Gauss Reduction in Higher Dimensions

  • Oleg Karpenkov
Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 26)


In this chapter we continue to study integer conjugacy classes of integer matrices in general dimension. Namely we are aiming to contribute to the following problem: describe the set of integer conjugacy classes in \(\operatorname{SL}(n,{\mathbb{Z}})\). Gauss‘s reduction theory gives a complete geometric description of conjugacy classes for the case n=2, as we have already discussed in Chap.  7. In the multidimensional case the situation is more complicated. It is relatively simple to check whether two given matrices are integer conjugate, but to distinguish conjugacy classes is a much harder task. Using multidimensional Gauss’s reduction theory, we give the solution to this problem for matrices whose characteristic polynomials are irreducible over the field of rational numbers. We study questions related to the three-dimensional case in more detail.


Conjugacy Class Characteristic Polynomial Fundamental Domain Real Eigenvalue Diophantine Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 4.
    H. Appelgate, H. Onishi, The similarity problem for 3×3 integer matrices. Linear Algebra Appl. 42, 159–174 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 25.
    K. Briggs, Klein polyhedra.
  3. 27.
    A.D. Bryuno, V.I. Parusnikov, Klein polyhedra for two Davenport cubic forms. Math. Notes - Ross. Akad. 56(3–4), 994–1007 (1995). Russian version: Mat. Zametki 56(4), 9–27 (1994) MathSciNetGoogle Scholar
  4. 28.
    J. Buchmann, A generalization of Voronoĭ’s unit algorithm. I. J. Number Theory 20(2), 177–191 (1985) MathSciNetCrossRefGoogle Scholar
  5. 29.
    J. Buchmann, A generalization of Voronoĭ’s unit algorithm. II. J. Number Theory 20(2), 192–209 (1985) MathSciNetCrossRefGoogle Scholar
  6. 46.
    H. Davenport, On the product of three homogeneous linear forms. I. Proc. Lond. Math. Soc. 13, 139–145 (1938) Google Scholar
  7. 47.
    H. Davenport, On the product of three homogeneous linear forms. II. Proc. Lond. Math. Soc. (2) 44, 412–431 (1938) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 48.
    H. Davenport, On the product of three homogeneous linear forms. III. Proc. Lond. Math. Soc. (2) 45, 98–125 (1939) MathSciNetCrossRefGoogle Scholar
  9. 49.
    H. Davenport, Note on the product of three homogeneous linear forms. J. Lond. Math. Soc. 16, 98–101 (1941) MathSciNetCrossRefGoogle Scholar
  10. 50.
    H. Davenport, On the product of three homogeneous linear forms. IV. Proc. Camb. Philos. Soc. 39, 1–21 (1943) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 53.
    J.W. Demmel, Applied Numerical Linear Algebra (Society for Industrial and Applied Mathematics, Philadelphia, 1997) zbMATHCrossRefGoogle Scholar
  12. 73.
    F.J. Grunewald, Solution of the conjugacy problem in certain arithmetic groups, in Word Problems II, Oxford, 1976. Stud. Logic Foundations Math., vol. 95 (North-Holland, Amsterdam, 1980), pp. 101–139 CrossRefGoogle Scholar
  13. 78.
    K. Hessenberg, Thesis, Technische Hochschule, Darmstadt, Germany, 1942 Google Scholar
  14. 89.
    O. Karpenkov, On the triangulations of tori associated with two-dimensional continued fractions of cubic irrationalities. Funct. Anal. Appl. 38(2), 102–110 (2004). Russian version: Funkc. Anal. Prilozh. 38(2), 28–37 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 90.
    O. Karpenkov, On two-dimensional continued fractions of hyperbolic integer matrices with small norm. Russ. Math. Surv. 59(5), 959–960 (2004). Russian version: Usp. Mat. Nauk 59(5), 149–150 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 92.
    O. Karpenkov, Three examples of three-dimensional continued fractions in the sense of Klein. C. R. Math. Acad. Sci. Paris 343(1), 5–7 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 95.
    O. Karpenkov, Constructing multidimensional periodic continued fractions in the sense of Klein. Math. Comput. 78(267), 1687–1711 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 99.
    O. Karpenkov, Multidimensional Gauss reduction theory for conjugacy classes of SL(n,Z). Preprint (2012) Google Scholar
  19. 118.
    E.I. Korkina, Two-dimensional continued fractions. The simplest examples. Tr. Mat. Inst. Steklova 209, 143–166 (1995) MathSciNetGoogle Scholar
  20. 119.
    E.I. Korkina, The simplest 2-dimensional continued fraction. J. Math. Sci. 82(5), 3680–3685 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 123.
    G. Lachaud, Polyèdre d’Arnol’d et voile d’un cône simplicial: analogues du théorème de Lagrange. C. R. Acad. Sci., Sér. 1 Math. 317(8), 711–716 (1993) MathSciNetzbMATHGoogle Scholar
  22. 125.
    G. Lachaud, Voiles et Polyhedres de Klein (Act. Sci. Ind., Hermann, 2002) Google Scholar
  23. 135.
    A. Markoff, Sur les formes quadratiques binaires indéfinies. Math. Ann. 15(3–4), 381–406 (1879) zbMATHCrossRefGoogle Scholar
  24. 142.
    J.-O. Moussafir, Voiles et Polyédres de Klein: Geometrie, Algorithmes et Statistiques. Docteur en sciences thèse, Université Paris IX—Dauphine, 2000 Google Scholar
  25. 153.
    R. Okazaki, On an effective determination of a Shintani’s decomposition of the cone \(\mathbf{R}^{n}_{+}\). J. Math. Kyoto Univ. 33(4), 1057–1070 (1993) MathSciNetzbMATHGoogle Scholar
  26. 154.
    J.M. Ortega, H.F. Kaiser, The LL T and QR methods for symmetric tridiagonal matrices. Comput. J. 6, 99–101 (1963/1964) MathSciNetCrossRefGoogle Scholar
  27. 161.
    V.I. Parusnikov, Klein’s polyhedra for the seventh extremal cubic form. Technical report, preprint 79, Keldysh Institute of the RAS, Moscow (1999) Google Scholar
  28. 162.
    V.I. Parusnikov, Klein polyhedra for the fourth extremal cubic form. Math. Notes - Ross. Akad. 67(1–2), 87–102 (2000). Russian version: Mat. Zametki 67(1), 110–128 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 163.
    V.I. Parusnikov, Klein polyhedra for three extremal cubic forms. Math. Notes - Ross. Akad. 77(4), 566–583 (2005). Russian version: Mat. Zametki 77(3–4), 523–538 (2000) MathSciNetGoogle Scholar
  30. 185.
    T. Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-positive integers. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 23(2), 393–417 (1976) MathSciNetzbMATHGoogle Scholar
  31. 193.
    J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, 3rd edn. Texts in Applied Mathematics, vol. 12 (Springer, New York, 2002). Translated from the German by R. Bartels, W. Gautschi and C. Witzgall zbMATHGoogle Scholar
  32. 198.
    L.N. Trefethen, D. Bau, Numerical Linear Algebra (Society for Industrial and Applied Mathematics, Philadelphia, 1997) zbMATHCrossRefGoogle Scholar
  33. 205.
    G.F. Voronoĭ, On a Generalization of the Algorithm of Continued Fraction. Collected Works in Three Volumes (USSR Ac. Sci, Kiev, 1952) (in Russian) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Oleg Karpenkov
    • 1
  1. 1.Dept. of Mathematical SciencesUniversity of LiverpoolLiverpoolUK

Personalised recommendations