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Sendov's conjecture holds for every degree n≥ 10200000

  • Teng Zhang
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Abstract

Sendov's conjecture, first formulated in 1958, asserts that if a complex polynomial f of degree n ≥ 2 has all its zeros in the closed unit disk z ∈ ℂ : |z| ≤ 1, then, for every zero λ0 of f, there exists a critical point ζ of f such that |ζ−λ0| ≤ 1. The conjecture was previously known to hold for polynomials of degree n ≤ 8, as well as in several special higher-degree cases. In 2022, using compactness methods, balayage, and the argument principle, Tao~\cite{Tao22} proved that there exists an absolute constant n0 such that Sendov's conjecture holds for all n ≥ n0. However, Tao's argument does not provide an explicit admissible value of n0. In the present paper, we make Tao's result effective and prove that one may take n0 = 10200000.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026sendovs,
  title = {{Sendov's conjecture holds for every degree $n\ge 10^200000$}},
  author = {Teng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20256},
  eprint = {2609.20256},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20256},
}
RIS
TY  - UNPB
TI  - Sendov's conjecture holds for every degree n≥ 10^200000
AU  - Teng Zhang
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.20256
UR  - https://arxiv.org/abs/2609.20256
AB  - Sendov's conjecture, first formulated in 1958, asserts that if a complex polynomial f of degree n ≥ 2 has all its zeros in the closed unit disk z ∈ ℂ : |z| ≤ 1, then, for every zero λ_0 of f, there exists a critical point ζ of f such that |ζ−λ_0| ≤ 1. The conjecture was previously known to hold for polynomials of degree n ≤ 8, as well as in several special higher-degree cases. In 2022, using compactness methods, balayage, and the argument principle, Tao~\cite{Tao22} proved that there exists an absolute constant n_0 such that Sendov's conjecture holds for all n ≥ n_0. However, Tao's argument does not provide an explicit admissible value of n_0. In the present paper, we make Tao's result effective and prove that one may take n_0 = 10^200000.
ER  -