Solving simultaneously Thue equations in the almost totally imaginary case

TL;DR

This paper develops an effective method to bound solutions of certain Thue equations associated with algebraic numbers of degree at least three, especially in the almost totally imaginary case, extending previous results.

Contribution

It introduces a new effective bound for solutions of Thue equations linked to algebraic numbers with specific conjugate properties, focusing on the almost totally imaginary case.

Findings

Provides explicit bounds for solutions of Thue equations in the almost totally imaginary case.

Extends previous work by considering algebraic numbers with at most one real conjugate.

Offers a computational approach to solve these Diophantine equations.

Abstract

Let $\alpha$ be an algebraic number of degree $d\ge 3$ having at most one real conjugate and let $K$ be the algebraic number field ${\mathbf Q}(\alpha)$. For any unit $\epsilon$ of $K$ such that ${\mathbf Q}(\alpha\epsilon)=K$, we consider the irreducible polynomial $f_\epsilon(X)\in{\mathbf Z}[X]$ such that $f_\epsilon(\alpha\epsilon)=0$. Let $F_\epsilon(X,Y)\ = Y^df_\epsilon(X/Y)\in{\mathbf Z}[X,Y]$ be the associated binary form. For each positive integer $m$, we exhibit an effectively computable bound for the solutions $(x,y,\epsilon)$ of the diophantine equation $|F_\epsilon(x,y)|\leq m$.