Twisted Thue equations with multiple exponents in fixed number fields

Tobias Hilgart; Volker Ziegler

doi:10.5802/jtnb.1290

Tobias Hilgart ¹ ; Volker Ziegler ¹

¹ Hellbrunnerstraße 34 Austria

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 2, pp. 621-635.

Résumé
Abstract

Soit $K$ un corps de nombres de degré $d \geq 3 .$ On fixe $s \leq d - 2$ éléments multiplicativement indépendants et remplissant certaines conditions techniques, qui se réduisent à une condition d’indépendance $ℚ$ -linéaire si on admet la conjecture de Schanuel. Nous considérons l’équation de Thue tordue

| N_{K / ℚ} (X - γ_{1}^{t_{1}} \dots γ_{s}^{t_{s}} Y) | = 1,

et prouvons qu’il n’existe qu’un nombre fini de solutions $(x, y; t_{1}, \dots, t_{s})$ dans $ℤ^{2} \times ℕ^{s}$ avec $x y \neq 0$ et $ℚ (γ_{1}^{t_{1}} \dots γ_{s}^{t_{s}}) = K$ . Ces solutions sont effectivement calculables.

Let $K$ be a number field of degree $d \geq 3$ and fix $s \leq d - 2$ multiplicatively independent $γ_{1}, \dots, γ_{s} \in K^{*}$ that fulfil some technical requirements, which can be vastly simplified to $ℚ$ -linearly independence, given Schanuel’s conjecture. We then consider the twisted Thue equation

| N_{K / ℚ} (X - γ_{1}^{t_{1}} \dots γ_{s}^{t_{s}} Y) | = 1,

and prove that it has only finitely many solutions $(x, y; t_{1}, \dots, t_{s})$ in $ℤ^{2} \times ℕ^{s}$ with $x y \neq 0$ and $ℚ (γ_{1}^{t_{1}} \dots γ_{s}^{t_{s}}) = K$ , all of which are effectively computable.

Reçu le : 2022-12-14
Accepté le : 2023-03-16
Publié le : 2024-11-13

DOI : 10.5802/jtnb.1290

Classification : 11N56, 14G42
Mots clés : Multiplicative and norm form equations, Exponential Diophantine equations

Affiliations des auteurs :

Tobias Hilgart ¹ ; Volker Ziegler ¹

¹ Hellbrunnerstraße 34 Austria

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Tobias Hilgart; Volker Ziegler. Twisted Thue equations with multiple exponents in fixed number fields. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 2, pp. 621-635. doi : 10.5802/jtnb.1290. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1290/

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