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Counting solutions to trinomial Thue equations: a different approach


Author: Emery Thomas
Journal: Trans. Amer. Math. Soc. 352 (2000), 3595-3622
MSC (2000): Primary 11D41, 11J68; Secondary 11Y50
DOI: https://doi.org/10.1090/S0002-9947-00-02437-5
Published electronically: March 16, 2000
MathSciNet review: 1641119
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Abstract:

We consider the problem of counting solutions to a trinomial Thue equation -- that is, an equation

\begin{equation*}\vert F(x,y)\vert = 1,\tag{$*$ } \end{equation*}

where $F$ is an irreducible form in $Z[x,y]$ with degree at least three and with three non-zero coefficients. In a 1987 paper J. Mueller and W. Schmidt gave effective bounds for this problem. Their work was based on a series of papers by Bombieri, Bombieri-Mueller and Bombieri-Schmidt, all concerned with the ``Thue-Siegel principle" and its relation to $(*)$. In this paper we give specific numerical bounds for the number of solutions to $(*)$ by a somewhat different approach, the difference lying in the initial step -- solving a certain diophantine approximation problem. We regard this as a real variable extremal problem, which we then solve by elementary calculus.


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Additional Information

Emery Thomas
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720–3840

DOI: https://doi.org/10.1090/S0002-9947-00-02437-5
Keywords: Thue equation, Thue-Siegel principle, diophantine approximation, trinomial equation, counting solutions.
Received by editor(s): May 23, 1997
Received by editor(s) in revised form: July 29, 1998
Published electronically: March 16, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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