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 $y$ 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 $(x,y) \in {{\mathbf {Z}}^2}$ in terms of the height of $F$ and the degrees in $x$ and $y$ of $F$.