Noncommutative plurisubharmonic polynomials part I: Global assumptions

We consider symmetric polynomials, p, in the noncommutative (nc) free variables {x1,x2,...,xg}. We define the nc complex hessian of p as the second directional derivative (replacing xT by y). We call an nc symmetric polynomial nc plurisubharmonic (nc plush) if it has an nc complex hessian that is positive semidefinite when evaluated on all tuples of n×n matrices for every size n; i.e.,. for all X,H ε (R^n×n)^g for every n≥1. In this paper, we classify all symmetric nc plush polynomials as convex polynomials with an nc analytic change of variables; i.e., an nc symmetric polynomial p is nc plush if and only if it has the form. where the sums are finite and fj, kj, F are all nc analytic. In this paper, we also present a theory of noncommutative integration for nc polynomials and we prove a noncommutative version of the Frobenius theorem. A subsequent paper (J.M. Greene, preprint [6]), proves that if the nc complex hessian, q, of p takes positive semidefinite values on an "nc open set" then q takes positive semidefinite values on all tuples X, H. Thus, p has the form in Eq. (0.1). The proof, in J.M. Greene (preprint) [6], draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions.