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Iterates of polynomials over $\F_q(t)$ and their Galois groups. (arXiv:2305.05380v2 [math.NT] UPDATED)

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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.

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