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

Abstract

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