Contact Info
Pure MathematicsUniversity of Waterloo
200 University Avenue West
Waterloo, Ontario, Canada
N2L 3G1
Departmental office: MC 5304
Phone: 519 888 4567 x33484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca
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Anton Mosunov, Department of Pure Mathematics, University of Waterloo
"Generalizations of the Gap Principle and the ThueSiegel Principle, With Applications to Diophantine Equations"
We develop generalizations of two wellknown principles from the theory of Diophantine approximation, namely the gap principle and the ThueSiegel 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 ThueMahler 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.
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Departmental office: MC 5304
Phone: 519 888 4567 x33484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca