Abstract
AbstractWe study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Baraket al. [Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes.Proceedings of the 43rd annual ACM symposium on Theory of computing, STOC 11, (ACM, NY 2011), 519–528] in which they were used to answer questions regarding point configurations. In this work, we derive near-optimal rank bounds for these matrices and use them to obtain asymptotically tight bounds in many of the geometric applications. As a consequence of our improved analysis, we also obtain a new, linear algebraic, proof of Kelly’s theorem, which is the complex analog of the Sylvester–Gallai theorem.
Highlights
The classical Sylvester–Gallai (SG) theorem states the following: Given any finite set of points in the Euclidean plane, not all on the same line, there exists a line passing through exactly two of the points
The following complex variant of the Sylvester–Gallai theorem was proved by Kelly [18] in response to a question of Serre: Given any finite set of points in Cd, not all on the same complex two-dimensional plane, there exists a line passing through exactly two of the points
We say that A is a (q, k, t)-design matrix if the following hold
Summary
The classical Sylvester–Gallai (SG) theorem states the following: Given any finite set of points in the Euclidean plane, not all on the same line, there exists a line passing through exactly two of the points. Theorem 1.5, which removes the dependence on q, allows us to get meaningful lower bounds on the rank of square design matrices. Theorem 5.1 is proved, as in [2], by reduction to the rank bound for design matrices.
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