A few results concerning the Hurwitz stability of polytopes of complex polynomials
Let \(f_1,f_2,\dots, f_m\) be complex polynomials with \[ n-1 = \mathrm{deg} f_1 =\dots =\mathrm{deg}f_k< \mathrm{deg}f_{k+1}= \dots = \mathrm{deg}f_m =n. \] Let further \[ C_{f_1,f_2,\dots, f_m}=\left\{\sum\limits_{i=1}^n \alpha_i f_i\;: \alpha_i \geq 0 \; (i=1, \dots , m),\; \sum\limits_{i=1}^m \alpha_i=1 \right\} \] be the polytope of the polynomials \(f_1,f_2,\dots ,f_m\). Let \(H\) be a nontrivial supporting hyperplane of \(C_{f_1,f_2,\dots, f_m}\). The one-dimensional polytopes \(C_{f_1,f_2,\dots, f_m}\cap H\) are called exposed edges. Under the above assumptions it is proved that the polytope \(C_{f_1,f_2,\dots, f_m}\) is Hurwitz stable if and only if all its exposed edges are stable. In the case of two complex polynomials of arbitrary degrees also a necessary and sufficient condition is given for the (Hurwitz) stability of segments.
scientific article; zbMATH DE number 6012749