Arithmetic circuits: A chasm at depth

We show that, over ℚ, if an n-variate polynomial of degree d=^nO(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 σπσ circuit) of size exp(O(√ d log n log d log s)) (respectively, of size exp(O(√ d log n log s)). In particular this yields a σπσ circuit of size exp(ω(√ d · log d)) computing the d × d determinant Det_d. It also means that if we can prove a lower bound of exp(ω(√ d · log d)) on the size of any σπσ circuit computing the d × 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 σπσ 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 σπσ 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(Ω(d)).