On two-dimensional continued fractions for the integer hyperbolic matrices with small norm

TL;DR

This paper classifies two-dimensional continued fractions for cubic irrationalities generated by small norm matrices, revealing a link between matrix properties and solutions to specific integer equations.

Contribution

It introduces a new classification criterion for matrices of Frobenius type based on the existence of integer solutions to certain equations.

Findings

Classified matrices with small norm ($|*| \u2264 6$) for two-dimensional continued fractions.

Established a criterion linking irreducible characteristic polynomials to Frobenius type matrices.

Provided a method to identify matrices of Frobenius type via integer solutions to polynomial equations.

Abstract

In this note we classify two-dimensional continued fractions for cubic irrationalities constructed by matrices with not large norm ($|*| \le 6$). The classification is based on the following new result: the class of matrices with an irreducible characteristic polynomial over the field of rational numbers is the class of matrices of frobenius type iff there exists an integer solution for a certain equation with integer coefficients.