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
"Absolute Bounds on the Number of Solutions of Certain Equations of Thue and ThueMahler Type"
Let $F(X, Y)$ be an irreducible polynomial with integer coefficients of degree at least three. In 1909 it was proved by Thue that the Diophantine equation
$$F(x, y) = m,$$
where $m$ is a fixed integer, has only finitely many solutions in integers $x$ and $y$. This equation is known as the Thue equation. Thue's proof involved the combination of two principles, which are known in the theory of Diophantine approximation as the gap principle and the ThueSiegel principle.
In this talk, we present a generalization of the gap principle, which enabled us to establish absolute bounds on the number of solutions of certain equations of Thue and ThueMahler type. As an example, let $F$ be of degree at least seventeen. Further, suppose that the field extension $\mathbb Q(\alpha)/\mathbb Q$ is Galois, where $\alpha$ is a root of $F(X, 1)$. We prove that the Thue equation above has at most 24 solutions in coprime integers $x$ and $y$, provided that $m$ is sufficiently large and there exists a prime power $p^k$ dividing $m$ such that $p^k \geq m^{d/(2d16.2)}$.
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Departmental office: MC 5304
Phone: 519 888 4567 x33484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca
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