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The set of stable primes for polynomial sequences with large Galois group


Author: Andrea Ferraguti
Journal: Proc. Amer. Math. Soc. 146 (2018), 2773-2784
MSC (2010): Primary 11R32, 11R45, 20E08
DOI: https://doi.org/10.1090/proc/13958
Published electronically: February 16, 2018
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Abstract: Let $ K$ be a number field with ring of integers $ \mathcal {O}_K$, and let $ \{f_k\}_{k\in \mathbb{N}}$ be a sequence of monic polynomials in $ \mathcal {O}_K[x]$ such that for every $ n\in \mathbb{N}$, the composition $ f^{(n)}=f_1\circ f_2\circ \ldots \circ f_n$ is irreducible. In this paper we show that if the size of the Galois group of $ f^{(n)}$ is large enough (in a precise sense) as a function of $ n$, then the set of primes $ \mathfrak{p}\subseteq \mathcal {O}_K$ such that every $ f^{(n)}$ is irreducible modulo $ \mathfrak{p}$ has density zero. Moreover, we prove that the subset of polynomial sequences such that the Galois group of $ f^{(n)}$ is large enough has density 1, in an appropriate sense, within the set of all polynomial sequences.


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Additional Information

Andrea Ferraguti
Affiliation: Centre for Mathematical Sciences, University of Cambridge, DPMMS, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom
Email: af612@cam.ac.uk

DOI: https://doi.org/10.1090/proc/13958
Keywords: Arboreal Galois representations, number fields, stable primes, natural density
Received by editor(s): April 25, 2017
Received by editor(s) in revised form: September 14, 2017
Published electronically: February 16, 2018
Additional Notes: The author was supported by Swiss National Science Foundation grant number 168459.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2018 American Mathematical Society

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