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Number Theory Seminar

Tuesday, June 4, 2019 — 1:30 PM EDT

Anton Mosunov, Department of Pure Mathematics, University of Waterloo

"Absolute Bounds on the Number of Solutions of Certain Equations of Thue and Thue-Mahler 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 Thue-Siegel 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 Thue-Mahler 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 $p^k \geq m^{d/(2d-16.2)}$.

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