The Method Of Thue-Siegel For Binary Quartic Forms

TL;DR

This paper applies the Thue-Siegel method, utilizing hypergeometric functions, to establish upper bounds on the number of integral solutions for specific quartic binary forms, advancing understanding of their solution sets.

Contribution

It introduces a novel application of the Thue-Siegel method with hypergeometric functions to bound solutions of quartic binary forms.

Findings

Derived upper bounds for solutions to |F(x,y)|=1

Extended bounds to inequalities |F(x,y)| ≤ h

Applied method to a specific family of irreducible quartic forms

Abstract

We will use Thue-Siegel method, based on Pad\'e approximation via hypergeometric functions, to give upper bounds for the number of integral solutions to the equation $|F(x, y)| = 1$ as well as the inequalities $|F(x, y)| \leq h$, for a certain family of irreducible quartic binary forms.