Total nonnegativity of finite Hurwitz matrices and root location of polynomials

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2017
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Zusammenfassung

In 1970, B.A. Asner, Jr., proved that for a real quasi-stable polynomial, i.e., a polynomial whose zeros lie in the closed left half-plane of the complex plane, its finite Hurwitz matrix is totally nonnegative, i.e., all its minors are nonnegative, and that the converse statement is not true. In this work, we explain this phenomenon in detail, and provide necessary and sufficient conditions for a real polynomial to have a totally nonnegative finite Hurwitz matrix.

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Fachgebiet (DDC)
510 Mathematik
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Hurwitz matrix, totally nonnegative matrix, stable polynomial, quasi-stable polynomial, R-function
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ISO 690ADM, Mohammad, Jürgen GARLOFF, Mikhail TYAGLOV, 2017. Total nonnegativity of finite Hurwitz matrices and root location of polynomials
BibTex
@techreport{Adm2017-11-09T23:17:28ZTotal-40767,
  year={2017},
  series={Konstanzer Schriften in Mathematik},
  title={Total nonnegativity of finite Hurwitz matrices and root location of polynomials},
  number={368},
  author={Adm, Mohammad and Garloff, Jürgen and Tyaglov, Mikhail}
}
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    <dcterms:abstract xml:lang="eng">In 1970, B.A. Asner, Jr., proved that for a real quasi-stable polynomial, i.e., a polynomial whose zeros lie in the closed left half-plane of the complex plane, its finite Hurwitz matrix is totally nonnegative, i.e., all its minors are nonnegative, and that the converse statement is not true. In this work, we explain this phenomenon in detail, and provide necessary and sufficient conditions for a real polynomial to have a totally nonnegative finite Hurwitz matrix.</dcterms:abstract>
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