对称虚位移下的非单调实根集合
A Nonmonotone Real-Rootedness Set for Symmetric Imaginary Shifts
AI总结：
该研究构造了八次偶有理多项式，证明对称虚位移对应的实根集合既非区间也非上集，验证了相关零点数量与带形范围，还给出尾部结论的初等证明与精确验证器。
AI中文摘要：
对于实多项式$F$且$\omega\geq0$，定义$A_\omega(z)=(F(z+i\omega)+F(z-i\omega))/2$，并记$\Omega_F=\{\omega\geq0:A_\omega\text{仅有实零点}\}$。我们构造了一个显式的八次偶有理多项式，使得$6/25$和$12/25$属于$\Omega_F$，但$3/10$不属于。精确的Sturm证书（斯图姆凭证）显示三者分别有8个、4个和8个不同实零点，因此$\Omega_F$既不是区间也不是上集。$F$的所有零点都位于带形$|\operatorname{Im}z|\leq11/25$内，经典的带形收缩定理给出其最终尾部$[11/25,\infty)\subset\Omega_F$，我们还给出了该尾部结论的直接初等证明以及标准库形式的精确验证器。
英文摘要：
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.
