@inproceedings{d6326c96b40a40cea776639ecc51912d,
title = "Polynomials Vanishing on Cartesian Products: The Elekes-Szab{\'o} Theorem Revisited",
abstract = "Let F ε 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 Szab{\'o} [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 Erdos-type problems in combinatorial geometry, and we discuss several applications of this kind.",
keywords = "Combinatorial geometry, Incidences, Polynomials",
author = "Raz, {Orit E.} and Micha Sharir and {De Zeeuw}, Frank",
year = "2015",
month = jun,
day = "1",
doi = "https://doi.org/10.4230/LIPIcs.SOCG.2015.522",
language = "الإنجليزيّة",
series = "Leibniz International Proceedings in Informatics, LIPIcs",
publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",
pages = "522--536",
editor = "Janos Pach and Lars Arge",
booktitle = "31st International Symposium on Computational Geometry, SoCG 2015",
address = "ألمانيا",
note = "31st International Symposium on Computational Geometry, SoCG 2015 ; Conference date: 22-06-2015 Through 25-06-2015",
}