Quasi-convex free polynomials

Abstract:

Let $\mathbb R\langle x \rangle$ denote the ring of polynomials in $g$ freely noncommuting variables $x=(x_1,\dots ,x_g)$. There is a natural involution $*$ on $\mathbb R\langle x \rangle$ determined by $x_j^*=x_j$ and $(pq)^*=q^* p^*$, and a free polynomial $p\in \mathbb R\langle x \rangle$ is symmetric if it is invariant under this involution. If $X=(X_1,\dots ,X_g)$ 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.