Finite Orbits of a Polynomial Automorphism on Ane Three Space with Applications

Abstract

In the Painlev´e VI differential equation, a certain group action governs the behavior of solutions as they are analytically continued around a pole. It turns out that the finite groups correspond to the algebraic solutions. In this paper, we will study the finite groups of this action in affine three-space. A complete classification of all finite orbits will be done in a single-parameter case, and an overview of the classification for a more complex case will also be studied. Remarkably, there is a deep connection between these finite orbits and the transitivity of another action in a Diophantine equation. We will investigate how classifying the finite orbits will give obstructions to the action being transitive in a finite field.