Familles d'\'equations de Thue associ\'ees \`a un sous-groupe de rang 1 d'unit\'es totalement r\'eelles d'un corps de nombres

TL;DR

This paper studies a family of Thue equations derived from binary forms associated with a number field and a totally real unit, providing effective bounds on solutions involving these forms.

Contribution

It introduces a method to bound solutions of Thue equations twisted by powers of a totally real unit in a number field.

Findings

Effective bounds for solutions of twisted Thue equations.

Application to binary forms associated with number fields.

Explicit bounds involving parameters a, x, y, and m.

Abstract

Let $F$ be an irreducible binary form attached to a number field $K$ of degree $\geq 3$. Let $\epsilon\not\in \{-1, 1\}$ be a totally real unit of $K$. By twisting $F$ with the powers $\epsilon^a$ of $\epsilon$, ($a\in{\mathbf Z}$), we obtain an infinite family $F_a$ of binary forms. Let $m\in{\mathbf Z}$. We give an effective bound for $\max\{|a|, \log|x|, \log|y|\}$ when $a,x,y$ are rational integers satisfying $F_a(x,y)=m$ with $xy\not=0$.