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Mathematics of Computation

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ISSN 1088-6842 (online) ISSN 0025-5718 (print)

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Constructing multidimensional periodic continued fractions in the sense of Klein
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by O. N. Karpenkov PDF
Math. Comp. 78 (2009), 1687-1711 Request permission


We consider the geometric generalization of ordinary continued fractions to the multidimensional case introduced by F. Klein in 1895. A multidimensional periodic continued fraction is the union of sails with some special group acting freely on these sails. This group transposes the faces. In this article, we present a method of constructing “approximate” fundamental domains of algebraic multidimensional continued fractions and an algorithm testing whether this domain is indeed fundamental or not. We give some polynomial estimates on the number of the operations for the algorithm. In conclusion we present an example of a fundamental domain calculation for a two-dimensional series of two-dimensional periodic continued fractions.
  • V. I. Arnold, Continued fractions, Moscow: Moscow Center of Continuous Mathematical Education, (2002).
  • V. I. Arnold, Higher-dimensional continued fractions, Regul. Chaotic Dyn. 3 (1998), no. 3, 10–17 (English, with English and Russian summaries). J. Moser at 70 (Russian). MR 1704965, DOI 10.1070/rd1998v003n03ABEH000076
  • Z. I. Borevich and I. R. Shafarevich, Teoriya chisel, 3rd ed., “Nauka”, Moscow, 1985 (Russian). MR 816135
  • K. Briggs, Klein polyhedra,, (2002).
  • A. D. Bryuno and V. I. Parusnikov, Klein polyhedra for two Davenport cubic forms, Mat. Zametki 56 (1994), no. 4, 9–27, 156 (Russian, with Russian summary); English transl., Math. Notes 56 (1994), no. 3-4, 994–1007 (1995). MR 1330372, DOI 10.1007/BF02362367
  • Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
  • O. N. German, Sails and Hilbert bases, Tr. Mat. Inst. Steklova 239 (2002), no. Diskret. Geom. i Geom. Chisel, 98–105 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 4(239) (2002), 88–95. MR 1975137
  • O. N. German, Sails and norm minima of lattices, Mat. Sb. 196 (2005), no. 3, 31–60 (Russian, with Russian summary); English transl., Sb. Math. 196 (2005), no. 3-4, 337–365. MR 2144275, DOI 10.1070/SM2005v196n03ABEH000883
  • C. Hermite, Letter to C. D. J. Jacobi, J. Reine Angew. Math. 40, (1839), p. 286.
  • A. Ya. Hinchin, Continued fractions, Moscow: FISMATGIS, (1961).
  • O. N. Karpenkov, On the triangulations of tori associated with two-dimensional continued fractions of cubic irrationalities, Funktsional. Anal. i Prilozhen. 38 (2004), no. 2, 28–37, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 38 (2004), no. 2, 102–110. MR 2086625, DOI 10.1023/B:FAIA.0000034040.08573.22
  • O. N. Karpenkov, On two-dimensional continued fractions of hyperbolic integer matrices with small norm, Uspekhi Mat. Nauk 59 (2004), no. 5(359), 149–150 (Russian); English transl., Russian Math. Surveys 59 (2004), no. 5, 959–960. MR 2125934, DOI 10.1070/RM2004v059n05ABEH000778
  • O. N. Karpenkov, On examples of two-dimensional periodic continued fractions, preprint, Cahiers du Ceremade, UMR 7534, Université Paris-Dauphine, (2004).
  • O. N. Karpenkov, Classification of three-dimensional multistoried completely hollow convex marked pyramids, Uspekhi Mat. Nauk 60 (2005), no. 1(361), 169–170 (Russian); English transl., Russian Math. Surveys 60 (2005), no. 1, 165–166. MR 2145668, DOI 10.1070/RM2005v060n01ABEH000816
  • F. Klein, Über eine geometrische Auffassung der gewöhnlichen Kettenbruchentwicklung, Nachr. Ges. Wiss. Göttingen Math-Phys. Kl., 3, (1895), pp. 357–359.
  • F. Klein, Sur une représentation géométrique de développement en fraction continue ordinaire, Nouv. Ann. Math. 15(3), (1896), pp. 327–331.
  • M. L. Kontsevich and Yu. M. Suhov, Statistics of Klein polyhedra and multidimensional continued fractions, Pseudoperiodic topology, Amer. Math. Soc. Transl. Ser. 2, vol. 197, Amer. Math. Soc., Providence, RI, 1999, pp. 9–27. MR 1733869, DOI 10.1090/trans2/197/02
  • E. I. Korkina, The simplest 2-dimensional continued fraction, International Geometrical Colloquium, Moscow 1993.
