2017/04/07 by Andrea Ferraguti, Ferraguti, Andrea
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1704.02204
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arxiv created 2017/04/07 · openalex publication_date 2017/04/07 · arxiv updated 2017/04/10 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Let K be a number field with ring of integers \mathcal OK, and let \fk\k∈ \mathbb N⊆ \mathcal OK[x] be a sequence of monic polynomials such that for every n∈ \mathbb 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 \mathfrak p⊆\mathcal OK such that every f(n) is irreducible modulo \mathfrak 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.