Hypersurface; Lemma (botany); Complex plane

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 combine tools from algebraic geometry and differential geometry to prove a key technical lemma that controls the number of complex curves that can be contained inside a real hypersurface. This lemma may be of independent interest to other researchers proving incidence theorems over ${\mathbb C}$.<br/>The proof was significantly simplified, and now relies on the Picard-Lindelof theorem, rather than on foliations<br/>