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DOI: 10.4230/LIPIcs.SOCG.2015.522
URN: urn:nbn:de:0030-drops-51031
URL: http://drops.dagstuhl.de/opus/volltexte/2015/5103/
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Raz, Orit E. ; Sharir, Micha ; de Zeeuw, Frank

Polynomials Vanishing on Cartesian Products: The Elekes-Szabó Theorem Revisited

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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.

BibTeX - Entry

@InProceedings{raz_et_al:LIPIcs:2015:5103,
  author =	{Orit E. Raz and Micha Sharir and Frank de Zeeuw},
  title =	{{Polynomials Vanishing on Cartesian Products: The Elekes-Szab{\'o} Theorem Revisited}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{522--536},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Lars Arge and J{\'a}nos Pach},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5103},
  URN =		{urn:nbn:de:0030-drops-51031},
  doi =		{10.4230/LIPIcs.SOCG.2015.522},
  annote =	{Keywords: Combinatorial geometry, incidences, polynomials}
}

Keywords: Combinatorial geometry, incidences, polynomials
Seminar: 31st International Symposium on Computational Geometry (SoCG 2015)
Issue Date: 2015
Date of publication: 11.06.2015


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