Thue's inequalities and the hypergeometric method

TL;DR

This paper develops upper bounds for primitive solutions to certain polynomial inequalities using hypergeometric methods, with applications to binomial Thue's inequalities, extending classical approaches by Siegel, Thue, and Evertse.

Contribution

It introduces new bounds for solutions to inequalities involving algebraic forms, applying refined hypergeometric techniques to Thue's inequalities.

Findings

Established upper bounds for primitive solutions to specific inequalities.

Applied the method to binomial Thue's inequalities with explicit bounds.

Extended classical methods with modern refinements for algebraic inequalities.

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.