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

Abstract:

Let $k$ be a number field with cyclotomic closure $k^{\mathrm {c}}$, and let $h \in k^{\mathrm {c}}(x)$. For $A \ge 1$ a real number, we show that \[ \{ \alpha \in k^{\mathrm {c}} : h(\alpha ) \in \overline {\mathbb Z} \text { has house at most } A \} \] 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 $A=1$.