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

TL;DR

This paper demonstrates that for broad families of integer polynomials over number fields, the set of primes where all iterates remain irreducible is very sparse, supporting the idea that such polynomials are often postcritically finite.

Contribution

It extends previous results by showing that the set of stable primes has density zero for very general families of polynomials, including 100% of polynomials of any odd degree.

Findings

Stable primes form a density zero set for broad polynomial families.

Results apply to 100% of polynomials of any given odd degree.

Supports conjecture linking stable primes to postcritically finite polynomials.

Abstract

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 $100\%$ 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.