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- Title
Galois groups and prime divisors in random quadratic sequences.
- Authors
DOYLE, JOHN R.; HEALEY, VIVIAN OLSIEWSKI; HINDES, WADE; JONES, RAFE
- Abstract
Given a set $S=\{x^2+c_1,\dots,x^2+c_s\}$ defined over a field and an infinite sequence $\gamma$ of elements of S , one can associate an arboreal representation to $\gamma$ , generalising the case of iterating a single polynomial. We study the probability that a random sequence $\gamma$ 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 $\mathbb{Z}[t]$ , and we conjecture a similar positive-probability result for suitable sets over $\mathbb{Q}$. 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 generalises the post-critically finite case in single-polynomial iteration.
- Subjects
PROBABILITY theory; LOGICAL prediction; POLYNOMIALS; DIVISOR theory
- Publication
Mathematical Proceedings of the Cambridge Philosophical Society, 2024, Vol 176, Issue 1, p95
- ISSN
0305-0041
- Publication type
Article
- DOI
10.1017/S0305004123000439