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1 December 2016 Polynomials vanishing on Cartesian products: The Elekes–Szabó theorem revisited
Orit E. Raz, Micha Sharir, Frank De Zeeuw
Duke Math. J. 165(18): 3517-3566 (1 December 2016). DOI: 10.1215/00127094-3674103

Abstract

Let FC[x,y,z] be a constant-degree polynomial, and let A,B,CC 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 R, 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 Erdős-type problems in combinatorial geometry, and we discuss several applications of this kind.

Citation

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Orit E. Raz. Micha Sharir. Frank De Zeeuw. "Polynomials vanishing on Cartesian products: The Elekes–Szabó theorem revisited." Duke Math. J. 165 (18) 3517 - 3566, 1 December 2016. https://doi.org/10.1215/00127094-3674103

Information

Received: 28 April 2015; Revised: 18 February 2016; Published: 1 December 2016
First available in Project Euclid: 7 September 2016

zbMATH: 1365.52023
MathSciNet: MR3577370
Digital Object Identifier: 10.1215/00127094-3674103

Subjects:
Primary: 52C10
Secondary: 05D99

Rights: Copyright © 2016 Duke University Press

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Vol.165 • No. 18 • 1 December 2016
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