A Nonmonotone Real-Rootedness Set for Symmetric Imaginary Shifts

Abstract

For a real polynomial $F$ and $\omega\geq 0$, set $A_\omega(z)=(F(z+i\omega)+F(z-i\omega))/2$ and $\Omega_F=\{\omega\geq0:A_\omega\text{ has only real zeros}\}$. We present an explicit rational even polynomial of degree eight for which $6/25$ and $12/25$ belong to $\Omega_F$, while $3/10$ does not. Exact Sturm certificates give respectively eight, four, and eight distinct real zeros. Consequently $\Omega_F$ is neither an interval nor an up-set. All zeros of $F$ lie in the strip $|\operatorname{Im}z|\leq11/25$, and the classical strip-contraction theorem gives the eventual tail $[11/25,\infty)\subset\Omega_F$. We also include a direct elementary proof of that tail and a standard-library exact verifier.