Stable Noncommutative Polynomials and Their Determinantal Representations

Volčič, Jurij

doi:10.1137/18m1206734

Cited by 9 publications

(4 citation statements)

References 41 publications

(31 reference statements)

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“…By Theorem 1.1, it suffices to present an algorithm for checking whether property (iv) of Theorem 1.1 holds, that is, whether q L is invertible on the interior of D p L . To this end we first prove general statements about (rectangular) affine linear pencils being of full rank on the interior of a free spectrahedron (see also [KPV17,Vol,Pas18,GGOW16] for related results).…”

Section: Proof Of Theorem 11 and Algorithms: Corollary 13mentioning

confidence: 99%

Noncommutative polynomials describing convex sets

Helton,

Klep,

McCullough

et al. 2018

Preprint

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The free closed semialgebraic set D f determined by a hermitian noncommutative polynomial f P M δ pCăx, x ˚ąq is the closure of the connected component of tpX, X ˚q | f pX, X ˚q ą 0u containing the origin. When L is a hermitian monic linear pencil, the free closed semialgebraic set D L is the feasible set of the linear matrix inequality LpX, X ˚q ľ 0 and is known as a free spectrahedron. Evidently these are convex and it is well-known that a free closed semialgebraic set is convex if and only it is a free spectrahedron. The main result of this paper solves the basic problem of determining those f for which D f is convex. The solution leads to an effective probabilistic algorithm that not only determines if D f is convex, but if so, produces a minimal hermitian monic pencil L such that D f " D L . Of independent interest is a subalgorithm based on a Nichtsingulärstellensatz presented here: given a linear pencil r L and a hermitian monic pencil L, it determines if r L takes invertible values on the interior of D L . Finally, it is shown that if D f is convex for an irreducible hermitian f P Căx, x ˚ą, then f has degree at most two, and arises as the Schur complement of an L such that D f " D L .

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Section: Proof Of Theorem 11 and Algorithms: Corollary 13mentioning

confidence: 99%

Noncommutative polynomials describing convex sets

Helton,

Klep,

McCullough

et al. 2018

Preprint

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“…In terms of the imaginary projection I(p), we can express the stability of p as the condition I(p) ∩ R n >0 = ∅. Stable polynomials have applications in many branches of mathematics including combinatorics ( [6] and see [8] for the connection of the imaginary projection to combinatorics), differential equations [4], optimization [32], probability theory [5], and applied algebraic geometry [34]. Further application areas include theoretical computer science [22,23], statistical physics [3], and control theory [24], see also the surveys [28] and [35].…”

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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“…Recently, there has been wide-spread research interest in stable polynomials and the geometry of polynomials, accompanied by a variety of new connections to other branches of mathematics (including combinatorics [6], differential equations [4], optimization [27], probability theory [5], applied algebraic geometry [28], theoretical computer science [22,23] and statistical physics [3]). See also the surveys of Pemantle [25] and Wagner [29].…”

Section: Introductionmentioning

confidence: 99%

Conic stability of polynomials and positive maps

Dey¹,

Gardoll²,

Theobald³

2019

Preprint

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Given a proper cone K ⊆ R n , a multivariate polynomial f ∈ C[z] = C[z 1 , . . . , z n ] is called K-stable if it does not have a root whose vector of the imaginary parts is contained in the interior of K. If K is the non-negative orthant, then Kstability specializes to the usual notion of stability of polynomials.We study conditions and certificates for the K-stability of a given polynomial f , especially for the case of determinantal polynomials as well as for quadratic polynomials. A particular focus is on psd-stability. For cones K with a spectrahedral representation, we construct a semidefinite feasibility problem, which, in the case of feasibility, certifies K-stability of f . This reduction to a semidefinite problem builds upon techniques from the connection of containment of spectrahedra and positive maps.In the case of psd-stability, if the criterion is satisfied, we can explicitly construct a determinantal representation of the given polynomial. We also show that under certain conditions, for a K-stable polynomial f , the criterion is at least fulfilled for some scaled version of K.

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Stable Noncommutative Polynomials and Their Determinantal Representations

Cited by 9 publications

References 41 publications

Noncommutative polynomials describing convex sets

Noncommutative polynomials describing convex sets

Imaginary Projections: Complex Versus Real Coefficients

Conic stability of polynomials and positive maps

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