PreprintarXiv licencearXiv
Beyond Sendov's conjecture: the quadratic Tang--Zhang inequality
Abstract
Very recently, Lech Mazur proved the celebrated Sendov conjecture, and Terence Tao subsequently distilled the main ideas of the proof in a blog post. In this paper, we establish a quantitative strengthening of Sendov's conjecture, namely the quadratic Tang--Zhang inequality. Let p be a polynomial of degree n≥2 whose zeros lie in the closed unit disk, and let ζ1,…,ζn−1 denote its critical points, counted with multiplicity. We prove that, for every zero a of p, ∑j=1n−11/(|a−ζj|2)≥ n−1. Moreover, equality holds if and only if p(z)=c(zn−ω) for some c∈ℂ∖{0} and |ω|=1. We also provide a Lean 4 formalization of the main results.
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BibTeX
@misc{zhang2026beyond,
title = {{Beyond Sendov's conjecture: the quadratic Tang--Zhang inequality}},
author = {Teng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19126},
eprint = {2609.19126},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19126},
}
RIS
TY - UNPB
TI - Beyond Sendov's conjecture: the quadratic Tang--Zhang inequality
AU - Teng Zhang
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.19126
UR - https://arxiv.org/abs/2609.19126
AB - Very recently, Lech Mazur proved the celebrated Sendov conjecture, and Terence Tao subsequently distilled the main ideas of the proof in a blog post. In this paper, we establish a quantitative strengthening of Sendov's conjecture, namely the quadratic Tang--Zhang inequality. Let p be a polynomial of degree n≥2 whose zeros lie in the closed unit disk, and let ζ_1,…,ζ_n−1 denote its critical points, counted with multiplicity. We prove that, for every zero a of p, ∑_j=1^n−11/(|a−ζ_j|^2)≥ n−1. Moreover, equality holds if and only if p(z)=c(z^n−ω) for some c∈ℂ∖{0} and |ω|=1. We also provide a Lean 4 formalization of the main results.
ER -