Recently Joshua Zahl and I have uploaded to arXiv a paper about incidence in . In this post I briefly survey what was previously known concerning such incidences, and then describe what the new paper is about.
Let be a set of curves and let be a set of points, both in for some field . We say that the arrangement has degrees of freedom and multiplicity-type if
- For any points from , there are at most curves from that contain all of them.
- Any pair of curves from intersect in at most points.
The following theorem is currently the best known general incidence bound in (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 be a set of points and let be a set of algebraic curves of degree at most , both in . Suppose that has degrees of freedom and multiplicity type . Then
In the complex plane, things progressed rather slowly. Csaba Tóth derived a matching incidence bound for the special case of complex lines in (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.