Continued Fraction approach to Gauss Reduction Theory



Karpenkov, Oleg ORCID: 0000-0002-3358-6998
(2021) Continued Fraction approach to Gauss Reduction Theory. .

[img] PDF
rm.pdf - Author Accepted Manuscript

Download (111kB) | Preview

Abstract

Jordan Normal Forms serve as excellent representatives of conjugacy classes of matrices over closed fields. Once we knows normal forms, we can compute functions of matrices, their main invariant, etc. The situation is much more complicated if we search for normal forms for conjugacy classes over fields that are not closed and especially for rings. In this paper we study PGL(2,Z)-conjugacy classes of GL(2,Z) matrices. For the ring of integers Jordan approach has various limitations and in fact it is not effective. The normal forms of conjugacy classes of GL(2,Z) matrices are provided by alternative theory, which is known as Gauss Reduction Theory. We introduce a new techniques to compute reduced forms In Gauss Reduction Theory in terms of the elements of certain continued fractions. Current approach is based on recent progress in geometry of numbers. The proposed technique provides an explicit computation of periods of continued fractions for the slopes of eigenvectors.

Item Type: Conference or Workshop Item (Unspecified)
Additional Information: 18 pages
Uncontrolled Keywords: 15A21, 11H06, 15A36, math.NT, math.NT
Divisions: Faculty of Science and Engineering > School of Physical Sciences
Depositing User: Symplectic Admin
Date Deposited: 22 Mar 2024 15:56
Last Modified: 22 Mar 2024 16:21
DOI: 10.1007/978-3-030-89716-1_7
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3179812