Abstract
For a class of polynomials \(f \in {\mathbb {Z}}[X]\), 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 \(Q \rightarrow \infty \). For this class of polynomials this gives a more precise version of a recent result of Ferraguti (Proc Am Math Soc 146:2773–2784, 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.
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Acknowledgements
The authors thank Andrea Ferraguti for feedback on an early version of the paper and pointing out an imprecision in the initial statement of Theorem 1.2 and supplying the example at the end of Sect. 1.2. The authors are also grateful to the anonymous referee for helpful comments and suggestions. During the preparation of this work, L.M. was supported by the Austrian Science Fund (FWF): Project P31762, A. O. was supported by the Australian Research Council (ARC): Grant DP180100201 and I. S. was supported by the Australian Research Council (ARC): Grant DP170100786.
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Mérai, L., Ostafe, A. & Shparlinski, I.E. Dynamical irreducibility of polynomials modulo primes. Math. Z. 298, 1187–1199 (2021). https://doi.org/10.1007/s00209-020-02630-5
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DOI: https://doi.org/10.1007/s00209-020-02630-5