Representation of Small Integers by Binary Forms

TL;DR

This paper derives upper bounds on the number of integer solutions to Thue inequalities and equations involving binary forms, with bounds independent of the parameter m under certain conditions related to the discriminant.

Contribution

It provides new bounds on solutions to Thue inequalities and equations that are independent of m when m is below specific thresholds related to the discriminant.

Findings

Upper bounds for solutions to Thue inequalities independent of m for small m.

Upper bounds for solutions to Thue equations independent of m for small m.

Results depend on the discriminant and degree of the binary form.

Abstract

We establish some upper bounds for the number of integer solutions to the Thue inequality $|F(x , y)| \leq m$, where $F$ is a binary form of degree $n \geq 3$ and with non-zero discriminant $D$, and $m$ is an integer. Our upper bounds are independent of $m$, when $m$ is smaller than $|D|^{\frac{1}{4(n-1)}}$. We also consider the Thue equation $|F(x , y)| = m$ and give some upper bounds for the number of its integral solutions. In the case of equation, our upper bounds will be independent of integer $m$, when $ m < |D|^{\frac{1}{2(n-1)}}$.