Polynomials Vanishing on Cartesian Products: The Elekes-Szabó Theorem Revisited

Let F ∈ C[x, y, z] be a constant-degree polynomial, and let A,B,C ⊂ C with |A| = |B| = |C| = n. We show that F vanishes on at most O(n11/6) points of the Cartesian product A×B×C (where the constant of proportionality depends polynomially on the degree of F ), unless F has a special group-related form. This improves a theorem of Elekes and Szabó [2], and generalizes a result of Raz, Sharir, and Solymosi [9]. The same statement holds over R. When A,B,C have different sizes, a similar statement holds, with a more involved bound replacing O(n11/6). 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. 1998 ACM Subject Classification G.2 Discrete Mathematics