Interpolation of depth-3 arithmetic circuits with two multiplication gates

In this paper we consider the problem of constructing a small arithmetic circuit for a polynomial for which we have oracle access. Our focus is on n-variate polynomials, over a finite field F, that have depth-3 arithmetic circuits with two multiplication gates of degree d. We obtain the following results: 1. Multilinear case: When the circuit is multilinear (multiplication gates compute multilinear polynomials) we give an algorithm that outputs, with probability 1 - o(1), all the depth-3 circuits two multiplication gates computing the same polynomial. The running time of the algorithm is poly(n, |F|). 2. General case: When the circuit is not multilinear we give a quasi-polynomial (in n, |F|) time algorithm that outputs, with probability 1-o(1), a succinct representation of the polynomial. In particular, if the depth-3 circuit for the polynomial is not of small depth- 3 rank (namely, after removing the g.c.d. of the two multiplication gates, the remaining linear functions span a not small linear space) then we output the depth-3 circuit itself. In case that the rank is small we output a depth-3 circuit with a quasi-polynomial number of multiplication gates.