We show that the rank of a depth-3 circuit (over any field) that is simple, minimal and zero is at most k^3log d. The previous best rank bound known was 2^{O(k^2)}(log d)^{k-2} by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we also provide a simple and minimal identity of rank Omega(klog d)). Our rank bound significantly improves (dependence on k exponentially reduced) the best known deterministic black-box identity tests for depth-3 circuits by Karnin and Shpilka (CCC 2008). Our techniques also shed light on the factorization pattern of nonzero depth-3 circuits, most strikingly: the rank of linear factors of a simple, minimal and nonzero depth-3 circuit (over any field) is at most k^3log d. The novel feature of this work is a new notion of maps between sets of linear forms, called ideal matchings, used to study depth-3 circuits. We prove interesting structural results about depth-3 identities using these techniques. We believe that these can lead to the goal of a deterministic polynomial time identity test for these circuits.
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