  • Elena Korkina, La périodicité des fractions continues multidimensionnelles, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 8, 777–780 (French, with English and French summaries). MR 1300940
  • E. I. Korkina, Two-dimensional continued fractions. The simplest examples, Trudy Mat. Inst. Steklov. 209 (1995), no. Osob. Gladkikh Otobrazh. s Dop. Strukt., 143–166 (Russian). MR 1422222
  • E. I. Korkina, The simplest $2$-dimensional continued fraction, J. Math. Sci. 82 (1996), no. 5, 3680–3685. Topology, 3. MR 1428725, DOI 10.1007/BF02362573
  • Gilles Lachaud, Polyèdre d’Arnol′d et voile d’un cône simplicial: analogues du théorème de Lagrange, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 8, 711–716 (French, with English and French summaries). MR 1244417
  • G. Lachaud, Voiles et Polyèdres de Klein, preprint n 95-22, Laboratoire de Mathématiques Discrètes du C.N.R.S., Luminy (1995).
  • Gilles Lachaud, Sails and Klein polyhedra, Number theory (Tiruchirapalli, 1996) Contemp. Math., vol. 210, Amer. Math. Soc., Providence, RI, 1998, pp. 373–385. MR 1478504, DOI 10.1090/conm/210/02797
  • A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515–534. MR 682664, DOI 10.1007/BF01457454
  • Hermann Minkowski, Généralisation de la théorie des fractions continues, Ann. Sci. École Norm. Sup. (3) 13 (1896), 41–60 (French). MR 1508923
  • Zh.-O. Mussafir, Sails and Hilbert bases, Funktsional. Anal. i Prilozhen. 34 (2000), no. 2, 43–49, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 34 (2000), no. 2, 114–118. MR 1773843, DOI 10.1007/BF02482424
  • J.-O. Moussafir, Voiles et Polyédres de Klein: Geometrie, Algorithmes et Statistiques, docteur en sciences thése, Université Paris IX - Dauphine, (2000), see also˜msfr/
  • Ryotaro Okazaki, On an effective determination of a Shintani’s decomposition of the cone $\mathbf R^n_{+}$, J. Math. Kyoto Univ. 33 (1993), no. 4, 1057–1070. MR 1251215, DOI 10.1215/kjm/1250519129
  • V.I. Parusnikov, Klein’s polyhedra for the third extremal ternary cubic form, preprint 137 of Keldysh Institute of the RAS, Moscow, (1995).
  • V.I. Parusnikov, Klein’s polyhedra for the fifth extremal cubic form, preprint 69 of Keldysh Institute of the RAS, Moscow, (1998).
  • V.I. Parusnikov, Klein’s polyhedra for the seventh extremal cubic form, preprint 79 of Keldysh Institute of the RAS, Moscow, (1999).
  • V. I. Parusnikov, Klein polyhedra for the fourth extremal cubic form, Mat. Zametki 67 (2000), no. 1, 110–128 (Russian, with Russian summary); English transl., Math. Notes 67 (2000), no. 1-2, 87–102. MR 1763552, DOI 10.1007/BF02675796
  • Takuro Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 2, 393–417. MR 427231
  • B. F. Skubenko, Minima of a decomposible cubic form of three variables, Sci. Seminar Notes LOMI, 168, (1988), Analytic Number Theory and Theory of Functions, 9, Leningrad, “Nauka”.
  • B. F. Skubenko, Minima of decomposible forms of degree $n$ of $n$ variables for $n \ge 3$, Sci. Seminar Notes LOMI, 183, (1990), Modular functions and quadratic forms, 1, Leningrad, “Nauka”.
  • E. Thomas and A. T. Vasquez, On the resolution of cusp singularities and the Shintani decomposition in totally real cubic number fields, Math. Ann. 247 (1980), no. 1, 1–20. MR 565136, DOI 10.1007/BF01359864
  • Hiroyasu Tsuchihashi, Higher-dimensional analogues of periodic continued fractions and cusp singularities, Tohoku Math. J. (2) 35 (1983), no. 4, 607–639. MR 721966, DOI 10.2748/tmj/1178228955
  • G. F. Voronoi, On a generalization of continued fraction algorithm, USSR Ac. Sci., 1, (1952), pp. 197–391.
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Additional Information
  • O. N. Karpenkov
  • Affiliation: Poncelet Laboratory (UMI 2615 of CNRS and Independent University of Moscow)
  • Email:
  • Received by editor(s): May 12, 2005
  • Received by editor(s) in revised form: June 9, 2008
  • Published electronically: October 24, 2008
  • Additional Notes: The author was supported by SS-1972.2003.1 and RFBR-05-01-01012a grants.
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 78 (2009), 1687-1711
  • MSC (2000): Primary 11J70; Secondary 11Y16
  • DOI:
  • MathSciNet review: 2501070