Average Zsigmondy sets, dynamical Galois groups, and the Kodaira-Spencer map

TL;DR

This paper investigates bounds on dynamical Zsigmondy sets over global function fields and number fields, showing average emptiness and analyzing Galois groups of polynomial iterates with applications to arithmetic dynamics.

Contribution

It establishes uniform bounds on Zsigmondy sets for wandering points, proves average emptiness, and analyzes Galois groups of polynomial iterates in various number field contexts.

Findings

Bounded size of Zsigmondy sets depending on $\

Average emptiness of Zsigmondy sets when ordered by height

Galois groups of iterates form finite index subgroups of wreath products

Abstract

Let $K$ be a global function field and let $\phi\in K[x]$. For all wandering basepoints $b\in K$, we show that there is a bound on the size of the elements of the dynamical Zsigmondy set $\mathcal{Z}(\phi,b)$ that depends only on $\phi$, the poles of the $b$, and $K$. Moreover, when we order $b\in\mathcal{O}_{K,S}$ by height, we show that $\mathcal{Z}(\phi,b)$ is empty on average. As an application, we prove that the inverse limit of the Galois groups of iterates of $\phi(x)=x^d+f$ is a finite index subgroup of an iterated wreath product of cyclic groups. Finally, we establish similar results on Zsigmondy sets when $K$ is the field of rational numbers or $K$ is a quadratic imaginary field subject to an added stipulation: either zero has finite orbit under iteration of $\phi$ or the Vojta conjecture for algebraic points on curves holds.