TY - JOUR

T1 - Noncommutative Polynomials Describing Convex Sets

AU - Helton, J. William

AU - Klep, Igor

AU - McCullough, Scott

AU - Volčič, Jurij

N1 - Funding Information:
J. William Helton: Research supported by the NSF Grant DMS-1500835. Igor Klep: Supported by the Slovenian Research Agency Grants J1-8132, N1-0057 and P1-0222. Partially supported by the Marsden Fund Council of the Royal Society of New Zealand. Scott McCullough: Research supported by NSF Grants DMS-1361501 and DMS-1764231. Jurij Volčič: Research supported by the Deutsche Forschungsgemeinschaft (DFG) Grant No. SCHW 1723/1-1.
Publisher Copyright:
© 2020, SFoCM.

PY - 2021/4/1

Y1 - 2021/4/1

N2 - The free closed semialgebraic set Df determined by a hermitian noncommutative polynomial f∈ M δ(C< x, x∗>) 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< x, x∗> , 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∈ M δ(C< x, x∗>) 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< x, x∗> , then f has degree at most two, and arises as the Schur complement of an L such that Df= DL.

KW - Convex set

KW - Free locus

KW - Linear matrix inequality (LMI)

KW - Noncommutative rational function

KW - Semialgebraic set

KW - Spectrahedron

UR - http://www.scopus.com/inward/record.url?scp=85087067478&partnerID=8YFLogxK

U2 - 10.1007/s10208-020-09465-w

DO - 10.1007/s10208-020-09465-w

M3 - Article

AN - SCOPUS:85087067478

SN - 1615-3375

VL - 21

SP - 575

EP - 611

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

IS - 2

ER -