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Noncommutative Plurisubharmonic Polynomials Part II: Local Assumptions

We say that a symmetric noncommutative polynomial in the noncommutative free variables (x_1, x_2, ..., x_g) is noncommutative plurisubharmonic on a noncommutative open set if it has a noncommutative complex hessian that is positive semidefinite when evaluated on open sets of matrix tuples of sufficiently large size. In this paper, we show that if a noncommutative polynomial is noncommutative plurisubharmonic on a noncommutative open set, then the polynomial is actually noncommutative plurisubharmonic everywhere and 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. In the paper by Greene, Helton, and Vinnikov, it is shown that if p is noncommutative plurisubharmonic everywhere, then p has the form above. In other words, the paper by Greene, Helton, and Vinnikov makes a global assumption while the current paper makes a local assumption, but both reach the same conclusion. This paper uses a Gram-like matrix representation of noncommutative polynomials. A careful analysis of this Gram matrix plus the main theorem in the paper by Greene, Helton, and Vinnikov ultimately force the form in the equation above.