The full shadow of a polynomial I. The limiting core and the transient set

Abstract

The full shadow of a complex polynomial $P$ of degree $d\ge 2$ is the closure of all zeros of $(P^n{)}^{(m)}$, $n\ge 1$, $0\le m<dn$. 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 $m/n\to \alpha \in (0,d)$, 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.

42 pages, 7 figures