Galois groups and prime divisors in random quadratic sequences

2021/08/25 by John R. Doyle, Doyle, John R., Vivian Olsiewski Healey +5 · 1 citation
Mathematics · #11F80. Secondary: 11D99 #37P15 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #Primary: 11R32

paper · pdf · doi:10.48550/arxiv.2108.11233

openalex publication_date 2021/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a set S=\x²+c₁,…,x²+c_s\ defined over a field and an infinite sequence γ of elements of S, one can associate an arboreal representation to γ, generalizing the case of iterating a single polynomial. We study the probability that a random sequence γ produces a ``large-image'' representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets S defined over ℤ[t], and we conjecture a similar positive-probability result for suitable sets over ℚ. As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify all S possessing a particular kind of obstruction that generalizes the post-critically finite case in single-polynomial iteration.

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• Preperiodic integers for x^d+c in large degree

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