Periodic Jacobi-Perron expansions associated with a unit

Brigitte Adam[1]; Georges Rhin[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 (2011)

  • Volume: 23, Issue: 3, page 527-539
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Adam, Brigitte, and Rhin, Georges. "Periodic Jacobi-Perron expansions associated with a unit." Journal de Théorie des Nombres de Bordeaux 23.3 (2011): 527-539. <http://eudml.org/doc/219814>.

@article{Adam2011,
abstract = {We prove that, for any unit $\epsilon $ 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.},
affiliation = {2, rue clos du pré 57530 Courcelles-Chaussy, France; UMR CNRS 7122 Département de Mathématiques UFR MIM Université de Metz Ile du Saulcy 57045 Metz Cedex 01, France},
author = {Adam, Brigitte, Rhin, Georges},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Jacobi-Perron algorithms},
language = {eng},
month = {11},
number = {3},
pages = {527-539},
publisher = {Société Arithmétique de Bordeaux},
title = {Periodic Jacobi-Perron expansions associated with a unit},
url = {http://eudml.org/doc/219814},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Adam, Brigitte
AU - Rhin, Georges
TI - Periodic Jacobi-Perron expansions associated with a unit
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/11//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 3
SP - 527
EP - 539
AB - We prove that, for any unit $\epsilon $ 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.
LA - eng
KW - Jacobi-Perron algorithms
UR - http://eudml.org/doc/219814
ER -

References

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  2. B. Adam and G. Rhin, Algorithme des fractions continues et de Jacobi-Perron. Bull. Austral. Math. Soc. 53 (1996), 341–350. Zbl0858.11069MR1381775
  3. L. Bernstein, The Jacobi-Perron Algorithm, Its Theory and Application. Lecture Notes in Mathematics, 207, Springer-Verlag, Berlin-New York, 1971. Zbl0213.05201MR285478
  4. L. Bernstein, Einheitenberechnung in kubischen Körpern mittels des Jacobi-Perronschen Algorithmus aus der Rechenanlage. J. Reine Angew. Math. 244 (1970), 201–220. Zbl0205.35301MR271064
  5. L. Bernstein, A 3-Dimensional Periodic Jacobi-Perron Algorithm of Period Length 8. J. of Number Theory 4 (1972), no.1, 48–69. Zbl0244.10006MR294259
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  7. H. Cohen, A Course in Computational Algebraic Number Theory. Graduate Texts in Maths, 138, Springer Verlag, 2007. Zbl0786.11071MR1228206
  8. E. Dubois and R. Paysant-Le Roux, Une application des nombres de Pisot à l’algorithme de Jacobi-Perron. Monatschefte für Mathematik 98 (1984), 145–155. Zbl0543.10023MR776351
  9. E. Dubois, A. Farhane and R. Paysant-Le Roux, The Jacobi-Perron Algorithm and Pisot numbers. Acta Arith. 111 (2004), no 3, 269–275. Zbl1051.11037MR2039226
  10. J. C. Lagarias, The Quality of the Diophantine Approximations found by the Jacobi-Perron Algorithm and Related Algorithms. Monatschefte für Math. 115 (1993), 299–328. Zbl0790.11059MR1230366
  11. C. Levesque and G. Rhin, Two Families of Periodic Jacobi Algorithms with Period Lengths Going to Infinity. Jour. of Number Theory 37 (1991), no. 2, 173–180. Zbl0723.11032MR1092604
  12. C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, GP-Pari version 2.3.4 (2009). 
  13. O. Perron, Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Ann. 64 (1907), 1–76. Zbl38.0262.01MR1511422
  14. H. J. Stender, Eine Formel für Grundeinheiten in reinen algebraischen Zahlkörpern dritten, vierten und sechsten Grades. Jour. of Number Theory 7 (1975), no. 2, 235–250. Zbl0308.12001MR369317
  15. http://www.math.univ-metz.fr/~rhin 

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