Arithmetic Circuits: A Chasm at Depth 3.

We show that, over Q, if an n-variate polynomial of degree d = n(O(1)) is computable by an arithmetic circuit of size s (respectively, by an arithmetic branching program of size s), then it can also be computed by a depth-3 circuit (i.e., a Sigma Pi Sigma circuit) of size exp(O(root d log n log d log s)) (respectively, of size exp(O(root d log n log s)). In particular this yields a S.S circuit of size exp(O(root d. log d)) computing the d x d determinant Detd. It also means that if we can prove a lower bound of exp(omega(root d . log d)) on the size of any Sigma Pi Sigma circuit computing the d x d permanent Perm(d), then we get superpolynomial lower bounds for the size of any arithmetic branching program computing Perm(d). We then give some further results pertaining to derandomizing polynomial identity testing and circuit lower bounds. The Sigma Pi Sigma circuits that we construct have the property that (some of) the intermediate polynomials have degree much higher than d. Indeed such a counterintuitive construction is unavoidable-it is known that in any Sigma Pi Sigma circuit C computing either Det(d) or Perm(d), if every multiplication gate has fanin at most d (or any constant multiple thereof), then C must have size at least exp(Omega(d)).