Polynomials Vanishing on Cartesian Products: The Elekes-Szabó Theorem Revisited
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}).
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.
Combinatorial geometry
incidences
polynomials
522-536
Regular Paper
Orit E.
Raz
Orit E. Raz
Micha
Sharir
Micha Sharir
Frank
de Zeeuw
Frank de Zeeuw
10.4230/LIPIcs.SOCG.2015.522
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode