Proved
Known for n < 9 ↗
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026
Theorem · version 1
Sendov for degree at most eight
Statement
For every polynomial p of degree n with 2 ≤ n ≤ 8 whose zeros all satisfy |z| ≤ 1, each zero a has a critical point ζ with |ζ − a| ≤ 1.
Assumptions
- Complex polynomials of degree 2 to 8 with all zeros in the closed unit disk.
Scope
The low-degree cases, settled before the general proof.
Evidence
Source reports
Connections
In this problem · publication v6 · 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 →
- §1, before Theorem 1.2; reference [1] → · Original passage ↗
Recorded wording
The introduction records that the conjecture is known for all n < 9, citing Brown and Xiang’s proof for polynomials of degree at most eight.
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
Known for n < 9 ↗
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-low-degree
- Statement version
- 1
- SHA-256
- adb307b4116e91800ed3a27cf605798ae2dcaa3fbe0e926891eab6a7bb602c47