Improved rank bounds for design matrices and a new proof of Kelly's theorem

TL;DR

This paper establishes near-optimal rank bounds for complex design matrices with small support intersections and uses these results to provide a new linear algebraic proof of Kelly's theorem, enhancing understanding of geometric configurations.

Contribution

The paper derives near-optimal rank bounds for complex design matrices and offers a new linear algebraic proof of Kelly's theorem, improving previous bounds and proofs.

Findings

Derived near-optimal rank bounds for complex design matrices

Obtained asymptotically tight bounds for geometric applications

Provided a new linear algebraic proof of Kelly's theorem

Abstract

We 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 Barak et. al. (BDWY11) 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.