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Representation of unity by binary forms


Author: Shabnam Akhtari
Journal: Trans. Amer. Math. Soc. 364 (2012), 2129-2155
MSC (2010): Primary 11D45, 11J68
DOI: https://doi.org/10.1090/S0002-9947-2011-05507-8
Published electronically: December 2, 2011
MathSciNet review: 2869201
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Abstract: In this paper, it is shown that if $ F(x , y)$ is an irreducible binary form with integral coefficients and degree $ n \geq 3$, then provided that the absolute value of the discriminant of $ F$ is large enough, the equation $ F(x , y) = \pm 1$ has at most $ 11n-2$ solutions in integers $ x$ and $ y$. We will also establish some sharper bounds when more restrictions are assumed. These upper bounds are derived by combining methods from classical analysis and geometry of numbers. The theory of linear forms in logarithms plays an essential role in studying the geometry of our Diophantine equations.


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Additional Information

Shabnam Akhtari
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
Address at time of publication: Centre de recherches mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, Québec, Canada H3C 3J7
Email: akhtari@mpim-bonn.mpg.de

DOI: https://doi.org/10.1090/S0002-9947-2011-05507-8
Keywords: Thue equations, linear forms in logarithms
Received by editor(s): July 20, 2009
Received by editor(s) in revised form: June 9, 2010, and September 14, 2010
Published electronically: December 2, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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