Stability of multivariate complex diamond polynomials

Recently, attention has been focused on the (open left half plane) stability of a family of polynomials having complex coefficients with their real and imaginary parts each varying in a diamond. It has been found that the stability of the family of polynomials requires the checking of 16 one-dimensional edges of the diamond. This result is extended to n-variate case. It is proved that checking the scattering Hurwitz property of certain 16n one-dimensional edges of the diamond can guarantee the scattering Hurwitz property of the whole diamond family of n-variate complex polynomials