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 $0<|F(x,y)|\le h$, 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 $\alpha \delta -\beta \gamma \ne 0$, and $r\ge 3$ and $h$ are integers. As an important application, we pay special attention to the binomial Thue's inequaities $|a{x}^r-b{y}^r|\le c$. The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.

45 pages