The Szemeredi-Trotter Theorem in the Complex Plane

Csaba D. Toth

Published 2003-05-20, updated 2014-05-16Version 6

It is shown that $n$ points and $e$ lines in the complex Euclidean plane ${\mathbb C}^2$ determine $O(n^{2/3}e^{2/3}+n+e)$ point-line incidences. This bound is the best possible, and it generalizes the celebrated theorem by Szemer\'edi and Trotter about point-line incidences in the real Euclidean plane ${\mathbb R}^2$.