The set of stable primes for polynomial sequences with large Galois group

Abstract:

Let $K$ be a number field with ring of integers $\mathcal {O}_K$, and let $\{f_k\}_{k\in \mathbb {N}}$ be a sequence of monic polynomials in $\mathcal {O}_K[x]$ such that for every $n\in \mathbb {N}$, the composition $f^{(n)}=f_1\circ f_2\circ \ldots \circ f_n$ is irreducible. In this paper we show that if the size of the Galois group of $f^{(n)}$ is large enough (in a precise sense) as a function of $n$, then the set of primes $\mathfrak {p}\subseteq \mathcal {O}_K$ such that every $f^{(n)}$ is irreducible modulo $\mathfrak {p}$ has density zero. Moreover, we prove that the subset of polynomial sequences such that the Galois group of $f^{(n)}$ is large enough has density 1, in an appropriate sense, within the set of all polynomial sequences.