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 $\{f_k{\}}_{k\in \mathbb{N}}\subseteq {\mathcal{O}}_K[x]$ be a sequence of monic polynomials such that for every $n\in \mathbb{N}$, the composition $f^{(n)}=f_1\circ f_2\circ \dots \circ f_n$ 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.

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References in corpus (1)

• The density of prime divisors in the arithmetic dynamics of quadratic polynomials