Primitive Divisors in Arithmetic Dynamics

TL;DR

This paper proves that for a broad class of rational functions, the numerators of iterates at a point typically have a new prime divisor not seen in earlier iterates, extending classical results to arithmetic dynamics.

Contribution

It establishes the existence of primitive divisors in the numerators of iterates of rational functions over number fields, generalizing classical number theory results to dynamical systems.

Findings

Most numerators have a new prime divisor for large n

Results extend to functions over number fields with periodic points

Provides a dynamical analogue of Zsigmondy's theorem

Abstract

Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in lowest terms. We prove that for all but finitely many n > 0, the numerator A_n has a primitive divisor, i.e., there is a prime p such that p divides A_n and p does not divide A_i for all i < n. More generally, we prove an analogous result when F is defined over a number field and 0 is a periodic point for F.