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. We show that for arbitrary $n\geq4$ there exists a Hurwitz stable polynomial of degree $n$ which does not have a Hadamard factorization.