Thue's inequalities and the hypergeometric method

Abstract

Following a method originally due to Siegel, we establish upper bounds for the number of primitive integer solutions to inequalities of the shape $0<|F(x, y)| \leq h$, where $F(x , y) =(\alpha x + \beta y)^r -(\gamma x + \delta y)^r \in \mathbb{Z}[x ,y]$, $\alpha$, $\beta$, $\gamma$ and $\delta$ are algebraic constants with $\alpha\delta-\beta\gamma \neq 0$, and $r \geq 3$ and $h$ are integers. As an important application, we pay special attention to the binomial Thue's inequaities $|ax^r - by^r| \leq c$. The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.