Iterates of polynomials over $\F_q(t)$ and their Galois groups

Abstract

A conjecture of Odoni stated over Hilbertian fields $K$ of characteristic zero asserts that for every positive integer $d$, there exists a polynomial $f\in K[x]$ of degree $d$ such that for every positive integer $n$, each iterate $f^{\circ n}$ of $f$ is irreducible and the Galois group of the splitting field of $f^{\circ n}$ is isomorphic to $[S_d]^{n}$, the $n$ folded iterated wreath product of the symmetric group $S_{d}$. We prove an analogue this conjecture over $\F_q(t)$, the field of rational functions in $t$ over a finite field $\F_q$ of characteristic $p>0$. We present some examples and see that most polynomials in $\F_q[t][x]$ satisfy these conditions.