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Sylvester-Gallai for Arrangements of Subspaces

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Zeev Dvir and Guangda Hu. Sylvester-Gallai for Arrangements of Subspaces. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 29-43, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.SOCG.2015.29

Abstract

In this work we study arrangements of k-dimensional subspaces V_1,...,V_n over the complex numbers. Our main result shows that, if every pair V_a, V_b of subspaces is contained in a dependent triple (a triple V_a, V_b, V_c contained in a 2k-dimensional space), then the entire arrangement must be contained in a subspace whose dimension depends only on k (and not on n). The theorem holds under the assumption that the subspaces are pairwise non-intersecting (otherwise it is false). This generalizes the Sylvester-Gallai theorem (or Kelly's theorem for complex numbers), which proves the k=1 case. Our proof also handles arrangements in which we have many pairs (instead of all) appearing in dependent triples, generalizing the quantitative results of Barak et. al. One of the main ingredients in the proof is a strengthening of a theorem of Barthe (from the k=1 to k>1 case) proving the existence of a linear map that makes the angles between pairs of subspaces large on average. Such a mapping can be found, unless there is an obstruction in the form of a low dimensional subspace intersecting many of the spaces in the arrangement (in which case one can use a different argument to prove the main theorem).
Keywords
• Sylvester-Gallai
• Locally Correctable Codes

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References

1. Boaz Barak, Zeev Dvir, Avi Wigderson, and Amir Yehudayoff. Fractional Sylvester-Gallai theorems. Proceedings of the National Academy of Sciences, 110(48):19213-19219, 2013.
2. Boaz Barak, Zeev Dvir, Amir Yehudayoff, and Avi Wigderson. Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes. In Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing, STOC'11, pages 519-528, 2011.
3. Franck Barthe. On a reverse form of the Brascamp-Lieb inequality. Inventiones mathematicae, 134(2):335-361, 1998.
4. P. Borwein and W. O. J. Moser. A survey of Sylvester’s problem and its generalizations. Aequationes Mathematicae, 40(1):111-135, 1990.
5. Zeev Dvir. On matrix rigidity and locally self-correctable codes. computational complexity, 20(2):367-388, 2011.
6. Zeev Dvir, Shubhangi Saraf, and Avi Wigderson. Breaking the quadratic barrier for 3-LCC’s over the reals. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC'14, pages 784-793, 2014.
7. Zeev Dvir, Shubhangi Saraf, and Avi Wigderson. Improved rank bounds for design matrices and a new proof of Kelly’s theorem. Forum of Mathematics, Sigma, 2, 10 2014.
8. Noam Elkies, Lou M Pretorius, and Konrad J Swanepoel. Sylvester-Gallai theorems for complex numbers and quaternions. Discrete & Computational Geometry, 35(3):361-373, 2006.
9. P. Erdös, Richard Bellman, H. S. Wall, James Singer, and V. Thébault. Problems for solution: 4065-4069. The American Mathematical Monthly, 50(1):65-66, 1943.
10. Sten Hansen. A generalization of a theorem of Sylvester on the lines determined by a finite point set. Mathematica Scandinavica, 16:175-180, 1965.
11. L. M. Kelly. A resolution of the Sylvester-Gallai problem of J.-P. Serre. Discrete & Computational Geometry, 1(1):101-104, 1986.
12. Peter D. Lax. Linear algebra and its applications. Pure and Applied Mathematics. Wiley-Interscience, 2007.
13. E. Melchior. Uber vielseite der projektive ebene. Deutsche Math., 5:461-475, 1940.
14. J. J. Sylvester. Mathematical question 11851. Educational Times, 59:98, 1893.
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