# 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.

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.