雅可比端点束与中心二项式样本的严格交错性
Jacobi Endpoint Pencils and Sharp Interlacing for Centered Binomial Samples
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中文总结 AI 辅助
该研究针对中心二项式样本的商多项式，利用雅可比谱乘子理论，证明了其相邻多项式的严格交错性，确定了相关阈值并应用于戴德金ζ函数导数等对象。
中文摘要 AI 辅助
对于阶数不超过1的实整函数H，记B_{2n+1}[H]为其奇次中心二项式样本。去除由H的奇偶性强制产生的z=±1处的零点，并令x=z+z^{-1}，可得到实商C_n(x)。我们证明了在H的零点满足一致带条件时，对任意n，C_n与C_{n+1}可生成实根束。对于偶性H，最优一致半宽度为√(15/28)；对于奇性H，半宽度1即足够。该结论显著强于两个采样多项式各自的单位圆根性：后者成立时相邻商可能不交错。此结构结果是关于雅可比谱乘子的定理：对任意0<ν<2，商问题转化为保持端点束y^n(y+t)。我们得到了固定n和一致带的显式阈值，并确定了n=1时的精确阈值。证明结合了Bernstein变分缩减与H的零点轨道对应的有限雅可比矩阵的全非负性；未被完全亏格类论证覆盖的可能未配对外实对，直接在端点束上处理。对于满足固定n带条件的非多项式偶源，当相邻像在(0,4)内有单零点时，远程零点轨道变形可消除公共零点，这为与戴德金ζ函数导数及自对偶新形式嵌套临界值块相关的商族产生严格交错性。
英文摘要
For an even or odd real entire function $H$ of order at most one, let $B_{2n+1}[H]$ denote its centered binomial sample of odd degree. After removing the zero at $z=\pm1$ forced by the parity of $H$ and writing $x=z+z^{-1}$, one obtains a real quotient $C_n(x)$. We prove uniform strip conditions on the zeros of $H$ under which $C_n$ and $C_{n+1}$ generate a real-rooted pencil for every $n$. For even $H$ the optimal uniform half-width is $\sqrt{15/28}$, whereas for odd $H$ the half-width $1$ is sufficient. This conclusion is genuinely stronger than separate unit-circle-rootedness of the two sampled polynomials: the latter may hold while the adjacent quotients fail to interlace. The structural result is a theorem for Jacobi spectral multipliers. For every $0<ν<2$, the quotient problem becomes preservation of the endpoint pencil $y^n(y+t)$. We obtain explicit fixed-$n$ and uniform strip thresholds and determine the exact threshold for $n=1$. The proof combines Bernstein variation diminution with total nonnegativity of finite Jacobi matrices attached to the zero orbits of $H$; a possible unpaired outer real pair, which is not covered by the full defect-class argument, is treated directly on the endpoint pencil. For nonpolynomial even sources satisfying the fixed-$n$ strip condition, a remote-zero-orbit deformation removes common zeros whenever the adjacent images have simple zeros in $(0,4)$. This yields strict interlacing for quotient families associated with Dedekind zeta derivatives and with nested critical-value blocks of self-dual newforms.
