Uniform bounds on $S$-integral preperiodic points for chebyshev polynomials

TL;DR

This paper establishes uniform bounds on the number of $S$-integral preperiodic points for Chebyshev polynomials relative to a non-preperiodic point, as the degree of the number field varies.

Contribution

It provides the first uniform bounds on $S$-integral preperiodic points for Chebyshev polynomials over number fields of bounded degree.

Findings

Proves uniform bounds depending only on the degree and the set $S$.

Extends results in arithmetic dynamics to Chebyshev polynomials.

Shows finiteness of $S$-integral preperiodic points relative to a non-preperiodic point.

Abstract

Let $K$ be a number field with algebraic closure $\bar{K}$, let $S$ be a finite set of places of $K$ containing the archimedean places, and let $\varphi$ be Chebyshev polynomial. In this paper we prove uniformity results on the number of $S$-integral preperiodic points relative to a non-preperiodic point $\beta$, as $\beta$ varies over number fields of bounded degree.