Let be a constant-degree polynomial, and let be finite sets of size . We show that vanishes on at most points of the Cartesian product , unless 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 , and a similar statement holds when have different sizes (with a more involved bound replacing ). 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.
"Polynomials vanishing on Cartesian products: The Elekes–Szabó theorem revisited." Duke Math. J. 165 (18) 3517 - 3566, 1 December 2016. https://doi.org/10.1215/00127094-3674103