The size of arboreal images, I: exponential lower bounds for PCF and unicritical polynomials

Abstract

Let $f$ be a polynomial over a global field $K$. For each $α$ in $K$ and $N$ in $\mathbb{Z}_{\geq 0}$ denote by $K_N(f,α)$ the arboreal field $K(f^{-N}(α))$ and by $D_N(f,α)$ its degree over $K$. It is conjectured that $D_N(f,α)$ should grow as a double exponential function of $N$, unless $f$ is post-critically finite (PCF), in which case there are examples like $D_N(x^2,α) \leq 4^{N}$. There is evidence conditionally on Vojta's conjecture. However, before the present work, no unconditional non-trivial lower bound was known for post-critically infinite $f$. In the case $f$ is PCF, no non-trivial lower bound was known, not even under Vojta's conjecture. In this paper we give two simple methods that turn the finiteness of the critical orbit into an exploitable feature, also in the post-critically infinite case. First, assuming GRH for number fields, we establish for all PCF polynomials $f$ of degree at least $2$ and all $α$ outside of the critical orbits of $f$, the existence of a positive constant $c(f,α)$ such that $$D_N(f,α) \geq \text{exp}(c(f,α)N), $$ for all $N$, which is sharp up to, possibly, improve the constant $c(f,α)$. Second, we show unconditionally that if $f$ is post-critically infinite over any number field $K$ and unicritical, then, for each $α$ in $K$, there exists a positive constant $c(f,α)$ such that $$D_N(f,α) \geq \text{exp}(c(f,α)N), $$ for all $N$. The main input here is to work modulo a suitably chosen prime and use a construction available for PCF unicritical polynomials with periodic critical orbit.