非对易多重次调和多项式第一部分：整体假设
Noncommutative Plurisubharmonic Polynomials Part I: Global Assumptions
摘要：
我们考虑非交换自由变元（x_1，x_2，.，x_g）。我们定义了p的非交换复海森矩阵，我们称一个非交换对称多项式为非交换多重次调和多项式，如果它有一个非交换复海森矩阵，当对每个大小n的n × n矩阵的所有元组求值时，它是半正定的。本文证明了对称非交换多重次调和多项式是具有非交换解析变量变换的非交换凸多项式，即，一个非交换对称多项式p是非交换多重次调和的当且仅当它具有形式p = \sum f_j^T f_j + \sum k_j^T + F + F^T其中和是有限的并且f_j，k_j，F都是非交换解析的。我们还提出了一个理论的非交换积分的非交换多项式，我们证明了一个非交换版本的Frobenius定理。随后的文件格林证明，如果非交换复杂的海森的p采取半正定值的“非交换开集“然后非交换复杂的海森采取半正定值的所有矩阵元组。因此，p具有上面的形式。在随后的文件中的证明借鉴了本文中的大部分定理，以及涉及非交换二次函数表示的非常不同的技术。
英文摘要：
We consider symmetric polynomials, p, in the noncommutative free variables (x_1, x_2, ..., x_g). We define the noncommutative complex hessian of p and we call a noncommutative symmetric polynomial noncommutative plurisubharmonic if it has a noncommutative complex hessian that is positive semidefinite when evaluated on all tuples of n x n matrices for every size n. In this paper, we show that the symmetric noncommutative plurisubharmonic polynomials are precisely the noncommutative convex polynomials with a noncommutative analytic change of variables; i.e., a noncommutative symmetric polynomial, p, is noncommutative plurisubharmonic if and only if it has the form p = \sum f_j^T f_j + \sum k_j k_j^T + F + F^T where the sums are finite and f_j, k_j, F are all noncommutative analytic. We also present a theory of noncommutative integration for noncommutative polynomials and we prove a noncommutative version of the Frobenius theorem. A subsequent paper by Greene proves that if the noncommutative complex hessian of p takes positive semidefinite values on a "noncommutative open set" then the noncommutative complex hessian takes positive semidefinite values on all matrix tuples. Thus, p has the form above. The proof in the subsequent paper draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions.
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