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Stable and real-zero polynomials in two variables

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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).

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Notes

  1. We note that a less common terminology was used in Grinshpan et al. (2013): stable polynomials were called semi-stable, and strongly stable polynomials were called stable.

  2. To see that this assumption is indeed generic, notice first that it means that all the roots of the resultant determinant \(r(z_1)\) of \(p(z_1,\cdot )\) and \(\overleftarrow{p}(z_1,\cdot )\) are simple—i.e., the discriminant of \(r(z_1)\) is not zero—and that \(z_1=0\) is not a root of \(r(z_1)\), and second that the coefficients of \(r(z_1)\) are polynomials in the coefficients of \(p\) and of \(\overleftarrow{p}\), i.e., polynomials in the coefficients of \(p\) and in their conjugates.

  3. Alternatively, one may use the formula

    $$\begin{aligned} \frac{ p(z_1, z_2) \overline{p(1/\bar{z_1}, z_2 )} - \overleftarrow{p} (z_1, z_2) \overline{\overleftarrow{p} (1/\bar{z_1}, z_2 )}}{1-|z_2|^2} = v_{n_2-1}(z_2) Q(z_1) v_{n_2 -1} (z_2)^* \end{aligned}$$

    to come to the same conclusion. This formula can be easily checked by hand, but also appears in many sources; see, e.g., (Kailath et al. (1978), Section 4).

  4. This was previously noticed in (Speyer (2005), Lemma 8) and (Plaumann and Vinzant (2013), Lemma 3.4).

  5. Alternatively, we can prove that a zero \(a\) of \(P(x_2)\) is not a pole of \(M(x_2)\) similarly to the proof of the claim in the proof of Theorem 2.1. It is well known that \(\det B(a) = 0\) if and only if the polynomials \(\check{p}_{a}\) and \(\check{q}_{a}\) have a common zero \(\lambda \); let us assume that \(\lambda \) is a simple zero of both \(\check{p}_{a}\) and \(\check{q}_{a}\), then it is also well known that the left kernel of \(B(a)\) is spanned by \(v_{d-1}(\lambda )\) (all these facts follow quite easily from (4.12)–(4.13)). Since \(B(a) = P(a) P(\bar{a})^*\), and since \(v_{d-1}(\lambda ) C(a) = \lambda v_{d-1}(\lambda )\), it follows that the one-dimensional left kernel of \(P(a)\) is the left eigenspace of \(C(a)\), implying as in the proof of Theorem 2.1 that \(a\) is not a pole of \(M(x_2)\).

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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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Authors and Affiliations

  1. Department of Mathematics, Drexel University, 3141 Chestnut St., Philadelphia, PA, 19104, USA

    Anatolii Grinshpan, Dmitry S. Kaliuzhnyi-Verbovetskyi & Hugo J. Woerdeman

  2. Department of Mathematics, Ben-Gurion University of the Negev, 84105 , Beer-Sheva, Israel

    Victor Vinnikov

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  1. Anatolii Grinshpan
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  2. Dmitry S. Kaliuzhnyi-Verbovetskyi
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  3. Victor Vinnikov
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Correspondence to Dmitry S. Kaliuzhnyi-Verbovetskyi.

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A.G., D.K.-V., H.W. were partially supported by NSF Grant DMS-0901628. D.K.-V. and V.V. were partially supported by BSF Grant 2010432. V.V. was partially supported by the Institute for Mathematical Sciences of the National University of Singapore within the framework of the program “Inverse Moment Problems: the Crossroads of Analysis, Algebra, Discrete Geometry and Combinatorics”.

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Grinshpan, A., Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V. et al. Stable and real-zero polynomials in two variables. Multidim Syst Sign Process 27, 1–26 (2016). https://doi.org/10.1007/s11045-014-0286-3

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