Journal of Mathematical Sciences

, Volume 141, Issue 6, pp 1663–1667 | Cite as

To solving multiparameter problems of algebra. 9. The ΨF-q method for factorizing invariant polynomials and its applications

  • V. N. Kublanovskaya


A new method (the ΨF-q method) for computing the invariant polynomials of a q-parameter (q ≥ 1) polynomial matrix F is suggested. Invariant polynomials are computed in factored form, which permits one to analyze the structure of the regular spectrum of the matrix F, to isolate the divisors of each of the invariant polynomials whose zeros belong to the invariant polynomial in question, to find the divisors whose zeros belong to at least two of the neighboring invariant polynomials, and to determine the heredity levels of points of the spectrum for each of the invariant polynomials. Applications of the ΨF-q method to representing a polynomial matrix F(λ) as a product of matrices whose spectra coincide with the zeros of the corresponding divisors of the characteristic polynomial and, in particular, with the zeros of an arbitrary invariant polynomial or its divisors are considered. Bibliography: 5 titles.


Factor Form Characteristic Polynomial Factorization Method Polynomial Matrix Great Common Divisor 
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  1. 1.
    V. N. Kublanovskaya, “Methods and algorithms for solving spectral problems for polynomial and rational matrices,” Zap. Nauchn. Semin. POMI, 238, 3–329 (1997).Google Scholar
  2. 2.
    V. N. Kublanovskaya and V. B. Khazanov, Numerical Methods for Solving Parametric Problems of Algebra. Part I. One-Parameter Problems [in Russian], Nauka, St.Petersburg (2004).Google Scholar
  3. 3.
    V. N. Kublanovskaya, “To solving multiparameter problems of algebra. 5,” Zap. Nauchn. Semin. POMI, 309, 144–153 (2004).zbMATHGoogle Scholar
  4. 4.
    V. B. Khazanov, “Methods for solving some parameter problems of algebra,” Zap. Nauchn. Semin. POMI, 323, 164–181 (2005).zbMATHMathSciNetGoogle Scholar
  5. 5.
    V. N. Kublanovskaya, “To solving multiparameter problems of algebra. 8,” Zap. Nauchn. Semin. POMI, 334, 149–164 (2006).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • V. N. Kublanovskaya
    • 1
  1. 1.St.Petersburg Department of the Steklov Mathematical InstituteSt.PetersburgRussia

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