Polynomials vanishing on Cartesian products: The Elekes–Szabó theorem revisited
Orit E. Raz, Micha Sharir, Frank de Zeeuw · Duke Mathematical Journal · 2016
Let F∈C[x,y,z] be a constant-degree polynomial, and let A,B,C⊂C be finite sets of size n. We show that F vanishes on at most O(n11/6) points of the Cartesian product A×B×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 R, and a similar statement holds when A,B,C have different sizes (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.