# Dynamical irreducibility of polynomials modulo primes

## TL;DR

This paper investigates the distribution of primes for which all polynomial iterations remain irreducible modulo p, showing that such primes are rare for certain polynomial classes, with explicit decay bounds and under GRH.

## Contribution

It provides explicit bounds on the density decay of primes with irreducible polynomial iterations for specific polynomial classes, refining previous results and under GRH.

## Key findings

- Primes with all iterations irreducible are of density zero for the studied classes.
- Explicit bounds on the decay rate of such primes in intervals [1, Q].
- Under GRH, stronger bounds on the prime density are obtained.

## Abstract

For a class of polynomials $f \in \mathbb{Z}[X]$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set of primes $p$ such that all iterations of $f$ are irreducible modulo $p$ is of relative density zero. Furthermore, we give an explicit bound on the rate of the decay of the density of such primes in an interval $[1, Q]$ as $Q \to \infty$. For this class of polynomials this gives a more precise version of a recent result of A. Ferraguti (2018), which applies to arbitrary polynomials but requires a certain assumption about their Galois group. Furthermore, under the Generalised Riemann Hypothesis we obtain a stronger bound on this density.