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Gauss Reduction in Higher Dimensions

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

Abstract

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.

Keywords

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.
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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

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

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