ABC implies primitive prime divisors in arithmetic dynamic

TL;DR

This paper links the abc-conjecture to primitive prime divisors in the orbits of rational functions over number fields, establishing that, under certain conditions, primitive prime divisors appear infinitely often in these orbits.

Contribution

It proves that assuming the abc-conjecture, primitive prime divisors occur infinitely often in the iterates of rational functions over number fields, and unconditionally for certain function fields.

Findings

Conditional proof assuming abc-conjecture for number fields.

Unconditional proof for non-isotrivial functions over function fields.

Existence of primitive prime divisors in orbits of rational functions.

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.