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.

The low-degree cases, settled before the general proof.

Proved

Only the exact statement and passage cited here; not an independent OpenMath assessment.

Observed Oct 21, 2026

In this problem · publication v8

Special case · Proved

A range of degrees proved before the full statement.

Relationship not independently reviewed. This concerns its use in this problem.

§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.

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

Only the exact statement and passage cited here; not an independent OpenMath assessment.

Observed Oct 21, 2026 · Source passage →

Record details