Free Convex Sets Defined by Rational Expressions Have LMI Representations

\def\cD{{\mathcal D}} Suppose $p$ is a symmetric matrix whose entries are polynomials in freely noncommutative variables and $p(0)$ is positive definite. Let $\cD_p$ denote the component of zero of the set of those $g$-tuples $X=(X_1,\dots,X_g)$ of symmetric matrices (of the same size) such that $p(X)$ is positive definite. In another paper of the authors [{\it Every free convex basic semi-algebraic set has an LMI representation}, Annals of Mathematics, to appear] it was shown that if $\cD_p$ is convex and bounded, then $\cD_p$ can be described as the set of solutions of a linear matrix inequality (LMI). This article extends that result from matrices of polynomials to matrices of rational functions in free variables.\par As a refinement of a theorem of Kaliuzhnyi-Verbovetskyi and Vinnikov, it is also shown that a minimal symmetric descriptor realization $r$ for a symmetric free matrix-valued rational function $\mathfrak{r}$ in $g$ freely noncommuting variables $x=(x_1,\dots,x_g)$ precisely encodes the singularities of the rational function. This singularities result is an important ingredient in the proof of the LMI representation theorem stated above.