Illinois Journal of Mathematics

Thue equations and lattices

Jeffrey Lin Thunder

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We consider Diophantine equations of the kind $|F(x,y)|=m$, where $F(X,Y)\in\mathbb{Z}[X,Y]$ is a homogeneous polynomial of degree at least 3 that has non-zero discriminant, $m$ is a fixed positive integer and $x,y$ are relatively prime integer solutions. Our results improve upon previous theorems due to Bombieri and Schmidt and also Stewart. We further provide reasonable heuristics for conjectures of Schmidt and Stewart regarding such equations.

Article information

Illinois J. Math., Volume 59, Number 4 (2015), 999-1023.

Received: 15 March 2016
Revised: 4 November 2016
First available in Project Euclid: 27 February 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D75: Diophantine inequalities [See also 11J25] 11J25: Diophantine inequalities [See also 11D75]


Thunder, Jeffrey Lin. Thue equations and lattices. Illinois J. Math. 59 (2015), no. 4, 999--1023. doi:10.1215/ijm/1488186018.

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