Noncommutative polynomials describing convex sets
| Autor: | Helton, J. W., Klep, I., McCullough, S., Volčič, J. |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Found. Comput. Math. 21 (2021) 575--611 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s10208-020-09465-w |
| Popis: | The free closed semialgebraic set $D_f$ determined by a hermitian noncommutative polynomial $f$ is the closure of the connected component of $\{(X,X^*)\mid f(X,X^*)>0\}$ 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 $L(X,X^*)\geq 0$ 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 efficient 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 Nichtsingul\"arstellensatz presented here: given a linear pencil $L'$ and a hermitian monic pencil $L$, it determines if $L'$ takes invertible values on the interior of $D_L$. Finally, it is shown that if $D_f$ is convex for an irreducible hermitian polynomial $f$, then $f$ has degree at most two, and arises as the Schur complement of an $L$ such that $D_f=D_L$. Comment: v2: 37 pages, algorithm is now deterministic; v1: 36 pages, includes table of contents and index |
| Databáze: | arXiv |
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