Using Runge's Theorem to Determine Zariski Density of Integral Points in Two and Three Dimensions

Abstract

The German mathematician Carl Runge (1856-1927) came up with a theorem that said that any Diophantine equation in two variables satisfying a certain set of conditions has only finitely many integral solutions. This thesis will provide a detailed proof of this theorem and some examples in which we can apply it. This proof makes use of two theorems from abstract algebra: The Symmetric Function Theorem and Newton-Puiseux's Theorem. The statement and proof of these theorems will also be given. This thesis will then introduce the Zariski Topology in all dimensions and show the strong connection between the notion of Zariski density in two dimensions and the property of having finitely or infinitely many integral solutions to a given Diophantine equation in two variables. The concept of Zariski density makes it possible to formulate generalizations to Runge's Theorem in higher variables. After this introduction there will be an attempt of the writer to generalize Runge's Theorem such that it can be applied to Diophantine equations of three variables.