动态序列稳定素数的稀疏性

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

摘要

我们证明了一个动态序列（fn） n∈n $(f_n)_{n\in \mathbb {n}}$上的多项式密度为正的稳定素数集合必须有一个非常有限的，特别是虚可解的动态伽罗瓦群。此外，结合现有的启发式方法，我们的结果表明，一个多项式f$ f$，其所有迭代都是不可约模的素数的正密度子集，必须是线性函数，单项式和Dickson多项式的组合。

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Sparsity of stable primes for dynamical sequences

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.