On irreducible divisors of iterated polynomials

Abstract

Abstract. D. Gomez, A. Ostafe, A. P. Nicolás and D. Sadornil have recently shown that for almost all polynomials f ∈ F q [X] over the finite field of q elements, where q is an odd prime power, of their iterations eventually become reducible polynomials over F q . Here we combine their method together with some new ideas to derive series of finer results about the arithmetic structure of iterations of f . In particular, we prove that the nth iteration of f has a square-free divisor of degree of order at least n 1+o(1) as n → ∞ (uniformly over q).

Galois representations from pre-image trees: an arboreal survey

“…What, then, may be said about the group G arith ? Unfortunately, the extension L in (11) remains mysterious in general, particularly in the case where φ is post-critically finite. An outstanding contribution of [40] is the computation of L when φ is a post-critically finite quadratic polynomial defined over a general field K. In particular, [L : K] is finite when the orbit of the critical point of φ is pre-periodic and the post-critical set has at least 3 elements.…”

On the factorization of iterated polynomials

“…In this paper, we explore similar questions to the ones discussed in [5] over a more general class of iterated polynomials. For polynomials f, g ∈ F q [x], we define the following sequence of polynomials over F q :…”

“…In addition, we write A ≈ B if A ≪ B and A ≫ B. The main results in [5] are Theorems 9, 10 and 11 and they can be compiled as follows.…”

“…This paper deals with the asymptotic growth of certain arithmetic functions associated to the factorization of the iterated polynomials f (g (n) (x)) over Fq, such as the largest degree of an irreducible factor and the number of irreducible factors. In particular, we provide significant improvements on the results of D. Gómez-Pérez, A. Ostafe and I. Shparlinski (2014).…”

The arithmetic of consecutive polynomial sequences over finite fields

“…In Section 2, we introduce the main tools that we use to prove our results. In Section 3, we use a method introduced in [9], relying on the polynomial ABC theorem (proved first by Stothers [32], and then independently by Mason [19,20] and Silverman [29], see also [30]), to give a lower bound for D(f n ), n 1. In particular, we prove that if {f n } is of infinite length, for almost all integers n 1 we have…”