Galois groups and prime divisors in random quadratic sequences

TL;DR

This paper investigates the likelihood that random quadratic sequences generate large Galois groups, showing positive probability over certain polynomial sets and linking these to prime divisor density results.

Contribution

It establishes positive probability results for large Galois image in random quadratic sequences over polynomial sets and classifies obstructions to finite-index representations.

Findings

Positive probability of large Galois images for most polynomial sets over z[t]

Density-zero result for prime divisors of associated quadratic sequences

Classification of obstructions to finite-index arboreal representations

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 $\gamma$, generalizing 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 generalizes the post-critically finite case in single-polynomial iteration.