Cubic Thue inequalities with positive discriminant

TL;DR

This paper establishes an explicit upper bound on the number of solutions to cubic inequalities involving binary forms with positive discriminant, valid when the bound h is less than D^{1/4}.

Contribution

It provides a new explicit upper bound for solutions to cubic inequalities with positive discriminant, independent of h under certain conditions.

Findings

Upper bound for solutions when h < D^{1/4}

Bound is explicit and independent of h in the specified range

Applicable to cubic forms with positive discriminant

Abstract

We will give an explicit upper bound for the number of solutions to cubic inequality |F(x, y)| \leq h, where F(x, y) is a cubic binary form with integer coefficients and positive discriminant D. Our upper bound is independent of h, provided that h is smaller than D^{1/4}.