Avoiding algebraic integers of bounded house in orbits of rational functions over cyclotomic closures

TL;DR

This paper investigates the finiteness of algebraic integers with bounded house in orbits of rational functions over cyclotomic closures, extending previous results related to roots of unity to more general bounds.

Contribution

It generalizes Ostafe's result by establishing finiteness conditions for algebraic integers with bounded house in orbits of rational functions over cyclotomic closures.

Findings

Finiteness of algebraic integers with bounded house in certain orbits

Extension of previous results from roots of unity to broader bounds

Applicable to many rational functions over cyclotomic closures

Abstract

Let $k$ be a number field with cyclotomic closure $k^{\mathrm{cyc}}$, and let $h \in k^{\mathrm{cyc}}(x)$. For $A \ge 1$ a real number, we show that \[ \{ \alpha \in k^{\mathrm{cyc}} : h(\alpha) \in \overline{\mathbb Z} \text{ has house at most } A \} \] is finite for many $h$. We also show that for many such $h$ the same result holds if $h(\alpha)$ is replaced by orbits $h(h(\cdots h(\alpha)))$. This generalizes a result proved by Ostafe that concerns avoiding roots of unity, which is the case $A=1$.