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

## TL;DR

This paper investigates the distribution of primes for polynomial sequences with large Galois groups, showing that primes where all compositions remain irreducible are rare, and that sequences with large Galois groups are typical.

## Contribution

It establishes conditions under which the set of primes preserving irreducibility has density zero and proves that sequences with large Galois groups are generic among all polynomial sequences.

## Key findings

- Primes where all iterates are irreducible have density zero.
- Sequences with large Galois groups are prevalent (density 1).
- Large Galois groups imply scarcity of primes preserving irreducibility.

## 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 $f^{(n)}=f_1\circ f_2\circ\ldots\circ f_n$ 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.