  We introduce and study the notion of conic stability of multivariate complex
polynomials in $\mathbb{C}[z_1,\ldots, z_n]$, which naturally generalizes the
stability of multivariate polynomials. In particular, we generalize Borcea's
and Br\"and\'en's multivariate version of the Hermite-Kakeya-Obreschkoff
Theorem to the conic stability and provide a characterization in terms of a
directional Wronskian. And we generalize a major criterion for stability of
determinantal polynomials to stability with respect to the positive
semidefinite cone.
