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

Abstract

In this note we classify two-dimensional continued fractions for cubic irrationalities constructed by matrices with not large norm ($|\ast |\le 6$). 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.