It is clear that the multivariable complex polynomial $p(z) = \det (I - ZA)$, where $z=(z_1,\ldots,z_d)$, $Z$ is a diagonal matrix with the variables $z_1,\ldots,z_d$ on the diagonal (each variable can be repeated many times), and $A$ is a contraction, is stable, i.e., it has no zeroes on the unit polydisc in ${\mathbb C}^d$. I will discuss the converse question: does a stable multivariable complex polynomial admit such a determinantal representation? This question turns out to be related to von Neumann inequality for rational inner functions on the polydisc and to the generalized Lax conjecture in convex algebraic geometry. This talk is based on joint work with A. Grinshpan, D. Kalyuzhnyi-Verbovetskyi, and H. Woerdeman.