The Szemerédi-Trotter theorem in the complex plane
Csaba D. Tóth · 2012
This paper generalizes of the Szemerédi-Trotter theorem to the complex plane. Szemerédi and Trotter proved that the number of point-line incidences of n points and e lines in the real Euclidean plane is O(n 2/3 e 2/3 + n + e). This bound is tight. Although several short proofs were found to this theorem [14, 12], and many multidimensional generalizations were given, no tight bound has been known so far for incidences in higher dimensions. We extend the methods of Szemerédi and Trotter and prove that the number of point-line incidences of n points and e complex lines in the complex plane�2 is O(n