## Illinois Journal of Mathematics

### Thue equations and lattices

Jeffrey Lin Thunder

#### Abstract

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

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

Dates
Revised: 4 November 2016
First available in Project Euclid: 27 February 2017

https://projecteuclid.org/euclid.ijm/1488186018

Digital Object Identifier
doi:10.1215/ijm/1488186018

Mathematical Reviews number (MathSciNet)
MR3628298

Zentralblatt MATH identifier
06688853

#### Citation

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

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