Minimal models for rational functions in a dynamical setting

TL;DR

This paper introduces an algorithm for finding minimal models of rational functions under conjugation, explores rational functions with many integers in a single orbit, and discusses potential dynamical analogues of Szpiro's conjecture.

Contribution

It provides a practical algorithm for minimal models and reports new findings on rational functions with many integers in orbits, advancing understanding in dynamical systems.

Findings

Identified minimal quadratic rational functions with 8 integers in an orbit

Found minimal cubic rational functions with 10 integers in an orbit

Discussed elementary aspects of a dynamical Szpiro's conjecture

Abstract

We present a practical algorithm to compute models of rational functions with minimal resultant under conjugation by fractional linear transformations. We also report on a search for rational functions of degrees 2 and 3 with rational coefficients that have many integers in a single orbit. We find several minimal quadratic rational functions with 8 integers in an orbit and several minimal cubic rational functions with 10 integers in an orbit. We also make some elementary observations on possibilities of an analogue of Szpiro's conjecture in a dynamical setting and on the structure of the set of minimal models for a given rational function.