Incidence Bounds for Complex Algebraic Curves on Cartesian Products

Abstract

We prove bounds on the number of incidences between a set of algebraic curves in ${\mathbb C}^2$ and a Cartesian product $A\times B$ with finite sets $A,B\subset {\mathbb C}$. Similar bounds are known under various restrictive conditions, but we show that the Cartesian product assumption leads to a simpler proof and lets us remove these conditions. This assumption holds in a number of interesting applications, and with our bound these applications can be extended from ${\mathbb R}$ to ${\mathbb C}$. We also obtain more precise information in the bound, which is used in several recent papers (Raz et al., 31st international symposium on computational geometry (SoCG 2015), pp 522–536, 2015, [17], Valculescu, de Zeeuw, Distinct values of bilinear forms on algebraic curves, 2014, [25]). Our proof works via an incidence bound for surfaces in ${\mathbb R}^4$, which has its own applications (Raz, Sharir, 31st international symposium on computational geometry (SoCG 2015), pp 569–583, 2015, [15]). The proof is a new application of the polynomial partitioning technique introduced by Guth and Katz (Ann Math 181: 155–190, 2015, [11]).

The first author was supported by ERC Advanced Research Grant no. 321104, by Hungarian National Research Grant NK 104183, and by NSERC. The second author was partially supported by Swiss National Science Foundation Grants 200020-144531 and 200021-137574. Part of this research was performed while the authors visited the Institute for Pure and Applied Mathematics (IPAM) in Los Angeles, which is supported by the National Science Foundation.