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 \({\mathcal {D}}_f={\mathcal {D}}_L\).

由封闭式非交换多项式\（f \ in {\ text {M}} _ {{\ delta}}（{\ mathbb {C}}确定的自由封闭半代数集\ {{\ mathcal {D}} _ f \ \ mathop {<} x，x ^ * \ mathop {>}）\）是\（\ {（X，X ^ *）\ mid f（X，X ^ *）\ succ 0的连接组件的闭包\} \）包含原点。当L是厄米单调线性铅笔时，自由闭合半代数集\（{\数学{D}} _ L \）是线性矩阵不等式\（L（X，X ^ *）\ succeq 0 \）的可行集并被称为自由谱面体。显然，它们是凸的，并且众所周知，当且仅当它是一个自由谱面时，一个自由的封闭半代数集是凸的。本文解决了主要结果确定那些的基本问题˚F为其\（{\ mathcal {d}} _˚F\）是凸形的。该解决方案导致一种有效的算法，该算法不仅确定\（{\ mathcal {D}} _ f \）是否为凸形，而且如果是凸出的，则产生最小的埃尔米特式单笔铅笔L，使得\（{\ mathcal {D}} _ f = {\数学{D}} _ L \）。具有独立利益的是基于此处展示的Nichtsingulärstellensatz的子算法：给定线性铅笔\（{\ widetilde {L}} \）和厄米单峰铅笔L，它确定\（{\ widetilde {L}} \）是否在\（{\ mathcal {D}} _ L \）内部取可逆值。最后，表明如果\（{\ mathcal {D}} _ f \）对于不可约的埃尔米特\（f \ in {\ mathbb {C}} \ mathop {<} x，x ^ * \ mathop { >} \），则f的度数最多为2，并且作为L的Schur补码出现，使得\（{\ mathcal {D}} _ f = {\ mathcal {D}} _ L \）。