A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing

Abstract

In 1929 Siegel proved a celebrated theorem on finiteness for integral solutions of certain diophantine equations. This theorem applies to systems of polynomial equations which either (a) describe an irreducible curve whose projective closure has positive genus, or (b) describe an irreducible curve of genus zero with at least three points at infinity. For such systems, Siegel's theorem says that there are only finitely many solutions in the ring of integers of any given number field. Siegel's proof used the method of diophantine approximations, as pioneered by Thue in 1909 [T]. To give an example, if (x, y) is a large integral solution of the equation x3 2y3 = 1, then