Duke Mathematical Journal

Polynomials vanishing on Cartesian products: The Elekes–Szabó theorem revisited

Orit E. Raz, Micha Sharir, and Frank De Zeeuw

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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.

Article information

Duke Math. J., Volume 165, Number 18 (2016), 3517-3566.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52C10: Erdős problems and related topics of discrete geometry [See also 11Hxx]
Secondary: 05D99: None of the above, but in this section

combinatorial geometry incidences polynomials


Raz, Orit E.; Sharir, Micha; De Zeeuw, Frank. Polynomials vanishing on Cartesian products: The Elekes–Szabó theorem revisited. Duke Math. J. 165 (2016), no. 18, 3517--3566. doi:10.1215/00127094-3674103. https://projecteuclid.org/euclid.dmj/1473275860

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