Diagonal Circuit Identity Testing and Lower Bounds

In this paper we give the first deterministic polynomial time algorithm for testing whether a diagonal depth-3 circuit C(x1, . . . , xn) (i.e. C is a sum of powers of linear functions) is zero. We also prove an exponential lower bound showing that such a circuit will compute determinant or permanent only if there are exponentially many linear functions. Our techniques generalize to the following new results: 1. Suppose we are given a depth-3 circuit (over any field F) of the form: C(x1, . . . , xn) := k ∑ i=1 ` ei,1 i,1 · · · ` ei,s i,s where, the `i,j ’s are linear functions living in F[x1, . . . , xn]. We can test whether C is zero deterministically in poly (nk,max{(1 + ei,1) · · · (1 + ei,s) | 1 6 i 6 k}) field operations. This immediately gives a deterministic poly(nk2) time identity test for general depth-3 circuits of degree d. 2. We prove that if the above circuit C(x1, . . . , xn) computes the determinant (or permanent) of an m × m formal matrix with a “small” s = o ( m logm ) then k = 2. Our lower bounds work for all fields F. (Previous exponential lower bounds for depth-3 only work for nonzero characteristic.) 3. We present applications of our ideas to depth-4 circuits and an exponentially faster identity test for homogeneous diagonal circuits (deterministically in poly(n k log(d)) field operations over finite fields).