A short note on determinantal representation of stable polynomials

In this paper we provide some results that replace the condition ”real-zero” by the properties so-called x-substitution and y-substitution. We show that using these properties, we can still write the determinantal representation of a stable polynomial in terms of identity and Hermitian matrices.

On the Existence of Hurwitz Polynomials with no Hadamard Factorization

A Hurwitz stable polynomial of degree $n\geq1$ has a Hadamard factorization if it is a Hadamard product (i.e., element-wise multiplication) of two Hurwitz stable polynomials of degree $n$. It is known that Hurwitz stable polynomials of degrees less than four have a Hadamard factorization. It is shown that, for arbitrary $n\geq4$, there exists a Hurwitz stable polynomial of degree $n$ which does not have a Hadamard factorization.