Polynomials vanishing on Cartesian products: The Elekes–Szabó theorem revisited

Abstract

Let $F\in \mathbb{C}[x,y,z]$ be a constant-degree polynomial, and let $A,B,C\subset \mathbb{C}$ be finite sets of size $n$. We show that $F$ vanishes on at most $O\left(n^{11/6}\right)$ points of the Cartesian product $A\times B\times C$, unless $F$ has a special group-related form. This improves a theorem of Elekes and Szabó and generalizes a result of Raz, Sharir, and Solymosi. The same statement holds over $\mathbb{R}$, and a similar statement holds when $A,B,C$ have different sizes (with a more involved bound replacing $O\left(n^{11/6}\right)$). This result provides a unified tool for improving bounds in various Erdős-type problems in combinatorial geometry, and we discuss several applications of this kind.