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  • Noncommutative polynomials describing convex sets
    Helton, J. William, 1944- ...
    The free closed semialgebraic set ▫$\mathcal{D}_f$▫ determined by a hermitian noncommutative polynomial ▫$f\in \text{M}_{\delta} (\mathbb{C}\mathop{<}x,x^*\mathop{>})$▫ is the closure of the ... connected component of ▫$\{(X,X^*)\mid f(X,X^*)\succ 0\}$▫ containing the origin. When ▫$L$▫ is a hermitian monic linear pencil, the free closed semialgebraic set ▫$\mathcal{D}_L$▫ is the feasible set of the linear matrix inequality ▫$L(X,X^*)\succeq 0$▫ 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 ▫$\mathcal{D}_f$▫ is convex. The solution leads to an efficient algorithm that not only determines if ▫$\mathcal{D}_f$▫ is convex, but if so, produces a minimal hermitian monic pencil ▫$L$▫ such that ▫$\mathcal{D}_f=\mathcal{D}_L$▫. Of independent interest is a subalgorithm based on a Nichtsingulärstellensatz presented here: given a linear pencil ▫$\widetilde{L}$▫ and a hermitian monic pencil ▫$L$▫, it determines if ▫$\widetilde{L}$▫ takes invertible values on the interior of ▫$\mathcal{D}_L$▫. Finally, it is shown that if ▫$\mathcal{D}_f$▫ is convex for an irreducible hermitian ▫$f\in \mathbb{C}\mathop{<}x,x^*\mathop{>}$▫, then ▫$f$▫ has degree at most two, and arises as the Schur complement of an ▫$L$▫ such that ▫$\mathcal{D}_f=\mathcal{D}_L$▫.
    Vir: Foundations of computational mathematics. - ISSN 1615-3375 (Vol. 21, iss. 2, Apr. 2021, str. 575-611)
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2021
    Jezik - angleški
    COBISS.SI-ID - 48249347

    Povezava(-e):

    https://link.springer.com/article/10.1007/s10208-020-09465-w
    DOI

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