Incidences in the complex plane

Recently Joshua Zahl and I have uploaded to arXiv a paper about incidence in {\mathbb C}^2  . In this post I briefly survey what was previously known concerning such incidences, and then describe what the new paper is about.

zahl_photo
Joshua Zahl.

Let \Gamma  be a set of curves and let {\cal P}  be a set of points, both in {\mathbb F}^2  for some field \mathbb F  . We say that the arrangement ({\cal P},\Gamma)  has k  degrees of freedom and multiplicity-type s  if

  • For any k  points from \cal P  , there are at most s  curves from \Gamma  that contain all of them.
  • Any pair of curves from \Gamma  intersect in at most s  points.
The following theorem is currently the best known general incidence bound in {\mathbb R}^2  (better bounds are known for some special cases, such as circles and parabolas). For a proof, see for example this paper by Pach and Sharir.

Theorem 1. Let \cal P  be a set of m  points and let \Gamma  be a set of n  algebraic curves of degree at most D  , both in {\mathbb R}^2  . Suppose that ({\cal P},\Gamma)  has k  degrees of freedom and multiplicity type s  . Then

I({\cal P},\Gamma) = O_{k,s,D}\big(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n\big).

In the complex plane, things progressed rather slowly. Csaba Tóth derived a matching incidence bound for the special case of complex lines in {\mathbb C}^2  (i.e., the Szemerédi–Trotter bound). This is the longest case of refereeing that I know of, and if I recall correctly it took 13 years (!!!) for the journal to accept this paper (and it is still not published). In the meantime, a similar result was independently obtained by Solymosi and Tao, by using the polynomial method.

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