Polynomials vanishing on cartesian products: The elekes-szabó theorem revisited

Orit E. Raz, Micha Sharir, Frank De Zeeuw

Research output: Contribution to journalArticlepeer-review

24 Scopus citations

Abstract

Let F e C[x,y,z] be a constant-degree polynomial, and let A,B,C ⊂ C be finite sets of size n. We show that F vanishes on at most O(n11/6) points of the Cartesian product A × B × C, unless F 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 C, and a similar statement holds when A, B, C have different sizes (with a more involved bound replacing O.(n11/6)). This result provides a unified tool for improving bounds in various Erdös-type problems in combinatorial geometry, and we discuss several applications of this kind.

Original languageAmerican English
Pages (from-to)3517-3566
Number of pages50
JournalDuke Mathematical Journal
Volume165
Issue number18
DOIs
StatePublished - 2016
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2016.

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