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

Nitin Saxena, C. Seshadhri
2010 2010 IEEE 51st Annual Symposium on Foundations of Computer Science  
We study the problem of identity testing for depth-3 circuits of top fanin k and degree d (called ΣΠΣ(k, d) identities). We give a new structure theorem for such identities. A direct application of our theorem improves the known deterministic d k O(k) time black-box identity test over rationals (Kayal & Saraf, FOCS 2009) to one that takes d O(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
more » ... affirmatively the stronger rank conjectures posed by Dvir & Shpilka (STOC 2005) and Kayal & Saraf (FOCS 2009). Our techniques provide a unified framework that actually beats all known rank bounds and hence gives the best running time (for every field ) for black-box identity tests. Our main theorem (almost optimally) pins down the relation between higher dimensional Sylvester-Gallai theorems and the rank of depth-3 identities in a very transparent manner. The existence of this was hinted at by Dvir & Shpilka (STOC 2005), but first proven, for reals, by Kayal & Saraf (FOCS 2009). We introduce the concept of Sylvester-Gallai rank bounds for any field, and show the intimate connection between this and depth-3 identity rank bounds. We also prove the first ever theorem about high dimensional Sylvester-Gallai configurations over any field. Our proofs and techniques are very different from previous results and devise a very interesting ensemble of combinatorics and algebra. The latter concepts are ideal theoretic and involve a new Chinese remainder theorem. Our proof methods explain the structure of any depth-3 identity C: there is a nucleus of C that forms a low rank identity, while the remainder is a high dimensional Sylvester-Gallai configuration.
doi:10.1109/focs.2010.9 dblp:conf/focs/SaxenaS10 fatcat:hyibfk6frbftfikh7zzpkmiasq