THE SET OF STABLE PRIMES FOR POLYNOMIAL SEQUENCES WITH LARGE GALOIS GROUP.

Let K be a number field with ring of integers 𝓞K, and let {fk}k∈N be a sequence of monic polynomials in 𝓞K[x] such that for every n ∈ ℕ, the composition f(n) = f1 ◦ f2 ◦ . . . ◦ fn 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 p ⊆ 𝓞K such that every f(n) is irreducible modulo 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. [ABSTRACT FROM AUTHOR]