发表机构
Stockholm University（斯德哥尔摩大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明复多项式完整阴影由紧致极限核心与孤立瞬态零点构成，建立测度弱收敛和Hausdorff收敛，并展示无穷瞬态实例及算子谱实现。
AI 中文摘要
次数 $d\ge2$ 的复多项式 $P$ 的完整阴影定义为所有 $(P^n)^{(m)}$（$n\ge1$，$0\le m<dn$）的零点的闭包。我们证明它由一个包含 $P$ 的所有根和临界点的紧致连通极限核心，以及孤立的瞬态零点组成，每个瞬态零点仅出现在有限多个幂次中。该核心是当 $n\to\infty$ 时所有导数阶的零点集合并集的 Hausdorff 极限，也是其极限归一化根计数测度的支集。对于 $m/n\to\alpha\in(0,d)$，我们建立了归一化根计数测度的弱收敛以及零点集到极限支集的 Hausdorff 收敛，两者在系数扰动下均保持稳定。我们展示了具有无穷多个瞬态的四次多项式。由多项式微分器构造的一个有界算子实现了完整阴影、核心和瞬态，分别作为其谱、Fredholm 本质谱和离散谱。
英文摘要
The full shadow of a complex polynomial $P$ of degree $d\ge2$ is the closure of all zeros of $(P^n)^{(m)}$, $n\ge1$, $0\le m<dn$. We prove that it consists of a compact connected limiting core containing all roots and critical points of $P$, together with isolated transient zeros, each occurring at only finitely many powers. The core is the Hausdorff limit of the zero sets pooled over all derivative orders as $n\to\infty$ and the support of their limiting normalized root-counting measure. For $m/n\toα\in(0,d)$, we establish weak convergence of normalized root-counting measures and Hausdorff convergence of zero sets to the limiting supports, both stable under coefficient perturbations. We exhibit quartics with infinitely many transients. A bounded operator constructed from polynomial differentiators realizes the full shadow, core, and transients as its spectrum, Fredholm essential spectrum, and discrete spectrum, respectively.
Comments42 pages, 7 figures
