Incidence bounds for complex algebraic curves on Cartesian products

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]).