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 $x_j^{\ast }=x_j$ and $(pq{)}^{\ast }=q^{\ast }{p}^{\ast }$ 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 $p^{\ast }(X)=p(X{)}^{\ast }$. In particular, if $p$ is symmetric, then $p(X{)}^{\ast }=p(X)$. 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.