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 Pn, and its limiting core is the Hausdorff limit, as n→∞, of the set of zeros of all derivatives of Pn. 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 ε, the support of the limiting zero distribution at a fixed ratio of derivative order to degree develops an arc on the √ε 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≥2, there are polynomials with k+2 distinct root locations whose limiting cores and full shadows have at least binomk2 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.
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BibTeX
@misc{hagg2026full,
title = {{The full shadow of a polynomial II. Vanishing root masses and holes}},
author = {Christian Hägg and Boris Shapiro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40842},
eprint = {2610.40842},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40842},
}
RIS
TY - UNPB
TI - The full shadow of a polynomial II. Vanishing root masses and holes
AU - Christian Hägg
AU - Boris Shapiro
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40842
UR - https://arxiv.org/abs/2610.40842
AB - 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→∞, 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 ε, the support of the limiting zero distribution at a fixed ratio of derivative order to degree develops an arc on the √ε 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≥2, there are polynomials with k+2 distinct root locations whose limiting cores and full shadows have at least binomk2 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.
ER -