The full shadow of a polynomial II. Vanishing root masses and holes

Abstract

For a complex polynomial $P$, its full shadow is the closure of the set of zeros of all derivatives of all powers $P^n$, and its limiting core is the Hausdorff limit, as $n\to \infty$, of the set of zeros of all derivatives of $P^n$. These sets retain geometric information that weak limits of root-counting measures discard. We determine complete zero-set limits for arrays of polynomials whose roots approach finitely many locations. A cluster containing $o(D)$ roots of a degree-$D$ polynomial can survive outside the limiting support of the derivative zeros, which depends only on the positive limiting masses: under repeated differentiation, the zeros it produces follow a straight path, selected by an explicit min--max condition, and preserve the cluster's exact multiplicity. When such a location instead carries a small positive mass $\epsilon$, the support of the limiting zero distribution at a fixed ratio of derivative order to degree develops an arc on the $\sqrt{\epsilon }$ scale with a universal local semicircle profile. We use these mechanisms to prove that polynomial shadows can have robust and arbitrarily complicated topology. For every $k\ge 2$, there are polynomials with $k+2$ distinct root locations whose limiting cores and full shadows have at least $\left(\genfrac{}{}{0ex}{}{k}{2}\right)$ holes; this persists under coefficient perturbations and among square-free polynomials without a reflection axis. In a four-location family, the core and full-shadow areas tend to zero while one hole has area tending to $8/1221$.

arXiv comment38 pages, 7 figures