Periodic Jacobi-Perron expansions associated with a unit

Adam, Brigitte; Rhin, Georges

doi:10.5802/jtnb.776

Adam, Brigitte ¹; Rhin, Georges ²

¹ 2, rue clos du pré 57530 Courcelles-Chaussy, France
² UMR CNRS 7122 Département de Mathématiques UFR MIM Université de Metz Ile du Saulcy 57045 Metz Cedex 01, France

Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 3, pp. 527-539.

Abstract
Résumé

We prove that, for any unit $ϵ$ in a real number field $K$ of degree $n + 1$ , there exits only a finite number of n-tuples in $K^{n}$ which have a purely periodic expansion by the Jacobi-Perron algorithm. This generalizes the case of continued fractions for $n = 1$ . For $n = 2$ we give an explicit algorithm to compute all these pairs.

Nous démontrons que, pour toute unité $ϵ$ dans un corps de nombres réel $K$ de degré $n + 1$ , il existe seulement un nombre fini de $n$ -uples dans $K^{n}$ qui ont un développement purement périodique par l’algorithme de Jacobi-Perron. Ce résultat généralise le cas des fractions continues pour $n = 1$ . Pour $n = 2$ nous donnons un algorithme qui permet de calculer explicitement tous ces couples.

EuDML MR Zbl | 1 citation in Numdam

DOI: 10.5802/jtnb.776

Author's affiliations:

Adam, Brigitte ¹; Rhin, Georges ²

¹ 2, rue clos du pré 57530 Courcelles-Chaussy, France
² UMR CNRS 7122 Département de Mathématiques UFR MIM Université de Metz Ile du Saulcy 57045 Metz Cedex 01, France

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     title = {Periodic {Jacobi-Perron} expansions associated with a unit},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {527--539},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {23},
     number = {3},
     year = {2011},
     doi = {10.5802/jtnb.776},
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Adam, Brigitte;  Rhin, Georges. Periodic Jacobi-Perron expansions associated with a unit. Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 3, pp. 527-539. doi : 10.5802/jtnb.776. http://www.numdam.org/articles/10.5802/jtnb.776/

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