The $abc$-Conjecture implies uniform bounds on dynamical Zsigmondy sets

TL;DR

This paper demonstrates that assuming the $abc$-Conjecture leads to uniform bounds on dynamical Zsigmondy sets and the index of arboreal Galois representations across families of unicritical polynomials over number fields.

Contribution

It establishes that the $abc$-Conjecture implies uniform bounds on Zsigmondy sets and arboreal Galois representations for unicritical polynomials.

Findings

Upper bounds on Zsigmondy sets are uniform over polynomial families.

Existence of uniform bounds on the index of arboreal Galois representations.

Results depend on the assumption of the $abc$-Conjecture.

Abstract

We prove that the $abc$-Conjecture implies upper bounds on Zsigmondy sets that are uniform over families of unicritical polynomials over number fields. As an application, we use the $abc$-Conjecture to prove that there exist uniform bounds on the index of the associated arboreal Galois representations.