On the frequency of primes preserving dynamical irreducibility of polynomials

In this talk we address an open question in arithmetic dynamics regarding the frequency of primes modulo which all the iterates of a polynomial remain irreducible. More precisely, for a class of integer polynomials $f$, which in particular includes all quadratic polynomials, we show that, under some natural conditions, the set of primes $p$ such that all iterates of $f$ are irreducible modulo $p$ is of relative density zero. Our results rely on a combination of analytic (Selberg's sieve) and Diophantine (finiteness of solutions to certain hyperelliptic equations) tools, which we will briefly describe. Joint work with László Mérai and Igor Shparlinski (2021, 2024).