On Defining Integers in the Counting Hierarchy and Proving Arithmetic Circuit Lower Bounds

Let τ(n) denote the minimum number of arithmetic operations sufficient to build the integer n from the constant 1. We prove that if there are arithmetic circuits for computing the permanent of n by n matrices having size polynomial in n, then τ(n!) is polynomially bounded in log n. Under the same assumption on the permanent, we conclude that the Pochhammer-Wilkinson polynomials ∏n k=1(X − k) and the Taylor approximations ∑n k=0 1 k!X k and ∑n k=1 1 kX k of exp and log, respectively, can be computed by arithmetic circuits of size polynomial in log n (allowing divisions). This connects several so far unrelated conjectures in algebraic complexity.