Stable and real-zero polynomials in two variables

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

doi:10.1007/s11045-014-0286-3

Cited by 28 publications

(28 citation statements)

References 60 publications

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“…Remark 1.1 (Applications to Real Algebraic Geometry). Besides control theory, spectral factorization has several applications in real algebraic geometry -in particular it is a key step in a constructive proof of the existence of a definite determinantal representation for every hyperbolic polynomial in three variables [GKVVW16].…”

Section: Comparison To Related Workmentioning

confidence: 99%

Bit Complexity of Jordan Normal Form and Spectral Factorization

Dey¹,

Kannan²,

Ryder³

et al. 2021

Preprint

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We study the bit complexity of two related fundamental computational problems in linear algebra and control theory. Our results are: (1) An Õ(n ω+3 a+n 4 a 2 +n ω log(1/ǫ)) time algorithm for finding an ǫ−approximation to the Jordan Normal form of an integer matrix with a−bit entries, where ω is the exponent of matrix multiplication. (2) An Õ(n 6 d 6 a + n 4 d 4 a 2 + n 3 d 3 log(1/ǫ)) time algorithm for ǫ-approximately computing the spectral factorization P(x) = Q * (x)Q(x) of a given monic n × n rational matrix polynomial of degree 2d which satisfies P(x) 0 for all real x. The first algorithm is used as a subroutine in the second one.Despite its being of central importance, polynomial complexity bounds were not previously known for spectral factorization, and for Jordan form the best previous best boolean running time was an unspecified polynomial of degree at least twelve [Cai94]. Our algorithms are simple and judiciously combine techniques from numerical and symbolic computation, yielding significant advantages over either approach by itself.

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Section: Comparison To Related Workmentioning

confidence: 99%

Bit Complexity of Jordan Normal Form and Spectral Factorization

Dey¹,

Kannan²,

Ryder³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Several different determinantal representations are closely related to this one but not quite equivalent. There are determinantal representations for three variable hyperbolic polynomials, two variable real-zero polynomials, and two variable real-stable polynomials (see [2,3,7]). It turns out this formula can be derived from a determinantal representation for polynomials with no zeros on the bidisk D 2 = {(z 1 , z 2 ) : |z 1 |, |z 2 | < 1} from [2].…”

Section: Two Variable Szász Inequalitymentioning

confidence: 99%

Global Bounds on Stable Polynomials

Knese

2018

Complex Anal. Oper. Theory

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A classical inequality of Szász bounds polynomials with no zeros in the upper half plane entirely in terms of their first few coefficients. Borcea-Brändén generalized this result to several variables as a piece of their characterization of linear maps on polynomials preserving stability. In this paper, we improve Szász's original inequality, use determinantal representations to prove Szász type inequalities in two variables, and then prove that one can use the two variable inequality to prove an inequality for several variables.

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“…Indeed, it is not easy to compute determinantal representation numerically or symbolically using this method. Later, the problem of computing monic symmetric/Hermitian determinantal representation for a strictly RZ bivariate polynomial has been widely studied, for example one can see [Dix02], [PSV12], [Hen10], [GKVVW14].…”

Section: Introductionmentioning

confidence: 99%

Definite determinantal representations via orthostochastic matrices

Dey

2021

Journal of Symbolic Computation

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Determinantal polynomials play a crucial role in semidefinite programming problems. Helton-Vinnikov proved that real zero (RZ) bivariate polynomials are determinantal. However, it leads to a challenging problem to compute such a determinantal representation. We provide a necessary and sufficient condition for the existence of definite determinantal representation of a bivariate polynomial by identifying its coefficients as scalar products of two vectors where the scalar products are defined by orthostochastic matrices. This alternative condition enables us to develop a method to compute a monic symmetric/Hermitian determinantal representations for a bivariate polynomial of degree d. In addition, we propose a computational relaxation to the determinantal problem which turns into a problem of expressing the vector of coefficients of the given polynomial as convex combinations of some specified points. We also characterize the range set of vector coefficients of a certain type of determinantal bivariate polynomials.AMS Classification (2010). 15A75, 15B10, 15B51, 90C22.

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Stable and real-zero polynomials in two variables

Abstract: For every bivariate polynomial p(z 1 , z 2

Cited by 28 publications

References 60 publications

Bit Complexity of Jordan Normal Form and Spectral Factorization

Bit Complexity of Jordan Normal Form and Spectral Factorization

Global Bounds on Stable Polynomials

Definite determinantal representations via orthostochastic matrices

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