Primitive divisors in arithmetic dynamics

Abstract

AbstractLet ϕ(z) ∈ (z) be a rational function of degree d ≥ 2 with ϕ(0) = 0 and such that ϕ does not vanish to order d at 0. Let α ∈ have infinite orbit under iteration of ϕ and write ϕn(α) = An/Bn as a fraction in lowest terms. We prove that for all but finitely many n ≥ 0, the numerator An has a primitive divisor, i.e., there is a prime p such that p | An and p ∤ Ai for all i < n. More generally, we prove an analogous result when ϕ is defined over a number field and 0 is a preperiodic point for ϕ.