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

Abstract

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

arXiv comment11 pages; updated to version as published by PAMS

Cited by in corpus (3)

• On multiplicative dependence of values of rational functions and a generalisation of the Northcott theorem
• Perfect powers in value sets and orbits of polynomials
• Cyclotomic preperiodic points for morphisms in affine spaces and preperiodic points with bounded house and height