Rational preimages in families of dynamical systems

TL;DR

This paper investigates the number of rational preimages under rational functions over number fields, providing bounds, methods, and conjectures in the context of arithmetic dynamics.

Contribution

It introduces bounds for rational preimages in families of rational functions and connects these bounds to a new uniform boundedness conjecture.

Findings

Established bounds for the number of rational preimages

Applied unit equations and Runge's method to affine curves

Formulated a uniform boundedness conjecture relating to rational preimages

Abstract

Given a rational function of degree at least two defined over a number field k, we study the cardinality of the set of rational iterated preimages. We prove bounds for the cardinality of this set as the rational function varies in certain families. Our proofs are based on unit equations and a method of Runge for effectively determining integral points on certain affine curves. We also formulate and state a uniform boundedness conjecture for iterated preimages of rational functions and relate this conjecture to other well-known conjectures in arithmetic dynamics.