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Generalized Pisot Numbers and Matrix Decomposition

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Advances in Dynamical Systems and Control

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 69))

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.

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Notes

  1. 1.

    Throughout this paper, by \(\alpha \) we denote real irrational number.

  2. 2.

    By irreducible polynomial f(x) with integer coefficients, we understand such polynomial that if \(f(x)=g(x)h(x)\), where \(\mathrm {deg}(g(x))\le \mathrm {deg}(h(x))\), and then \(g(x)=\pm 1\), \(h(x)=\mp f(x)\). In particular, irreducibility of a polynomial means \((a_0,\ldots ,a_n)=1\).

  3. 3.

    Here we suppose that only null form belongs to all \(\mathbb {PK}_n[X,Y]\).

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Acknowledgments

This research was supported by the Russian Foundation for Basic Research (Grant Nos 15-01-01540, 15-41-03262, 15-41-03263).

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Authors and Affiliations

  1. Tula State Lev Tolstoy Pedagogical University, 125, Lenina pr., Tula, 300026, Russia

    Nikolai M. Dobrovol’skii, Nikolai N. Dobrovolsky, Irina N. Balaba & Irina Yu. Rebrova

  2. Moscow State Pedagogical University, 1/1 M. Pirogovskaya Str., Moscow, 119991, Russian Federation

    Dmitrii K. Sobolev & Valentina N. Soboleva

Authors
  1. Nikolai M. Dobrovol’skii
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  2. Nikolai N. Dobrovolsky
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  3. Irina N. Balaba
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  4. Irina Yu. Rebrova
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  5. Dmitrii K. Sobolev
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  6. Valentina N. Soboleva
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Correspondence to Nikolai M. Dobrovol’skii .

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Editors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    Victor A. Sadovnichiy

  2. Kyiv Polytechnic Institute, National Technical University of Ukraine, Kiev, Ukraine

    Mikhail Z. Zgurovsky

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Dobrovol’skii, N.M., Dobrovolsky, N.N., Balaba, I.N., Rebrova, I.Y., Sobolev, D.K., Soboleva, V.N. (2016). Generalized Pisot Numbers and Matrix Decomposition. In: Sadovnichiy, V., Zgurovsky, M. (eds) Advances in Dynamical Systems and Control. Studies in Systems, Decision and Control, vol 69. Springer, Cham. https://doi.org/10.1007/978-3-319-40673-2_5

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  • DOI: https://doi.org/10.1007/978-3-319-40673-2_5

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