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

Orit E. Raz; Micha Sharir; Frank De Zeeuw

doi:10.1215/00127094-3674103

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

Orit E. Raz, Micha Sharir, Frank De Zeeuw

Research output: Contribution to journal › Article › peer-review

28 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(n^11/6) 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.(n^11/6)). 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 language	English
Pages (from-to)	3517-3566
Number of pages	50
Journal	Duke Mathematical Journal
Volume	165
Issue number	18
DOIs	https://doi.org/10.1215/00127094-3674103
State	Published - 1 Jan 2016
Externally published	Yes

ASJC Scopus subject areas

General Mathematics

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title = "Polynomials vanishing on cartesian products: The elekes-szab{\'o} theorem revisited",

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 Szab{\'o} 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 Erd{\"o}s-type problems in combinatorial geometry, and we discuss several applications of this kind.",

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AU - Zeeuw, Frank De

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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.

AB - 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.

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ER -

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

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