On a family of unit equations over simplest cubic fields

Vukusic, Ingrid; Ziegler, Volker

doi:10.5802/jtnb.1223

Vukusic, Ingrid ¹ ; Ziegler, Volker ¹

¹ University of Salzburg Hellbrunnerstrasse 34/I A-5020 Salzburg, Austria

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 3, pp. 705-718.

Résumé
Abstract

Soient $a \in ℤ$ et $ρ$ une racine de $f_{a} (x) = x^{3} - a x^{2} - (a + 3) x - 1$ . On dit que le corps de nombres $K_{a} = ℚ (ρ)$ est un corps cubique élémentaire. Dans cet article, nous considérons la famille des équations en unités algébriques de la forme $u_{1} + u_{2} = n$ où $u_{1}, u_{2} \in ℤ {[ρ]}^{*}$ et $n \in ℤ$ . Nous résolvons complètement ces équations sous l’hypothèse $| n | \leq max \{1, {| a |}^{1 / 3}\}$ .

Let $a \in ℤ$ and $ρ$ be a root of $f_{a} (x) = x^{3} - a x^{2} - (a + 3) x - 1$ , then the number field $K_{a} = ℚ (ρ)$ is called a simplest cubic field. In this paper we consider the family of unit equations $u_{1} + u_{2} = n$ where $u_{1}, u_{2} \in ℤ {[ρ]}^{*}$ and $n \in ℤ$ . We completely solve the unit equations under the restriction $| n | \leq max {1, | a |^{1 / 3}}$ .

Reçu le : 2021-04-26
Accepté le : 2021-07-09
Publié le : 2023-01-27

DOI : 10.5802/jtnb.1223

Classification : 11D61, 11D75, 11J86
Mots clés : Diophantine equations, unit equations, simplest cubic fields

Affiliations des auteurs :

Vukusic, Ingrid ¹ ; Ziegler, Volker ¹

¹ University of Salzburg Hellbrunnerstrasse 34/I A-5020 Salzburg, Austria

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Vukusic, Ingrid; Ziegler, Volker. On a family of unit equations over simplest cubic fields. Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 3, pp. 705-718. doi : 10.5802/jtnb.1223. http://www.numdam.org/articles/10.5802/jtnb.1223/

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