Thue Equations and Lattices

TL;DR

This paper investigates solutions to certain homogeneous Diophantine equations, providing simplifications of previous results and heuristics for related conjectures, with implications for Thue equations and lattice theory.

Contribution

It offers simplified proofs of Stewart's results and develops heuristics for conjectures related to Thue equations involving lattices.

Findings

Simplified Stewart's results on Thue equations

Developed heuristics for Stewart's conjecture

Provided new insights into Diophantine equations with lattice structures

Abstract

We consider Diophantine equations of the kind $|F(x,y)|= m$, where $F(X,Y )\in \bz [X,Y]$ is a homogeneous polynomial of degree $d\ge 3$ that has non-zero discriminant and $m$ is a positive integer. We prove results that simplify those of Stewart and provide heuristics for a conjecture of Stewart.