A test for root-clustering transformability

Consider the problem of root clustering: given a square matrix
A
with spectrum
\sigma(A)
, for what region
S
in the complex plane is it possible to state a criterion (necessary and sufficient conditions) so that
\sigma(A) \in S ?
Recently it has been shown that one subclass Ω of
S
satisfies a certain transformability condition. In this note we test transformability via polynomial global nonnegativity.