Hadamard powers of polynomials with only real zeros

Let f(x) = Sigma(n)(i=1)a(i)x(i) be a polynomial with positive coefficients and p > 0. The pth Hadamard power of f(x) is the polynomial f([P])(x) = Sigma(n)(i=0)a(i)(p)x(i). It is conjectured that if f(x) has only real zeros, then so does f([P])(x) for p >= 1. We verify the conjecture when n = 3 and give a counterexample when n = 4. We also show that there exists a positive number P-n such that if f (x) has only real zeros, then so does f([P])(x) for p > P-n. (C) 2013 Elsevier Inc. All rights reserved.