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

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

On the Number of Ordinary Lines Determined by Sets in Complex Space

Kelly\u27s theorem states that a set of n points affinely spanning C^3 must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least 3n/2 ordinary lines, unless the configuration has n-1 points in a plane and one point outside the plane (in which case there are at least n-1 ordinary lines). In addition, when at most n/2 points are contained in any plane, we prove a theorem giving stronger bounds that take advantage of the existence of lines with four and more points (in the spirit of Melchior\u27s and Hirzebruch\u27s inequalities). Furthermore, when the points span four or more dimensions, with at most n/2 points contained in any three dimensional affine subspace, we show that there must be a quadratic number of ordinary lines

On a Conjecture of Kelly on (1,3)(1,3)-representation of Sylvester Gallai Designs

We give an exact criterion of a conjecture of L.M.Kelly to hold true which is stated as follows. If there is a finite family $\Sigma$ of mutually skew lines in $\mathbb{R}^l,l\geq 4$ such that the three dimensional affine span (hull) of every two lines in $\Sigma$, contains at least one more line of $\Sigma$, then we have that $\Sigma$ is entirely contained in a three dimensional space if and only if the arrangement of affine hulls is central. Finally, this article leads to an analogous question for higher dimensional skew affine spaces, that is, for $(2,5)$-representations of sylvester-gallai designs in $\mathbb{R}^6$, which is answered in the last section.Comment: 13 page

On some points-and-lines problems and configurations

We apply an old method for constructing points-and-lines configurations in the plane to study some recent questions in incidence geometry.Comment: 14 pages, numerous figures of point-and-line configurations; to appear in the Bezdek-50 special issue of Periodica Mathematica Hungaric

Strong Algebras and Radical Sylvester-Gallai Configurations

In this paper, we prove the following non-linear generalization of the classical Sylvester-Gallai theorem. Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$, and $\mathcal{F}=\{F_1,\cdots,F_m\} \subset \mathbb{K}[x_1,\cdots,x_N]$ be a set of irreducible homogeneous polynomials of degree at most $d$ such that $F_i$ is not a scalar multiple of $F_j$ for $i\neq j$. Suppose that for any two distinct $F_i,F_j\in \mathcal{F}$, there is $k\neq i,j$ such that $F_k\in \mathrm{rad}(F_i,F_j)$. We prove that such radical SG configurations must be low dimensional. More precisely, we show that there exists a function $\lambda : \mathbb{N} \to \mathbb{N}$, independent of $\mathbb{K},N$ and $m$, such that any such configuration $\mathcal{F}$ must satisfy $$\dim (\mathrm{span}_{\mathbb{K}}{\mathcal{F}}) \leq \lambda(d).$$ Our result confirms a conjecture of Gupta [Gup14, Conjecture 2] and generalizes the quadratic and cubic Sylvester-Gallai theorems of [S20,OS22]. Our result takes us one step closer towards the first deterministic polynomial time algorithm for the Polynomial Identity Testing (PIT) problem for depth-4 circuits of bounded top and bottom fanins. Our result, when combined with the Stillman uniformity type results of [AH20a,DLL19,ESS21], yields uniform bounds for several algebraic invariants such as projective dimension, Betti numbers and Castelnuovo-Mumford regularity of ideals generated by radical SG configurations.Comment: 62 pages. Comments are welcome

Topological representation of matroids from diagrams of spaces

Swartz proved that any matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a sphere. This was an unexpected extension from the oriented matroid case, but unfortunately the construction is not explicit. Anderson later provided an explicit construction, but had to use cell complexes of high dimensions that are homotopy equivalent to lower dimensional spheres. Using diagrams of spaces we give an explicit construction of arrangements in the right dimensions. Swartz asked if it is possible to arrange spheres of codimension two, and we provide a construction for any codimension. We also show that all matroids, and not only tropical oriented matroids, have a pseudo-tropical representation. We determine the homotopy type of all the constructed arrangements.Comment: 18 pages, 6 figures. Some more typos fixe