Upper bounds for the number of solutions to quartic Thue equations

TL;DR

This paper establishes upper bounds on the number of integral solutions to quartic Thue equations using logarithmic curves and linear forms in logarithms, with improvements based on form signatures.

Contribution

It introduces a new approach employing a logarithmic curve and refines previous bounds by considering the signature of the forms.

Findings

Provides explicit upper bounds for solutions to quartic Thue equations.

Enhances previous results by incorporating form signatures.

Utilizes linear forms in logarithms as a key analytical tool.

Abstract

We will give upper bounds for the number of integral solutions to quartic Thue equations. Our main tool here is a logarithmic curve $\phi(x, y)$ that allows us to use the theory of linear forms in logarithms. This manuscript improves the results of author's earlier work with Okazaki by giving special treatments to forms with respect to their signature.