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

Evan Chen

TL;DR

The work investigates when a rational function over a cyclotomic closure can avoid sending algebraic integers of bounded house to the closure of integers. By combining Loxton-type decompositions, torus torsion-geometry (via Zannier's results) and degree/term-bounds from composition theory (Fuchs–Zannier), the authors derive a precise criterion: $h$ is $P_A$-avoiding except if there exists an $A$-short witness expressing $h(S(x))$ as a Laurent sum with bounded terms. They further show that for large degree, most nonconstant maps are $P_A$-avoiding, and that maps with more than two poles are automatically $P_A$-avoiding; they extend Ostafe’s root-of-unity results to the bounded-house setting and obtain strong $P_A$-avoidance under suitable hypotheses. The results advance arithmetic dynamics over cyclotomic closures by linking orbit behavior to explicit algebraic witnesses, with potential implications for uniform finiteness phenomena and Diophantine avoidance in dynamical systems.

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 $h(α)$ is replaced by orbits $h(h(\cdots h(α)))$. This generalizes a result proved by Ostafe that concerns avoiding roots of unity, which is the case $A=1$.

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

TL;DR

The work investigates when a rational function over a cyclotomic closure can avoid sending algebraic integers of bounded house to the closure of integers. By combining Loxton-type decompositions, torus torsion-geometry (via Zannier's results) and degree/term-bounds from composition theory (Fuchs–Zannier), the authors derive a precise criterion: is -avoiding except if there exists an -short witness expressing as a Laurent sum with bounded terms. They further show that for large degree, most nonconstant maps are -avoiding, and that maps with more than two poles are automatically -avoiding; they extend Ostafe’s root-of-unity results to the bounded-house setting and obtain strong -avoidance under suitable hypotheses. The results advance arithmetic dynamics over cyclotomic closures by linking orbit behavior to explicit algebraic witnesses, with potential implications for uniform finiteness phenomena and Diophantine avoidance in dynamical systems.

Abstract

Let be a number field with cyclotomic closure , and let . For a real number, we show that is finite for many . We also show that for many such the same result holds if is replaced by orbits . This generalizes a result proved by Ostafe that concerns avoiding roots of unity, which is the case .

Paper Structure

This paper contains 11 sections, 14 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Rational functions and set avoidance
  3. Summary of results
  4. Outline
  5. Full statement of results on $P_A$-avoidance
  6. Background
  7. Tools from arithmetic geometry
  8. Results on compositions of rational functions
  9. Estimates on sizes of orbits
  10. Proof of results on $P_A$-avoidance
  11. Proof of results on strong $P_A$-avoidance

Key Result

Theorem 1

Let $h = p/q \in k^{\mathrm{c}}(x)$, where $p,q \in k^{\mathrm{c}}[x]$. Assume that $p(x) - y^m q(x)$ is irreducible over $k^{\mathrm{c}}$ for all positive integers $m \le \max(\deg p, \deg q)$. Then $h$ is $\mathbb U$-avoiding over $k^{\mathrm{c}}$.

Theorems & Definitions (28)

  • Definition 1.1
  • Theorem : From ref:zannier
  • Theorem : ref:ostafe
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 2.1: Loxton theorem, ref:zannier
  • Definition 2.2
  • Remark 2.3
  • ...and 18 more