Let F in Complex[x,y,z] be a constant-degree polynomial, and let A,B,C be sets of complex numbers with |A|=|B|=|C|=n. We show that F vanishes on at most O(n^{11/6}) points of the Cartesian product A x B x 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 Szabo [ES12], and generalizes a result of Raz, Sharir, and Solymosi [RSS14a]. The same statement holds over R. When A, B, C have different sizes, a similar statement holds, with a more involved bound replacing O(n^{11/6}).#R##N##R##N#This result provides a unified tool for improving bounds in various Erdos-type problems in combinatorial geometry, and we discuss several applications of this kind.
['Orit Raz', 'Micha Sharir', 'Frank de Zeeuw']
Polynomials Vanishing on Cartesian Products: The Elekes-Szabó Theorem Revisited