# Families of Thue Inequalities with Transitive Automorphisms

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## Date

2014-06-17

## Authors

An, Wenyong

## Advisor

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## Publisher

University of Waterloo

## Abstract

A family of parameterized Thue equations is defined as F_{t,s,...}(X, Y ) = m, m ∈ Z
where F_{t,s,...}(X,Y) is a form in X and Y with degree greater than or equal to 3 and integer coefficients that are parameterized by t, s, . . . ∈ Z. A variety of these families have been studied by different authors.
In this thesis, we study the following families of Thue inequalities
|sx3 −tx2y−(t+3s)xy2 −sy3|≤2t+3s, |sx4 −tx3y−6sx2y2 +txy3 +sy4|≤6t+7s,
|sx6 − 2tx5y − (5t + 15s)x4y2 − 20sx3y3 + 5tx2y4
+(2t + 6s)xy5 + sy6| ≤ 120t + 323s,
where s and t are integers. The forms in question are “simple”, in the sense that the roots of the underlying polynomials can be permuted transitively by automorphisms.
With this nice property and the hypergeometric functions, we construct sequences of good approximations to the roots of the underlying polynomials. We can then prove that under certain conditions on s and t there are upper bounds for the number of integer solutions to the above Thue inequalities.

## Description

## Keywords

Parameterized Diophantine equation, Thue equation and inequality, hypergeometric method