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Journal article
Stable and real-zero polynomials in two variables
Multidimensional systems and signal processing, v 27(1), pp 1-26
01 Jan 2016
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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 p( z1, z2) = det( I - K Z), where Z is an ( n1 + n2) x( n1 + n2) diagonal matrix with coordinate variables z1, z2 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 p( x1, x2) = det( I + x(1)A(1) + x2 A(2)), where A(1) and A2 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 poly-nomial in one variable, either on the circle ( trigonometric polynomial) or on the real line ( algebraic polynomial).
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Details
- Title
- Stable and real-zero polynomials in two variables
- Creators
- Anatolii Grinshpan - Drexel UniversityDmitry S. Kaliuzhnyi-Verbovetskyi - Drexel UniversityVictor Vinnikov - Ben-Gurion University of the NegevHugo J. Woerdeman - Drexel University
- Publication Details
- Multidimensional systems and signal processing, v 27(1), pp 1-26
- Publisher
- Springer Nature
- Number of pages
- 26
- Grant note
- Institute for Mathematical Sciences of the National University of Singapore; National University of Singapore 2010432 / BSF; US-Israel Binational Science Foundation DMS-0901628 / NSF; National Science Foundation (NSF)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000367895200001
- Scopus ID
- 2-s2.0-84953638074
- Other Identifier
- 991019168246604721
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- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Computer Science, Theory & Methods
- Engineering, Electrical & Electronic