The Szemerédi-Trotter theorem in the complex plane

Abstract

It is shown that n points and e lines in the complex Euclidean plane 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 Szemerédi and Trotter about point-line incidences in the real Euclidean plane ℝ 2.