Quasi-Convex Free Polynomials

TL;DR

This paper characterizes quasi-convex free polynomials, showing that symmetric polynomials with convex positivity sets are either quadratic convex or sums of squares, revealing a structural dichotomy in non-commutative polynomial convexity.

Contribution

It provides a classification of symmetric free polynomials with convex positivity sets, establishing that they are either quadratic convex or sums of hermitian squares, a significant structural insight.

Findings

Polynomials with convex positivity sets are at most quadratic.

Such polynomials are either convex or sums of hermitian squares.

The result applies to polynomials evaluated on symmetric matrix tuples.

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.