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Etude des interruptions dans l'algorithme de Jacobi–Perron

Published online by Cambridge University Press:  17 April 2009

Eugène Dubois ,
Ahmed Farhane  and
Roger Paysant-Le Roux
Eugène Dubois
Affiliation:
Laboratoire de Mathématiques Nicolas Oresme, UMR CNRS n∘ 6139, Université Caen, 14032 Caen-Cedex, France e-mail: eugene.dubois@math.unicaen.fr
Ahmed Farhane
Affiliation:
Département de Mathématiques et Informatique, Faculté des Sciences et Techniques de SETTAT, Maroc, e-mail: ahmed_farhane@yahoo.fr
Roger Paysant-Le Roux
Affiliation:
Laboratoire Nicolas ORESME, UMR CNRS n∘ 6139, Université de Caen14032 Caen-CedexFrance e-mail: Roger.Paysant-Lerous@math.unicaen.fr
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It is well known that in general the Jacobi–Perron algorithm (a multi-dimensional analogue of the continued fraction algorithm) may or might not acknowledge the dependence over ℚ of its arguments 1, α1,…, αn by truncating itself down to fewer arguments from some step onwards (if so, the algorithm is said to display an ‘interruption’). We show here that if n = 2 then 1, α1, α2 are linearly dependent over ℚ of and only if the Jacobi–Perron Algorithm displays an interruption. We give examples showing this is not so for any n ≥ 3.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 69 , Issue 2 , April 2004 , pp. 241 - 254
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Adam, B. and Rhin, G., ‘Algorithme des fractions continues et de Jacobi–Perron’, Bull. Aust. Math. Soc. 53 (1996), 341–350.CrossRefGoogle Scholar
[2]Bernstein, L., The Jacobi Perron. Its theory and application, Lecture Notes in Mathematics 207 (Springer-Verlag, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar
[3]Dubois, E., ‘Thèse de 3ème cycle’, (Caen 1970), C.R. Acad. Sci. Paris (1971), 564–566.Google Scholar
[4]Dubois, E., Fahrane, A. and Paysant-Le Roux, R., ‘The Jacobi–Perron algorithm and Pisot numbers’, Acta Arith. (to appear).Google Scholar
[5]Dubois, E. and Paysant-Le Roux, R., ‘Une application des nombres de Pisot à l'algorithme de Jacobi–Perron’, Monatsh. Math. 98 (1984), 145–155.Google Scholar
[6]Jacobi, C.G., ‘Allgemeine Theorie der kettenbruchaehnlichen Algorithmen, in welchen pede Zahl ans drei varhergehenden gebildlt wird’, J.f.d. reine Angew. Math. 69 (1869), 29–64.Google Scholar
[7]Paysant-Le Roux, R., ‘Thèse de 3ème’ cycle, (Caen 1970), C.R. Acad. Sci. Paris (1971), 649–652.Google Scholar
[8]Perron, O., ‘Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus’, Math. Ann. 64 (1907), 1–76.CrossRefGoogle Scholar
[9]Schweiger, F., The metrical theory of Jacobi–Perron Algorithm, Lecture Note in Mathematics 334 (Springer Verlag, Berlin, New York, 1973).CrossRefGoogle Scholar