Quasi-Convex Free Polynomials

Abstract: Let $\Rx$ denote the ring of polynomials in $g$ freely non-commuting variables $x=(x_1,...,x_g)$. There is a natural involution * on $\Rx$ determined by $x_j^*=x_j$ and $(pq)^=q^ p^*$ and a free polynomial $p\in\Rx$ is symmetric if it is invariant under this involution. If $X=(X_1,...,X_g)$ is a $g$ tuple of symmetric $n\times n$ 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 $n\times n$ matrix $A$ the set ${X:A-p(X)\succ 0}$ is convex, then $p$ has degree at most two and is itself convex, or $-p$ is a hermitian sum of squares.