PreprintCC BY 4.0arXiv
A Note on the Converse Sendov Problem
Abstract
For a polynomial of degree n whose zeros lie in the closed unit disk, we determine the largest possible distance from a prescribed critical point of modulus r to the nearest zero. If n is even, the sharp radius is √(1−r2); if n is odd, the sharp radius is strictly smaller for r∈(0,1) and depends on n. Equality cases are also determined. The proof is based on the logarithmic-derivative identity and elementary geometric considerations.
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BibTeX
@misc{grozev2026note,
title = {{A Note on the Converse Sendov Problem}},
author = {Dragomir Grozev and Nikolai Nikolov},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30284},
eprint = {2608.30284},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30284},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Note on the Converse Sendov Problem
AU - Dragomir Grozev
AU - Nikolai Nikolov
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30284
UR - https://arxiv.org/abs/2608.30284
AB - For a polynomial of degree n whose zeros lie in the closed unit disk, we determine the largest possible distance from a prescribed critical point of modulus r to the nearest zero. If n is even, the sharp radius is √(1−r^2); if n is odd, the sharp radius is strictly smaller for r∈(0,1) and depends on n. Equality cases are also determined. The proof is based on the logarithmic-derivative identity and elementary geometric considerations.
ER -