We present an algebraic-geometric approach for devising a deterministic polynomial time blackbox identity testing (PIT) algorithm for depth-4 circuits with bounded top fanin. Using our approach, we devise such an algorithm for the case when such circuits have bounded bottom fanin and satisfy a certain non-degeneracy condition. In particular, we present an algorithm that, given blackboxes to P 1 P d , Q 11 Q 1 d 1 , , Q k 1 Q k d k where k and the degrees of P i 's and Q i j 's are bounded, determines the membership of P 1 P d in the radical of the ideal generated by Q 11 Q 1 d 1 , , Q k 1 Q k d k in deterministic poly( n d max i ( d i ) )-time.

We also give a Dvir-Shpilka (STOC 2005)-like approach to resolve the degenerate case and, in the process, initiate a new direction in incidence geometry for non-linear varieties . This approach consists of a series of Sylvester-Gallai type conjectures for bounded-degree varieties and, if true, imply a complete derandomization in the bounded bottom fanin case. To the best of our knowledge, these problems have not been posed before.