Thue Equations and Lattices

We consider Diophantine equations of the kind $\mathrm{\mid }F(x,y)\mathrm{\mid }=m$, where $F(X,Y )\in \bz [X,Y]$isahomogeneouspolynomialofdegree$d\ge 3$ that has non-zero discriminant and $m$ is a positive integer. We prove results that simplify those of Stewart and provide heuristics for a conjecture of Stewart.