On the set of stable primes for postcritically infinite maps over number fields

Many interesting questions in arithmetic dynamics revolve, in one way or another, around the (local and/or global) reducibility behavior of iterates of a polynomial. We show that for very general families of integer polynomials $f$ (and, more generally, rational functions over number fields), the set of stable primes, i.e., primes modulo which all iterates of $f$ are irreducible, is a density zero set. Compared to previous results, our families cover a much wider ground, and in particular apply to $100\mathrm{\%}$ of polynomials of any given odd degree, thus adding evidence to the conjecture that polynomials with a “large” set of stable primes are necessarily of a very specific shape, and in particular are necessarily postcritically finite.