TY - JOUR

T1 - Stable and real-zero polynomials in two variables

AU - Grinshpan, Anatolii

AU - Kaliuzhnyi-Verbovetskyi, Dmitry S.

AU - Vinnikov, Victor

AU - Woerdeman, Hugo J.

N1 - Funding Information:
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”.
Publisher Copyright:
© 2014, Springer Science+Business Media New York.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - For every bivariate polynomial (Formula Presented.) of bidegree (Formula Presented.), which has no zeros in the open unit bidisk, we construct a determinantal representation of the form (Formula Presented.) diagonal matrix with coordinate variables (Formula Presented.) 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 (Formula Presented.), we provide a construction to build a representation of the form (Formula Presented.),where (Formula Presented.) 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).

AB - For every bivariate polynomial (Formula Presented.) of bidegree (Formula Presented.), which has no zeros in the open unit bidisk, we construct a determinantal representation of the form (Formula Presented.) diagonal matrix with coordinate variables (Formula Presented.) 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 (Formula Presented.), we provide a construction to build a representation of the form (Formula Presented.),where (Formula Presented.) 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).

KW - Determinantal representation

KW - Lax conjecture

KW - Multivariable polynomial

KW - Real-zero polynomial

KW - Self-reversive polynomial

KW - Stability radius

KW - Stable polynomial

UR - http://www.scopus.com/inward/record.url?scp=84953638074&partnerID=8YFLogxK

U2 - 10.1007/s11045-014-0286-3

DO - 10.1007/s11045-014-0286-3

M3 - Article

AN - SCOPUS:84953638074

SN - 0923-6082

VL - 27

SP - 1

EP - 26

JO - Multidimensional Systems and Signal Processing

JF - Multidimensional Systems and Signal Processing

IS - 1

ER -