Polynomials vanishing on Cartesian products: The Elekes-Szab\'o 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 $O(n^{11/6})$ 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 Szab\'o [Combinatorica, 2012], and generalizes a result of Raz, Sharir, and Solymosi [Amer. J. Math., to appear]. 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 $O(n^{11/6})$). This result provides a unified tool for improving bounds in various Erd\H os-type problems in combinatorial geometry, and we discuss several applications of this kind.