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Stable and real-zero polynomials in two variables

  • Anatolii Grinshpan
  • Dmitry S. Kaliuzhnyi-Verbovetskyi
  • Victor Vinnikov
  • Hugo J. Woerdeman
Article

Abstract

For every bivariate polynomial \(p(z_1, z_2)\) of bidegree \((n_1, n_2)\), with \(p(0,0)=1\), which has no zeros in the open unit bidisk, we construct a determinantal representation of the form
$$\begin{aligned} p(z_1,z_2)=\det (I - K Z ), \end{aligned}$$
where \(Z\) is an \((n_1+n_2)\times (n_1+n_2)\) diagonal matrix with coordinate variables \(z_1\), \(z_2\) on the diagonal and \(K\) is a contraction. We show that \(K\) may be chosen to be unitary if and only if \(p\) is a (unimodular) constant multiple of its reverse. Furthermore, for every bivariate real-zero polynomial \(p(x_1, x_2),\) with \(p(0,0)=1\), we provide a construction to build a representation of the form
$$\begin{aligned} p(x_1,x_2)=\det (I+x_1A_1+x_2A_2), \end{aligned}$$
where \(A_1\) and \(A_2\) are Hermitian matrices of size equal to the degree of \(p\). A key component of both constructions is a stable factorization of a positive semidefinite matrix-valued polynomial in one variable, either on the circle (trigonometric polynomial) or on the real line (algebraic polynomial).

Keywords

Determinantal representation Multivariable polynomial Stable polynomial Stability radius Self-reversive polynomial Real-zero polynomial Lax conjecture 

Notes

Acknowledgments

The authors wish to thank Greg Knese for helpful comments. Victor Vinnikov would like to thank Forschungsschwerpunkt Reelle Geometrie und Algebra at the University of Konstanz for its hospitality, and its members, especially Christoph Hanselka, Daniel Plaumann, and Markus Schweighofer, for useful discussions.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Anatolii Grinshpan
    • 1
  • Dmitry S. Kaliuzhnyi-Verbovetskyi
    • 1
  • Victor Vinnikov
    • 2
  • Hugo J. Woerdeman
    • 1
  1. 1.Department of MathematicsDrexel UniversityPhiladelphiaUSA
  2. 2.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael

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