Sylvester–Gallai Theorems for Complex Numbers and Quaternions |
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| Authors: | Noam Elkies Lou M. Pretorius Konrad J. Swanepoel |
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| Affiliation: | (1) Department of Mathematics, Harvard University, Cambridge, MA 02138, USA;(2) Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa;(3) Department of Mathematical Sciences, University of South Africa, PO Box 392, Pretoria 0003, South Africa |
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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. |
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