# Noncommutative plurisubharmonic polynomials part I: Global assumptions Academic Article

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### abstract

• We consider symmetric polynomials, p , in the noncommutative (nc) free variables { x 1 , x 2 , … , x g } . We define the nc complex hessian of p as the second directional derivative (replacing x T by y ) q ( x , x T ) [ h , h T ] : = ∂ 2 p ∂ s ∂ t ( x + t h , y + s k ) | t , s = 0 | y = x T , k = h T . 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., q ( X , X T ) [ H , H T ] ≽ 0 for all X , H ∈ ( 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 (0.1) p = ∑ f j T f j + ∑ k j k j T + F + F T where the sums are finite and f j , k j , 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.

### publication date

• January 1, 2011