Algebraic Geometric Techniques for Depth-4 PIT & Sylvester-Gallai Conjectures for Varieties

A From Sylvester-Gallai Configurations to Rank Bounds: Improved Blackbox Identity Test for Depth-3 Circuits

We study the problem of identity testing for depth-3 circuits of top fanin k and degree d. We give a new structure theorem for such identities that improves the known deterministic d O(k) -time blackbox identity test over rationals (Kayal & Saraf, FOCS 2009) to one that takes dO(k 2)-time. Our structure theorem essentially says that the number of independent variables in a real depth-3 identity...

Identity Testing for Depth-3 Circuits with Small Fan-in

4 History of the Sylvester-Gallai Problem 6 4.1 Melchior’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.2 Motzkin’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.3 Kelly and Moser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.4 Green and Tao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.5 How to describe 3-color ...

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 asymptot...

Moduli Theory and Arithmetic of Algebraic Varieties

This paper surveys a few applications of algebro-geometric moduli theory to issues concerning the distribution of rational points in algebraic varieties. A few well known arithmetic problems with their expected answers (the socalled “diophantine conjectures”) are introduced in section 2, explaining their connection with a circle of ideas, whose goal is to find a unifying theme in analytic, diff...

The Sylvester-Gallai Theorem, the Monochrome Line Theorem and Generalizations Report for a Seminar on the Sylvester-Gallai Theorem