LIPIcs.SOCG.2015.522.pdf
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Polynomials Vanishing on Cartesian Products: The Elekes-Szabó Theorem Revisited
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31st International Symposium on Computational Geometry (SoCG 2015)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Computational Geometry (SoCG) - License:
Creative Commons Attribution 3.0 Unported license
- Publication Date: 2015-06-12
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Orit E. Raz, Micha Sharir, and Frank de Zeeuw. Polynomials Vanishing on Cartesian Products: The Elekes-Szabó Theorem Revisited. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 522-536, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.SOCG.2015.522
Abstract
Let F in Complex[x,y,z] be a constant-degree polynomial, and let A,B,C be sets of complex numbers with |A|=|B|=|C|=n. We show that F vanishes on at most O(n^{11/6}) points of the Cartesian product A x B x 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 Szabo [ES12], and generalizes a result of Raz, Sharir, and Solymosi [RSS14a]. The same statement holds over R. When A, B, C have different sizes, a similar statement holds, with a more involved bound replacing O(n^{11/6}).
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.
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Keywords
- Combinatorial geometry
- incidences
- polynomials
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