Sparsity of stable primes for dynamical sequences

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-11-20 DOI:10.1112/blms.13191

Joachim König

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引用次数: 0

Abstract

We show that a dynamical sequence ${(f_{n})}_{n \in N}$ of polynomials over a number field whose set of stable primes is of positive density must necessarily have a very restricted, and, in particular, virtually prosolvable dynamical Galois group. Together with existing heuristics, our results suggest, moreover, that a polynomial $f$ all of whose iterates are irreducible modulo a positive density subset of the primes must necessarily be a composition of linear functions, monomials, and Dickson polynomials.