Noncommutative Polynomials Describing Convex Sets

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.