## Abstract

The free closed semialgebraic set D_{f} 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 D_{L} 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 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 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 D_{L}. Finally, it is shown that if D_{f} 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 D_{f}= D_{L}.

Original language | English |
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Pages (from-to) | 575-611 |

Number of pages | 37 |

Journal | Foundations of Computational Mathematics |

Volume | 21 |

Issue number | 2 |

DOIs | |

State | Published - 1 Apr 2021 |

Externally published | Yes |

## Keywords

- Convex set
- Free locus
- Linear matrix inequality (LMI)
- Noncommutative rational function
- Semialgebraic set
- Spectrahedron

## ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics