Comments on ‘On Hadamard powers of polynomials’

Let f ( x ) = a n x n + a n - 1 x n - 1 + ⋯ + a 1 x + a 0 be a polynomial with real positive coefficients and p ∈ R . The p th Hadamard power of f is the polynomial f [ p ] ( x ) : = a n p x n + a n - 1 p x n - 1 + ⋯ + a 1 p x + a 0 p . We give sufficient conditions for f [ p ] to be a Hurwitz polynomial (i.e., to be a stable polynomial) for all p > p 0 or p < p 1 with some positive p 0 and negative p 1 (without any assumption about stability of f ). Theorem 5 by Gregor and Tišer (Math Control Signals Syst 11:372–378, 1998 ) asserts that if f is a stable polynomial with positive coefficients then f [ p ] is stable for every p ≥ 1 . We construct a counterexample to this statement.