Proved
Sendov for n ≥ n₀ ↗
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Sendov’s conjecture for sufficiently high degree polynomials · Abstract; Theorem 1.2 →
Observed Oct 21, 2026
Theorem · version 1
Sendov for all sufficiently large degree
Statement
There is a constant n₀ such that for every polynomial p of degree n ≥ n₀ whose zeros all satisfy |z| ≤ 1, each zero a has a critical point ζ with |ζ − a| ≤ 1.
Assumptions
- Complex polynomials of degree at least n₀ with all zeros in the closed unit disk.
- n₀ is not explicit.
Scope
Large degree only; the degrees between 9 and n₀ were left open until the 2026 proof.
Evidence
Source reports
Connections
In this problem · publication v7 · Selected context
Sendov’s conjecture →
Special case · Proved
A range of degrees proved before the full statement.
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v8
Sendov’s conjecture →
Special case · Proved
A range of degrees proved before the full statement.
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Sendov’s conjecture for sufficiently high degree polynomials →
- Abstract; Theorem 1.2 → · Original passage ↗
Recorded wording
Tao proves that there is an absolute constant n₀ such that Sendov’s conjecture holds for every degree n ≥ n₀; the compactness argument gives no explicit n₀.
paraphrase · OpenMath supplies original paraphrases and links; no proof text is republished.
Terence Tao · Published · Publication date 2022
Acta Mathematica 229 (2022), 347–392, DOI 10.4310/acta.2022.v229.n2.a3 checked against Crossref; arXiv v2 pinned by content hash.
Revision & attribution
Source revision: arxiv-2012.04125v2. Observed Oct 21, 2026.
No AI contribution is asserted by this source record.
Source updated Aug 12, 2026.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Sendov for n ≥ n₀ ↗
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026 · Source passage →
Record details
- Claim ID
- claim-sendov-large-degree
- Statement version
- 1
- SHA-256
- 4645fb731506d26b9611f62fb3f4858410e42688bddd3e6a96b668a03340fe10