Diagonalizable thue equations -- revisited

Abstract

Let r,h∈N with r≥7 and let F(x,y)∈Z[x,y] be a binary form such that F(x,y)=(αx+βy)r−(γx+δy)r, where α, β, γ and δ are algebraic constants with αδ−βγ≠0. We establish upper bounds for the number of primitive solutions to the Thue inequality 0<|F(x,y)|≤h, improving an earlier result of Siegel and of Akhtari, Saradha & Sharma.