Polynomials Vanishing on Cartesian Products: The Elekes-Szabó Theorem Revisited

Orit E. Raz, Micha Sharir, Frank De Zeeuw

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 Szabó [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.

Original languageEnglish
Title of host publication31st International Symposium on Computational Geometry, SoCG 2015
EditorsJanos Pach, Janos Pach, Lars Arge
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages522-536
Number of pages15
ISBN (Electronic)9783939897835
DOIs
StatePublished - 1 Jun 2015
Event31st International Symposium on Computational Geometry, SoCG 2015 - Eindhoven, Netherlands
Duration: 22 Jun 201525 Jun 2015

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume34
ISSN (Print)1868-8969

Conference

Conference31st International Symposium on Computational Geometry, SoCG 2015
Country/TerritoryNetherlands
CityEindhoven
Period22/06/1525/06/15

Keywords

  • Combinatorial geometry
  • Incidences
  • Polynomials

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