On the frequency of primes preserving dynamical irreducibility of 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.