Rational Inner Functions on a Square-Matrix Polyball

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

doi:10.1007/978-3-319-51593-9_10

Cited by 4 publications

(1 citation statement)

References 20 publications

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“…Recently, various generalizations and variations of the stability notion have been studied, such as stability with respect to a polyball [13,14], conic stability [9,18], Lorentzian polynomials [7], or positively hyperbolic varieties [29]. Exemplarily, regarding the conic stability, a polynomial p ∈ C[z] is called K-stable for a proper cone K ⊂ R n if p(z) = 0, whenever z im ∈ int K, where int is the interior.…”

Section: Introductionmentioning

confidence: 99%

Imaginary Projections: Complex Versus Real Coefficients

Gardoll¹,

Namin²,

Theobald³

2021

Preprint

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Given a complex multivariate polynomial p ∈ C[z 1 , . . . , z n ], the imaginary projection I(p) of p is defined as the projection of the variety V(p) onto its imaginary part. We give a full characterization of the imaginary projections of conic sections with complex coefficients, which generalizes a classification for the case of real conics. More precisely, given a bivariate complex polynomial p ∈ C[z 1 , z 2 ] of total degree two, we describe the number and the boundedness of the components in the complement of I(p) as well as their boundary curves and the spectrahedral structure of the components. We further study the imaginary projections of some families of higher degree complex polynomials. In particular, we show a realizability result for strictly convex complement components which is in sharp contrast to the situation for real polynomials.

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Section: Introductionmentioning

confidence: 99%

Imaginary Projections: Complex Versus Real Coefficients

Gardoll¹,

Namin²,

Theobald³

2021

Preprint

View full text Add to dashboard Cite

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Imaginary projections: Complex versus real coefficients

Gardoll

Namin

Theobald

2023

Journal of Pure and Applied Algebra

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The non-tangential boundary behavior of the matrix-valued rational inner functions on bounded symmetric domain

Wang

Zhang

2023

Proc. Amer. Math. Soc.

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Knese [Proc. LMS 111(2015), 1261-1306] proved every rational inner function on polydisc has a non-tangential limit at every point of the Shilov boundary. We extended his result to the case of functions on general bounded symmetric domains. Namely, every rational inner function on a bounded symmetric domain has a non-tangential limit of modulus 1 1 at every point of the Shilov boundary. We also prove that every matrix-valued rational inner function on tube-type domain has a unitary non-tangential limit at every point of the Shilov boundary.

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Rational Inner Functions on a Square-Matrix Polyball

Cited by 4 publications

References 20 publications

Imaginary Projections: Complex Versus Real Coefficients

Imaginary Projections: Complex Versus Real Coefficients

Imaginary projections: Complex versus real coefficients

The non-tangential boundary behavior of the matrix-valued rational inner functions on bounded symmetric domain

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