Noncommutative Polynomials Describing Convex Sets

The free closed semialgebraic set $${\mathcal {D}}_f$$ D f determined by a hermitian noncommutative polynomial $$f\in {\text {M}}_{{\delta }}({\mathbb {C}}\mathop {<}x,x^*\mathop {>})$$ f ∈ M δ ( C < x , x ∗ > ) is the closure of the connected component of $$\{(X,X^*)\mid f(X,X^*)\succ 0\}$$ { ( X , X ∗ ) ∣ f ( X , X ∗ ) ≻ 0 } containing the origin. When L is a hermitian monic linear pencil, the free closed semialgebraic set $${\mathcal {D}}_L$$ D L is the feasible set of the linear matrix inequality $$L(X,X^*)\succeq 0$$ 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 $${\mathcal {D}}_f$$ D f is convex. The solution leads to an efficient algorithm that not only determines if $${\mathcal {D}}_f$$ D f is convex, but if so, produces a minimal hermitian monic pencil L such that $${\mathcal {D}}_f={\mathcal {D}}_L$$ D f = D L . Of independent interest is a subalgorithm based on a Nichtsingulärstellensatz presented here: given a linear pencil $${\widetilde{L}}$$ L ~ and a hermitian monic pencil L , it determines if $${\widetilde{L}}$$ L ~ takes invertible values on the interior of $${\mathcal {D}}_L$$ D L . Finally, it is shown that if $${\mathcal {D}}_f$$ D f is convex for an irreducible hermitian $$f\in {\mathbb {C}}\mathop {<}x,x^*\mathop {>}$$ f ∈ C < x , x ∗ > , then f has degree at most two, and arises as the Schur complement of an L such that $${\mathcal {D}}_f={\mathcal {D}}_L$$ D f = D L .