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Sylvester–Gallai Theorems for Complex Numbers and Quaternions

Abstract

A Sylvester-Gallai (SG) configuration is a finite set S of points such that the line through any two points in S contains a third point of S. According to the Sylvester-Gallai theorem, an SG configuration in real projective space must be collinear. A problem of Serre (1966) asks whether an SG configuration in a complex projective space must be coplanar. This was proved by Kelly (1986) using a deep inequality of Hirzebruch. We give an elementary proof of this result, and then extend it to show that an SG configuration in projective space over the quaternions must be contained in a three-dimensional flat.

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Correspondence to Noam Elkies or Lou M. Pretorius or Konrad J. Swanepoel.

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Elkies, N., Pretorius, L. & Swanepoel, K. Sylvester–Gallai Theorems for Complex Numbers and Quaternions. Discrete Comput Geom 35, 361–373 (2006). https://doi.org/10.1007/s00454-005-1226-7

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Keywords

  • Computational Mathematic
  • Projective Space
  • Elementary Proof
  • Complex Projective Space
  • Real Projective Space