Primitive prime divisors in the critical orbit of polynomial dynamical systems

Official URL: https://risc01.sabanciuniv.edu/record=b2767343_(Table of contents)

Let fd,c(x) = xd+c ∈ Q[x], d ≥ 2. We write fn d,c for fd,c ◦ fd,c ◦ · · · ◦ f d,c} n times . The critical orbit of fd,c(x) is the set Ofd,c(0) := {fn d,c(0) : n ≥ 0}. For a sequence {an : n ≥ 0}, a primitive prime divisor for an is a prime dividing an but not ak for any 1 ≤ k < n. A result of H. Krieger asserts that if the critical orbit Ofd,c(0) is infinite, then each element in Ofd,c(0) has at least one primitive prime divisor, except possibly for 23 elements. In addition, under certain conditions, R. Jones proved that the density of primitive prime divisors appearing in any orbit of fd,c(x) is always 0. Inspired by the previous results, we display an upper bound on the count of primitive prime divisors of a fixed iteration fn d,c(0). We also investigate primitive prime divisors in the critical orbit of fd,c(x) ∈ K[x], where K is a number field. We develop links between the existence of a primitive prime divisor in the critical orbit and the periodicity of the critical orbit of the reduction of fd,c in the residue field of K modulo the primitive prime divisor. Consequently, under certain assumptions, we calculate the density of primes that can appear as primitive prime divisors of fn2,c(0) for some c ∈ Q. Furthermore, we show that there is no uniform upper bound on the count of primitive prime divisors of fn d,c(0) that does not depend on c. In particular, given N > 0, there is c ∈ Q such that fn d,c(0) has at least N primitive prime divisors.