Norm-Constrained Determinantal Representations of Multivariable Polynomials

For every multivariable polynomial $p$ , with $p(0)=1$ , we construct a determinantal representation, $ p=\det (I - K Z )$ , where $Z$ is a diagonal matrix with coordinate variables on the diagonal and $K$ is a complex square matrix. Such a representation is equivalent to the existence of $K$ whose principal minors satisfy certain linear relations. When norm constraints on $K$ are imposed, we give connections to the multivariable von Neumann inequality, Agler denominators, and stability. We show that if a multivariable polynomial $q$ , $q(0)=0,$ satisfies the von Neumann inequality, then $1-q$ admits a determinantal representation with $K$ a contraction. On the other hand, every determinantal representation with a contractive $K$ gives rise to a rational inner function in the Schur–Agler class.