Generalized Pisot Numbers and Matrix Decomposition

Abstract

We consider the linear fractional transformations of polynomials and the linear transformations of homogeneous binary forms and study their properties. A definition of generalized Pisot number is given. This definition differs from definition of Pisot numbers by the absence of a requirement to be integer. In the case of totally real algebraic fields reduced generalized numbers Pisot are reduced algebraic irrationalities. It is shown that for arbitrary real algebraic irrationality \(\alpha \) of degree \(n\ge 2\), a sequence of residual fractions \(\alpha _m\) is a sequence of the reduced generalized numbers Pisot starting from some index \(m_0=m_0(\alpha )\). The asymptotic formula for conjugate numbers to residual fractions of generalized numbers Pisot is found. We study properties of the minimal polynomials of the residual fractions in the continued fraction expansion of the algebraic numbers. The recurrence formulas to find the minimum polynomials of the residual fractions using linear fractional transformations are given.