Galois groups and prime divisors in random quadratic sequences

Irreducible polynomials in quadratic semigroups

“…Then one can form the semigroup M S generated by S under composition, i.e., the set of all polynomials of the form f = θ 1 • • • • • θ n for some n ≥ 1 and some θ i ∈ S. These semigroups have gained increased interest in both dynamical systems and number theory in recent years, as they naturally generalize the process of iterating a single function; see [3,4,5,10,15,13,14] for results relating the structure of M S to preperiodic points, invariant probability measures, and height functions. In addition and of particular interest for this paper, the semigroup M S can be used to construct arboreal representations of Galois groups, i.e., Galois groups that arise as subgroups of the automorphism group of a tree; see [1,2,6,12,19] for results in the case of iterating a single function and see [8,9] for recent results in the more general situation. However, nearly all of the known results on arboreal representations rely on the underlying polynomials in the semigroup being irreducible (a natural assumption in Galois theory in general).…”

“…| θ n ∈ S} of infinite sequences of elements of S via the product measure. Then one can attach to each infinite sequence γ an arboreal representation ρ γ : Gal(Q/Q) → Aut(T γ,0 ), where T γ,0 is the infinite, regular, 2-ary rooted tree of preimages of 0 under the sequence of maps defining γ; see [7,8] for details. Then, it follows from Theorem 1.1 above that if the c i are integral and S contains at least 2 irreducible maps, then an arboreal representation acts transitively with positive probability, ν {γ ∈ Φ S : ρ γ Gal(Q/Q) acts transitively on T γ,0 } > 0.…”

“…Equivalently, a random sequence γ ∈ Φ S is stable over Q (i.e., for a sequence γ = (θ m ) m≥1 , the polynomials θ 1 • • • • • θ m are irreducible for all m ≥ 1) with positive probability [7,8].…”