On defining integers in the counting hierarchy and proving arithmetic circuit lower bounds

Abstract

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 ∏k=1n(X - k) and the Taylor approximations ∑k=0n and ∑k=1n 1/k Xk 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. © Springer-Verlag Berlin Heidelberg 2007.