ABC implies primitive prime divisors in arithmetic dynamics

Abstract

Let K be a number field, let f(x) in K(x) be a rational function of degree d> 1, and let z in K be a wandering point such that f^n(z) is nonzero for all n > 0. We prove that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that p | f^n(z) and p does not divide f^m(z) for all positive integers m < n. We prove the same result unconditionally for function fields of characteristic 0 when f is not isotrivial.