THUE EQUATIONS AND LATTICES

摘要

We consider Diophantine equations of the kind {F(x, y)} = m, where F(X,Y) ∈ Z[X,Y] is a homogeneous polynomial of degree at least 3 that has non-zero discriminant, m is a fixed positive integer and x,y are relatively prime integer solutions. Our results improve upon previous theorems due to Bombieri and Schmidt and also Stewart. We further provide reasonable heuristics for conjectures of Schmidt and Stewart regarding such equations.