Quantitative Estimates for the Size of the Zsigmondy Set in Arithmetic Dynamics

TL;DR

This paper provides quantitative bounds on the size of the Zsigmondy set for sequences generated by polynomial iteration over number fields, advancing understanding in arithmetic dynamics.

Contribution

It offers new quantitative estimates for the Zsigmondy set size in polynomial orbits over number fields, a novel contribution to arithmetic dynamics.

Findings

Derived explicit bounds for Zsigmondy set size

Applied results to polynomial iteration sequences

Enhanced understanding of primitive prime divisors in dynamics

Abstract

Let \( K \) be a number field. We provide quantitative estimates for the size of the Zsigmondy set of an integral ideal sequence generated by iterating a polynomial function \(\varphi(z) \in K[z]\) at a wandering point \(\alpha \in K.\)