Solving effectively some families of Thue Diophantine equations

TL;DR

This paper develops effective bounds for solutions to certain Thue Diophantine inequalities involving algebraic units and binary forms, especially for fields of degree at least 4, advancing the understanding of these equations.

Contribution

It provides new effective upper bounds for solutions of Thue inequalities associated with algebraic units in number fields of degree at least 4.

Findings

Established bounds for solutions of specific Thue inequalities.

Identified conditions on units ensuring the bounds are effective.

Extended previous results to higher degree number fields.

Abstract

Let $\alpha$ be an algebraic number of degree $d\ge 3$ and let $K$ be the algebraic number field $\Q(\alpha)$. When $\varepsilon$ is a unit of $K$ such that $\Q(\alpha\varepsilon)=K$, we consider the irreducible polynomial $f_\varepsilon(X) \in \Z[X]$ such that $f_\varepsilon(\alpha\varepsilon)=0$. Let $F_\varepsilon(X,Y)$ be the irrreducible binary form of degree $d$ associated to $f_{\varepsilon}(X) $ under the condition $F_{\varepsilon}(X,1)=f_{\varepsilon}(X)$. For each positive integer $m$, we want to exhibit an effective upper bound for the solutions $(x,y,\varepsilon)$ of the diophantine inequation $|F_\varepsilon(x,y)|\le m$. We achieve this goal by restricting ourselves to a subset of units $\varepsilon$ which we prove to be sufficiently large as soon as the degree of $K$ is $\geq 4$.