Generalizations of the Gap Principle and the Thue-Siegel Principle, with Applications to Diophantine Equations

Abstract

In this thesis we develop generalizations of two well-known principles from the theory of Diophantine approximation, namely the gap principle and the Thue-Siegel principle. Our results find their applications in the theory of Diophantine equations. Let α be a number that is algebraic over the field of rational numbers Q and let F(X, Y) be the homogenization of the minimal polynomial of α. In the special case when Q(α)/Q is a Galois extension of degree at least seven, we establish absolute bounds on the number of solutions of certain equations of Thue and Thue-Mahler type, which involve F(X, Y). Consequently, we give theoretical evidence in support of Stewart's conjecture (1991). More generally, if every conjugate β of α is such that the degree of β over Q(α) is small relative to the degree of α over Q, we establish bounds of the form Cγ, where C is an absolute constant and γ is a natural parameter associated with α that does not exceed the degree of α over Q.