Incidence bounds for complex algebraic curves on Cartesian products

TL;DR

This paper establishes bounds on incidences between algebraic curves in complex plane and Cartesian products, simplifying proofs and extending applications from real to complex numbers using polynomial partitioning.

Contribution

It introduces a new, simpler proof for incidence bounds in complex algebraic geometry leveraging polynomial partitioning, expanding applicability to complex sets.

Findings

Bounds on incidences between algebraic curves and Cartesian products in $\\mathbb{C}^2$

Simplified proof technique using polynomial partitioning

Extension of applications from real to complex numbers

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 conditions, but we show that the Cartesian product assumption leads to a simpler proof. This assumption holds in a number of interesting applications, and with our bound these applications can be extended from $\mathbb{R}$ to $\mathbb{C}$. The proof is a new application of the polynomial partitioning technique introduced by Guth and Katz.