Dynamical irreducibility of polynomials modulo primes
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Mathematische Zeitschrift
Dynamical irreducibility of polynomials modulo primes László Mérai1 · Alina Ostafe2 · Igor E. Shparlinski2 Received: 2 September 2019 / Accepted: 24 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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 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.
1 Introduction 1.1 Motivation For a polynomial f ∈ K[X ] over a field K we define the sequence of polynomials: f (0) (X ) = X , f (n) (X ) = f f (n−1) (X ) , n = 1, 2, . . . . The polynomial f (n) is called the n-th iterate of the polynomial f . Following the established terminology, see [1,2,8,9,14,16], one says that a polynomial f ∈ K[X ] is stable if all iterates f (n) (X ), n = 1, 2, . . ., are irreducible over K. However, we prefer to use the more informative terminology introduced by Heath-Brown and Micheli [12] and instead we call such polynomials dynamically irreducible.
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László Mérai [email protected] Alina Ostafe [email protected] Igor E. Shparlinski [email protected]
1
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, 4040 Linz, Austria
2
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
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L. Mérai et al.
For a polynomial f ∈ Q[X ] and a prime p we define f¯p ∈ F p [X ] to be the reduction of f modulo p. In this paper we consider the following question, see [3, Question 19.12]. Open Question 1.1 Let f ∈ Q[X ] be a dynamically irreducible polynomial of degree d ≥ 2. Is it true that the set of primes { p : f¯p is dynamically irreducible over F p }
(1.1)
is a finite set? For example, Jones [15, Conjecture 6.3] has conjectured that x 2 +1 is dynamically irreducible over F p if and only if p = 3. Ferraguti [7, Theorem 2.3] has shown that if the size of the Galois group Gal f (n) of f (n) is asymptotically close to its largest possible value then the set of primes (1.1) has density zero. It is natural to assume that this condition on the size of Gal f (n) is generically satisfied, however it may be difficult to verify it for concrete polynomials or find examples of such polynomials. Here we consider a special class of polynomials which includes trinomials of the form f (X ) = a X d + bX d−1 + c ∈ Z[X ] of even degree, and
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