Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Determination of bounds for the solutions to those binary Diophantine equations that satisfy the hypotheses of Runge’s theoremHTML articles powered by AMS MathViewer

by David Lee Hilliker and E. G. Straus
Trans. Amer. Math. Soc. 280 (1983), 637-657 Request permission

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 $F$.
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• Journal: Trans. Amer. Math. Soc. 280 (1983), 637-657
• MSC: Primary 11D41
• DOI: https://doi.org/10.1090/S0002-9947-1983-0716842-3
• MathSciNet review: 716842