Noncommutative Polynomials Describing Convex Sets

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Standard

Noncommutative Polynomials Describing Convex Sets. / Helton, J.W.; Klep, I.; McCullough, S.; Volčič, J.

I: Foundations of Computational Mathematics, Bind 21, 2021, s. 575–611.

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt

Harvard

Helton, JW, Klep, I, McCullough, S & Volčič, J 2021, 'Noncommutative Polynomials Describing Convex Sets', Foundations of Computational Mathematics, bind 21, s. 575–611. https://doi.org/10.1007/s10208-020-09465-w

APA

Helton, J. W., Klep, I., McCullough, S., & Volčič, J. (2021). Noncommutative Polynomials Describing Convex Sets. Foundations of Computational Mathematics, 21, 575–611. https://doi.org/10.1007/s10208-020-09465-w

Vancouver

Helton JW, Klep I, McCullough S, Volčič J. Noncommutative Polynomials Describing Convex Sets. Foundations of Computational Mathematics. 2021;21:575–611. https://doi.org/10.1007/s10208-020-09465-w

Author

Helton, J.W. ; Klep, I. ; McCullough, S. ; Volčič, J. / Noncommutative Polynomials Describing Convex Sets. I: Foundations of Computational Mathematics. 2021 ; Bind 21. s. 575–611.

Bibtex

@article{ab5c82f61db544d68305d44171fa7f80,
title = "Noncommutative Polynomials Describing Convex Sets",
abstract = "The free closed semialgebraic set Df determined by a hermitian noncommutative polynomial f∈Mδ(C) is the closure of the connected component of {(X,X∗)∣f(X,X∗)≻0} containing the origin. When L is a hermitian monic linear pencil, the free closed semialgebraic set DL is the feasible set of the linear matrix inequality L(X,X∗)⪰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 Df is convex. The solution leads to an efficient algorithm that not only determines if Df is convex, but if so, produces a minimal hermitian monic pencil L such that Df=DL. Of independent interest is a subalgorithm based on a Nichtsingul{\"a}rstellensatz presented here: given a linear pencil L˜ and a hermitian monic pencil L, it determines if L˜ takes invertible values on the interior of DL. Finally, it is shown that if Df is convex for an irreducible hermitian f∈C, then f has degree at most two, and arises as the Schur complement of an L such that Df=DL.",
author = "J.W. Helton and I. Klep and S. McCullough and J. Vol{\v c}i{\v c}",
year = "2021",
doi = "10.1007/s10208-020-09465-w",
language = "Udefineret/Ukendt",
volume = "21",
pages = "575–611",
journal = "Foundations of Computational Mathematics",
issn = "1615-3375",
publisher = "Springer",

}

RIS

TY - JOUR

T1 - Noncommutative Polynomials Describing Convex Sets

AU - Helton, J.W.

AU - Klep, I.

AU - McCullough, S.

AU - Volčič, J.

PY - 2021

Y1 - 2021

N2 - The free closed semialgebraic set Df determined by a hermitian noncommutative polynomial f∈Mδ(C) is the closure of the connected component of {(X,X∗)∣f(X,X∗)≻0} containing the origin. When L is a hermitian monic linear pencil, the free closed semialgebraic set DL is the feasible set of the linear matrix inequality L(X,X∗)⪰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 Df is convex. The solution leads to an efficient algorithm that not only determines if Df is convex, but if so, produces a minimal hermitian monic pencil L such that Df=DL. Of independent interest is a subalgorithm based on a Nichtsingulärstellensatz presented here: given a linear pencil L˜ and a hermitian monic pencil L, it determines if L˜ takes invertible values on the interior of DL. Finally, it is shown that if Df is convex for an irreducible hermitian f∈C, then f has degree at most two, and arises as the Schur complement of an L such that Df=DL.

AB - The free closed semialgebraic set Df determined by a hermitian noncommutative polynomial f∈Mδ(C) is the closure of the connected component of {(X,X∗)∣f(X,X∗)≻0} containing the origin. When L is a hermitian monic linear pencil, the free closed semialgebraic set DL is the feasible set of the linear matrix inequality L(X,X∗)⪰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 Df is convex. The solution leads to an efficient algorithm that not only determines if Df is convex, but if so, produces a minimal hermitian monic pencil L such that Df=DL. Of independent interest is a subalgorithm based on a Nichtsingulärstellensatz presented here: given a linear pencil L˜ and a hermitian monic pencil L, it determines if L˜ takes invertible values on the interior of DL. Finally, it is shown that if Df is convex for an irreducible hermitian f∈C, then f has degree at most two, and arises as the Schur complement of an L such that Df=DL.

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U2 - 10.1007/s10208-020-09465-w

DO - 10.1007/s10208-020-09465-w

M3 - Tidsskriftartikel

VL - 21

SP - 575

EP - 611

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

SN - 1615-3375

ER -

ID: 284012284