Primitive prime divisors in the critical orbit of z^d+c

Abstract

We prove the finiteness of the Zsigmondy set associated to the critical orbit of f(z) = z^d+c for rational values of c by finding an effective bound on the size of the set. For non-recurrent critical orbits, the Zsigmondy set is explicitly computed by utilizing effective dynamical height bounds. In the general case, we use Thue-style Diophantine approximation methods to bound the size of the Zsigmondy set when d >2, and complex-analytic methods when d=2.

arXiv commentThis version includes numerous typographical changes and expanded exposition, and a simplified proof of Theorem 6.1. 30 pages, to appear in International Math Research Notices