Hadamard powers of polynomials with only real zeros

Abstract

Let f(x)=∑i=0naixi be a polynomial with positive coefficients and p>0. The pth Hadamard power of f(x) is the polynomial f[p](x)=∑i=0naipxi. 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 Pn such that if f(x) has only real zeros, then so does f[p](x) for p>Pn.