On the Hurwitz stability of noninteger Hadamard powers of stable polynomials

Consider a polynomial f(z) = a(n) z(n) + ... + a(1) z + a(0) of positive coefficients that is stable (in the Hurwitz sense), i.e., every root of f lies in the open left half-plane of C. Due to Garloff and Wagner [J. Math. Anal. Appl. 202 (1996)], the pth Hadamard power of f: f[p](z) := a(p) (n) z(n) +...+ a(1)(p)z+ a(0)(p) is stable if p is a positive integer number. However, it turns out that f[p] does not need to be stable for all real p > 1. A counterexample is known for n = 8 and p = 1.139. On the other hand, f[p] is stable for n = 1, 2, 3, 4, and every p > 1. In this paper we fill the gap by showing that f[p] is stable for n = 5 and constructing counterexamples for n >= 6. Moreover, by means of Rouche's Theorem, we give some stability conditions for polynomials and two examples that complete and illustrate the results.(c) 2023 Published by Elsevier Inc.