On Defining Integers 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 of size polynomial in n for computing the permanent of n by n matrices, then τ(n!) is polynomially bounded in log n. Under the same assumption on the permanent, we conclude that the Pochhammer–Wilkinson polynomials $\Pi^{n}_{k=1}(X - k)$ and the Taylor approximations $\Sigma^{n}_{k=0}\frac{1}{k!}X^{k}$ and $\Sigma^{n}_{k=1}\frac{1}{k}X^{k}$ 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.