Diagonalizable Thue Equations -- revisited

TL;DR

This paper improves bounds on the number of primitive solutions to certain diagonalizable Thue inequalities involving algebraic forms, refining previous results by Siegel and others.

Contribution

It establishes tighter upper bounds for primitive solutions to specific diagonalizable Thue inequalities, advancing the understanding of their solution structure.

Findings

Improved upper bounds for primitive solutions to Thue inequalities.

Enhanced understanding of diagonalizable forms with algebraic coefficients.

Refinement of classical results by Siegel and Akhtari et al.

Abstract

Let $r,h\in\mathbb{N}$ with $r\geq 7$ and let $F(x,y)\in \mathbb{Z}[x ,y]$ be a binary form such that \[ F(x , y) =(\alpha x + \beta y)^r -(\gamma x + \delta y)^r, \] where $\alpha$, $\beta$, $\gamma$ and $\delta$ are algebraic constants with $\alpha\delta-\beta\gamma \neq 0$. We establish upper bounds for the number of primitive solutions to the Thue inequality $0<|F(x, y)| \leq h$, improving an earlier result of Siegel and of Akhtari, Saradha & Sharma.