From Sylvester-Gallai Configurations to Rank Bounds: Improved Black-Box 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. A direct application of our theorem improves the known deterministic d^{k^k}-time black-box identity test over rationals (Kayal & Saraf, FOCS 2009) to one that takes d^{k^2}-time. Our structure theorem essentially says that the number of independent variables in a real depth-3 identity is very small. This theorem affirmatively settles the strong rank conjecture posed by Dvir & Shpilka (STOC 2005). We devise a powerful algebraic framework and develop tools to study depth-3 identities. We use these tools to show that any depth-3 identity contains a much smaller nucleus identity that contains most of the "complexity" of the main identity. The special properties of this nucleus allow us to get almost optimal rank bounds for depth-3 identities.