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Sparsity of stable primes for dynamical sequences (2402.18772v2)

Published 29 Feb 2024 in math.NT, math.DS, and math.GR

Abstract: We show that a dynamical sequence $(f_n)_{n\in \mathbb{N}}$ of polynomials over a number field whose set of stable primes is of positive density must necessarily have a very restricted, and in particular ``near-solvable" 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.

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