Comments on \u2018On Hadamard powers of polynomials\u2019

Abstract

Let f(x)=a_n x^n + a_{n-1} x^{n-1} + cdots +a_1 x +a_0 be a polynomial with real positive coefficients and pin mathbb {R}. The pth Hadamard power of f is the polynomial f^{[p]}(x):=a_n^p x^n + a_{n-1}^p x^{n-1}+ cdots + 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 pge 1. We construct a counterexample to this statement.