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

TL;DR

This paper establishes finiteness results for primitive prime divisors in the critical orbit of polynomial functions z^d + c, providing explicit bounds and employing diverse mathematical techniques for different cases.

Contribution

It proves the finiteness of the Zsigmondy set for critical orbits of z^d + c and computes it explicitly in non-recurrent cases, using effective height bounds and Diophantine approximation.

Findings

Finiteness of the Zsigmondy set for critical orbits.

Explicit computation of the Zsigmondy set in non-recurrent cases.

Effective bounds established for the size of the Zsigmondy set.

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.