Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: David Lee Hilliker and E. G. Straus
Journal: Trans. Amer. Math. Soc. 280 (1983), 637-657
MSC: Primary 11D41
MathSciNet review: 716842
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11D41

Retrieve articles in all journals with MSC: 11D41


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0716842-3
PII: S 0002-9947(1983)0716842-3
Keywords: Algebraic function, Diophantine equation, Puiseux series
Article copyright: © Copyright 1983 American Mathematical Society