Permanent does not have succinct polynomial size arithmetic circuits of constant depth

We show that over fields of characteristic zero there does not exist a polynomial p(n)p(n) and a constant-free succinct   arithmetic circuit family {Φn}{Φn} using division by constants,3 where ΦnΦn has size at most p(n)p(n) and depth O(1)O(1), such that ΦnΦn computes the n×nn×n permanent. A circuit family {Φn}{Φn} is succinct if there exists a nonuniform   Boolean circuit family {Cn}{Cn} with O(logn) many inputs and size no(1)no(1) such that CnCn can correctly answer direct connection language queries about ΦnΦn – succinctness is a relaxation of uniformity.To obtain this result we develop a novel technique that further strengthens the connection between black-box derandomization of polynomial identity testing and lower bounds for arithmetic circuits. From this, we obtain the lower bound by giving an explicit construction, computable in the polynomial hierarchy, of a hitting set for arithmetic circuits.