Stable and real-zero polynomials in two variables

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

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


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).

Original languageEnglish
Pages (from-to)1-26
Number of pages26
JournalMultidimensional Systems and Signal Processing
Issue number1
StatePublished - 1 Jan 2016


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

ASJC Scopus subject areas

  • Software
  • Signal Processing
  • Information Systems
  • Hardware and Architecture
  • Computer Science Applications
  • Artificial Intelligence
  • Applied Mathematics


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