Sylvester-Gallai Theorems for Complex Numbers and Quaternions

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Sylvester-Gallai Theorems for Complex Numbers and Quaternions

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Title: Sylvester-Gallai Theorems for Complex Numbers and Quaternions
Author: Elkies, Noam; Swanepoel, Konrad J.; Pretorius, Lou M.

Note: Order does not necessarily reflect citation order of authors.

Citation: Elkies, Noam D., Lou M. Pretorius, and Konrad J. Swanepoel. 2006. Sylvester-gallai theorems for complex numbers and quaternions. Discrete and Computational Geometry 35, (3): 361-373.
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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.
Published Version: http://dx.doi.org/10.1007/s00454-005-1226-7
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:2794828

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  • FAS Scholarly Articles [7362]
    Peer reviewed scholarly articles from the Faculty of Arts and Sciences of Harvard University
 
 

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