On a family of unit equations over simplest cubic fields

Ingrid Vukusic; Volker Ziegler

doi:10.5802/jtnb.1223

Ingrid Vukusic ¹; Volker Ziegler ¹

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

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

Abstract
Résumé

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 $| n | \leq max {1, | a |^{1 / 3}}$ .

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 $| n | \leq max \{1, {| a |}^{1 / 3}\}$ .

Received: 2021-04-26
Accepted: 2021-07-09
Published online: 2023-01-27

DOI: 10.5802/jtnb.1223

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

Author's affiliations:

Ingrid Vukusic ¹; Volker Ziegler ¹

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

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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

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