  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.
