  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.
