SEMINAR:Primitive Prime Divisors in the Critical Orbit of Polynomial...

Speaker: Mohamed Wafik

Title: Primitive Prime Divisors in the Critical Orbit of Polynomial Dynamical Systems

Date/Time:18 May 2022 / 13:40 - 14:30

Zoom link: https://sabanciuniv.zoom.us/j/94368587993?pwd=WnpVZ3U1MTBBUGFHSTRRT0Flcmlhdz09

Passcode:658402

Abstract:Let f_d,c(x) = x^d + c ∈ Q[x], d ≥ 2. We write f_d,cⁿ for f_d,c ◦ f_d,c◦…◦ f_d,c (n times). The critical orbit of f_d,c is the set O_fd,c(0) := { f_d,cⁿ (0) : n ≥ 0}.

For a sequence {a_n : n ≥ 0}, a primitive prime divisor for a_k is a prime dividing a_k but not an for any 1 ≤ n < k. A result of H. Krieger asserts that if the critical orbit O_fd,c(0) is infinite, then each element in O_fd,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 f_d,c is always 0.

Inspired by the previous results, we display an upper bound on the count of primitive prime divisors of a fixed iteration f_d,cⁿ (0). Fixing n ≥ 2, we also, under certain assumptions, calculate the density of primes that can appear as primitive prime divisors of f_2,cⁿ (0) for some c ∈ Q. Further, we show that there is no uniform upper bound on the count of primitive prime divisors of f_d,cⁿ (0) that does not depend on c. In particular, given N > 0, there is c ∈ Q such that f_d,cⁿ (0) has at least N primitive prime divisors.