License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2017.15
URN: urn:nbn:de:0030-drops-71883
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7188/
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Basit, Abdul ; Dvir, Zeev ; Saraf, Shubhangi ; Wolf, Charles

On the Number of Ordinary Lines Determined by Sets in Complex Space

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LIPIcs-SoCG-2017-15.pdf (0.5 MB)

Abstract

Kelly's theorem states that a set of n points affinely spanning C^3 must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least 3n/2 ordinary lines, unless the configuration has n-1 points in a plane and one point outside the plane (in which case there are at least n-1 ordinary lines). In addition, when at most n/2 points are contained in any plane, we prove a theorem giving stronger bounds that take advantage of the existence of lines with four and more points (in the spirit of Melchior's and Hirzebruch's inequalities). Furthermore, when the points span four or more dimensions, with at most n/2 points contained in any three dimensional affine subspace, we show that there must be a quadratic number of ordinary lines.

BibTeX - Entry

@InProceedings{basit_et_al:LIPIcs:2017:7188,
  author =	{Abdul Basit and Zeev Dvir and Shubhangi Saraf and Charles Wolf},
  title =	{{On the Number of Ordinary Lines Determined by Sets in Complex Space}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{15:1--15:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Boris Aronov and Matthew J. Katz},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7188},
  URN =		{urn:nbn:de:0030-drops-71883},
  doi =		{10.4230/LIPIcs.SoCG.2017.15},
  annote =	{Keywords: Incidences, Combinatorial Geometry, Designs, Polynomial Method}
}

Keywords: Incidences, Combinatorial Geometry, Designs, Polynomial Method
Collection: 33rd International Symposium on Computational Geometry (SoCG 2017)
Issue Date: 2017
Date of publication: 20.06.2017

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