A Nonmonotone Real-Rootedness Set for Symmetric Imaginary Shifts

Abstract

For a real polynomial $F$ and $ω\geq 0$, set $A_ω(z)=(F(z+iω)+F(z-iω))/2$ and $Ω_F=\{ω\geq0:A_ω\text{ has only real zeros}\}$. We present an explicit rational even polynomial of degree eight for which $6/25$ and $12/25$ belong to $Ω_F$, while $3/10$ does not. Exact Sturm certificates give respectively eight, four, and eight distinct real zeros. Consequently $Ω_F$ 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 $[11/25,\infty)\subsetΩ_F$. We also include a direct elementary proof of that tail and a standard-library exact verifier.