# Dynamical uniform boundedness and the $abc$-conjecture

## TL;DR

This paper explores the connection between the uniform boundedness conjecture for unicritical polynomials and the $abc$-conjecture, establishing results under certain assumptions and degrees.

## Contribution

It demonstrates that the uniform boundedness conjecture for unicritical polynomials can be derived from the $abc$-conjecture, with degree-specific conditions.

## Key findings

- Results hold for degrees at least five under the standard $abc$-conjecture.
- The paper links dynamical systems conjectures with number theory conjectures.
- Provides conditional evidence supporting the uniform boundedness conjecture.

## Abstract

We address the Uniform Boundedness Conjecture of Morton and Silverman in the case of unicritical polynomials, assuming a generalization of the $abc$-conjecture. For unicritical polynomials of degree at least five, we require only the standard $abc$-conjecture.