Noncommutative plurisubharmonic polynomials part II: Local assumptions

This paper requires a technique that is not used in [1]. We use a Gram-like vector and matrix representation (called the border vector and middle matrix) for homogeneous degree 2 nc polynomials. We then analyze this representation for the nc complex hessian on an nc open set and positive semidefiniteness forces a very rigid structure on the border vector and middle matrix. This rigid structure plus the theorems in [1] ultimately force the form in Eq. (0.1).