From Sylvester-Gallai configurations to rank bounds: Improved black-box identity test for depth-3 circuits

Abstract

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 kO(k)-time black-box identity test over rationals (Kayal & Saraf, FOCS 2009) to one that takes dO(k2)-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. © 2010 IEEE.