Conic stability of polynomials and positive maps

Dey, Papri; Gardoll, Stephan; Theobald, Thorsten

doi:10.1016/j.jpaa.2020.106610

Cited by 4 publications

(3 citation statements)

References 30 publications

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“…For example (x 1 + x 3 ) 2 − x 2 2 is not a stable polynomial but psd-stable where K is the cone of positive semidefinite matrices. See [DGT21] for a comparison among stable, psd-stable and determinantal polynomials. By [JT18b], the hyperbolicity cones of a homogeneous polynomial f coincide with the components of I(f ) c , where I(f ) c denotes the complement of I(f ).…”

Section: 2mentioning

confidence: 99%

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Polynomials with Lorentzian Signature and computing Permanents

Dey¹

2022

Preprint

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We study the class of polynomials whose Hessians evaluated at any interior point of a closed convex cone K have Lorentzian signature. This class is a generalization to the remarkable class of Lorentzian polynomials. We prove that hyperbolic polynomials and conic stable polynomials belong to this class. Finally, we develop a method for computing permanents of nonsingular matrices which belong to a class that includes nonsingular k locally singular matrices via hyperbolic programming.

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Section: 2mentioning

confidence: 99%

“…The proof is based on the observation that a hyperbolic polynomial f ∈ C[z] is K-stable if and only if int K ⊆ I(f ) c . See [DGT21] for the details.…”

Section: 2mentioning

confidence: 99%

Polynomials with Lorentzian Signature and computing Permanents

Dey¹

2022

Preprint

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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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Combinatorics and preservation of conically stable polynomials

2023

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Given a closed, convex cone $$K\subseteq \mathbb {R}^n$$ K ⊆ R n , a multivariate polynomial $$f\in \mathbb {C}[\textbf{z}]$$ f ∈ C [ z ] is called K-stable if the imaginary parts of its roots are not contained in the relative interior of K. If K is the nonnegative orthant, K-stability specializes to the usual notion of stability of polynomials. We develop generalizations of preservation operations and of combinatorial criteria from usual stability toward conic stability. A particular focus is on the cone of positive semidefinite matrices (psd-stability). In particular, we prove the preservation of psd-stability under a natural generalization of the inversion operator. Moreover, we give conditions on the support of psd-stable polynomials and characterize the support of special families of psd-stable polynomials.

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Conic stability of polynomials and positive maps

Cited by 4 publications

References 30 publications

Polynomials with Lorentzian Signature and computing Permanents

Polynomials with Lorentzian Signature and computing Permanents

Imaginary Projections: Complex Versus Real Coefficients

Combinatorics and preservation of conically stable polynomials

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