Continued Fraction approach to Gauss Reduction Theory

TL;DR

This paper introduces a novel continued fraction method to compute reduced forms of matrices over integers, enhancing Gauss Reduction Theory with explicit period calculations of continued fractions for eigenvector slopes.

Contribution

It develops a new technique using continued fractions to effectively compute normal forms of GL(2,Z) matrices, improving upon existing Jordan normal form approaches over rings.

Findings

Provides explicit formulas for periods of continued fractions

Enhances computational methods in Gauss Reduction Theory

Offers a practical approach for classifying conjugacy classes

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.