Diagonalizable Thue Equations -- revisited

Abstract

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

14 pages