QUASI-CONVEX FREE POLYNOMIALS

Let R < x > denote the ring of polynomials in g freely noncommuting variables x = (x(1),..., x(g)). There is a natural involution * on R < x > determined by x(j)* = x(j) and (pq)* = q*p*, and a free polynomial p is an element of R < x > is symmetric if it is invariant under this involution. If X = (X-1,..., X-g) is a g tuple of symmetric n x n matrices, then the evaluation p(X) is naturally defined and further p*(X) = p(X)*. In particular, if p is symmetric, then p(X)* = p(X). The main result of this article says if p is symmetric, p(0) = 0 and for each n and each symmetric positive definite nxn matrix A the set {X : A-p(X) > 0} is convex, then p has degree at most two and is itself convex, or -p is a hermitian sum of squares.