Thue’s equations and rational approximations

Abstract

In this chapter we continue to deal only with solutions of diophantine equations in the classical sense, namely in integers of ℤ. We shall present a proof of the theorem of Thue, showing finiteness for equations f (X, Y) = c, where c ≠ 0 is constant and f is a homogeneous polynomial satisfying some natural necessary conditions. The proof will almost immediately follow from Thue’s celebrated result in diophantine approximation, providing a new, deeper, example of the fundamental link between these theories. To better illustrate the basic principles of Thue’s quite intricate, though elementary, proof, we shall limit ourselves to the original result, and only briefly recall the important sharpenings due to subsequent authors. Prior to details, we shall also present the main points of this argument. Finally, in the ‘Supplements’ we shall present some applications to the finiteness of integral points on other curves, a short proof of a theorem of Runge and a brief discussion of a function-field Thue Equation.