2019/05/28 by László Mérai, Mérai, László, Alina Ostafe +3
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1905.11657
arxiv created 2020/09/24 · arxiv updated 2020/09/25
For a class of polynomials f ∈ ℤ[X], which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set of primes p such that all iterations of f are irreducible modulo p is of relative density zero. Furthermore, we give an explicit bound on the rate of the decay of the density of such primes in an interval [1, Q] as Q → ∞. For this class of polynomials this gives a more precise version of a recent result of A. Ferraguti (2018), which applies to arbitrary polynomials but requires a certain assumption about their Galois group. Furthermore, under the Generalised Riemann Hypothesis we obtain a stronger bound on this density.