Determination of bounds for the solutions to those binary Diophantine equations that satisfy the hypotheses of Runge’s theorem

Abstract

In 1887 Runge [13] proved that a binary Diophantine equation F(x, y) = 0, with F irreducible, in a class including those in which the leading form of F is not a constant multiple of a power of an irreducible polynomial, has only a finite number of solutions. It follows from Runge's method of proof that there exists a computable upper bound for the absolute value of each of the integer solutions x and y. Runge did not give such a computation. Here we first deduce Runge's Theorem from a more general theorem on Puiseux series that may be of interest in its own right. Second, we extend the Puiseux series theorem and deduce from the generalized version a generalized form of Runge's Theorem in which the solutions x and v of the polynomial equation F(x, y) = 0 are integers, satisfying certain conditions, of an arbitrary algebraic number field. Third, we compute bounds for the solutions (i,f)EZ! in terms of the height of F and the degrees in x and y of F.