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DOI: 10.4230/LIPIcs.SoCG.2017.15

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

Authors: Abdul Basit, Zeev Dvir, Shubhangi Saraf, and Charles Wolf

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

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.

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Abdul Basit, Zeev Dvir, Shubhangi Saraf, and Charles Wolf. On the Number of Ordinary Lines Determined by Sets in Complex Space. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{basit_et_al:LIPIcs.SoCG.2017.15,
  author =	{Basit, Abdul and Dvir, Zeev and Saraf, Shubhangi and Wolf, Charles},
  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 =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.15},
  URN =		{urn:nbn:de:0030-drops-71883},
  doi =		{10.4230/LIPIcs.SoCG.2017.15},
  annote =	{Keywords: Incidences, Combinatorial Geometry, Designs, Polynomial Method}
}

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

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