自由代数中的凸正星形
The convex Positivstellensatz in a free algebra
摘要：
给定g元幺一线性束L，设D_L为其正性域，即使L（X）半正定的所有大小的对称矩阵的所有g元组X的集合。由于L是幺一线性束，所以DL是具有内部的凸集，反之，具有内部的凸有界非交换半代数集都是DL形式的。本文的主要结果建立了凸半代数集上的完备非交换Nichtnegativstellenetrix。也就是说，一个非交换多项式p在D_L上是半正定的当且仅当它有一个带最优次数界的加权平方和表示：p = s^* s + \sum_j f_j^* L f_j，其中s，f_j是次数不大于大于1/2 deg（p）的非交换多项式的向量。这个非对易的结果与对易的情况形成了鲜明的对比，在对易的情况下，对s，f_j的度数没有控制，并且假设只有p是非负的，而不是p严格为正，产生一个干净的Positivstellenance是如此罕见，以至于这种情况是值得注意的。
英文摘要：
Given a monic linear pencil L in g variables let D_L be its positivity domain, i.e., the set of all g-tuples X of symmetric matrices of all sizes making L(X) positive semidefinite. Because L is a monic linear pencil, D_L is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form D_L. The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative polynomial p is positive semidefinite on D_L if and only if it has a weighted sum of squares representation with optimal degree bounds: p = s^* s + \sum_j f_j^* L f_j, where s, f_j are vectors of noncommutative polynomials of degree no greater than 1/2 deg(p). This noncommutative result contrasts sharply with the commutative setting, where there is no control on the degrees of s, f_j and assuming only p nonnegative, as opposed to p strictly positive, yields a clean Positivstellensatz so seldom that such cases are noteworthy.
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