Combinatorics Seminar

We present a polynomial time randomized algorithm for reconstructing ΣΠΣ(2) circuits over characteristic zero fields, i.e. depth 3 circuits with fan in 2 at the top addition gate and having coefficients from a char zero field.

The algorithm needs only a black-box query access to the polynomial f(x1,…,xn) of degree d in n variables, computable by a ΣΠΣ(2) circuit C. In addition, we assume that the simple rank of this polynomial (essential number of variables after removing the gcd of the two multiplication gates) is bigger than a fixed constant. Our algorithm runs in time poly(n,d) and returns an equivalent ΣΠΣ(2) circuit(with high probability).

Our main techniques are based on the use of Quantitative Sylvester Gallai Theorems from the work of Barak et.al. to find a small collection of nice sub-spaces to project onto. The heart of our paper lies in subtle applications of the Quantitative Sylvester Gallai theorems to prove why projections w.r.t. the nice sub-spaces can be glued. We also use Brill's Equations to construct a small set of candidate linear forms (containing linear forms from both gates). Another important technique which comes very handy is the polynomial time randomized algorithm for factoring multivariate polynomials given by Kaltofen.