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≥2 is the closure of all zeros of (Pn)(m), n≥1, 0≤ 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→∞ and the support of their limiting normalized root-counting measure. For m/n→α∈(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.
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BibTeX
@misc{hagg2026full,
title = {{The full shadow of a polynomial I. The limiting core and the transient set}},
author = {Christian Hägg and Boris Shapiro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09602},
eprint = {2610.09602},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09602},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The full shadow of a polynomial I. The limiting core and the transient set
AU - Christian Hägg
AU - Boris Shapiro
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09602
UR - https://arxiv.org/abs/2610.09602
AB - The full shadow of a complex polynomial P of degree d≥2 is the closure of all zeros of (P^n)^(m), n≥1, 0≤ 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→∞ and the support of their limiting normalized root-counting measure. For m/n→α∈(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.
ER -