Anton Mosunov, Department of Pure Mathematics, University of Waterloo
"Generalizations of the Gap Principle and the Thue-Siegel Principle, With Applications to Diophantine Equations"
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 Diophantine equations. Let $\alpha$ be an algebraic number over $\mathbb Q$ and let $F(X, Y)$ be the homogenization of the minimal polynomial of $\alpha$. In the special case when $\mathbb Q(\alpha)/\mathbb Q$ 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 $\beta$ of $\alpha$ is such that the degree of $\beta$ over $\mathbb Q(\alpha)$ is small relative to the degree of $\alpha$ over $\mathbb Q$, we establish bounds of the form $C\gamma$, where $C$ is an absolute constant and $\gamma$ is a natural parameter associated to $\alpha$ that does not exceed the degree of $\alpha$ over $\mathbb Q$. We expect this parameter to be small, perhaps even bounded by an absolute constant.
MC 2009