On the frequency of primes preserving dynamical irreducibility of polynomials

TL;DR

This paper investigates the frequency of primes that preserve the irreducibility of polynomial iterations in arithmetic dynamics, providing improved quantitative estimates using sieve methods.

Contribution

It introduces the use of the Selberg sieve to enhance previous bounds on primes maintaining polynomial irreducibility, improving upon earlier results.

Findings

Selberg sieve yields better bounds than Brun sieve

Quantitative estimates on prime density for irreducibility preservation

Applicable to quadratic and certain higher-degree polynomials

Abstract

Towards a well-known open question in arithmetic dynamics, L. M\'erai, A. Ostafe and I. E. Shparlinski (2023), have shown, for a class of polynomials $f \in \mathbb Z[X]$, which in particular includes all quadratic polynomials, 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, with an explicit estimate on the rate of decay. This result relies on some bounds on character sums via the Brun sieve. Here we use the Selberg sieve and in some cases obtain a substantial quantitative improvement.