Sendov radius constant ($C_{69}$)

• History surveys. Tao notes that there is a long history of partial results and points to several surveys. [Tao2022-surveys]

• Milestone status. Tao records that the conjecture is known for all $n<9$, and proves it for all sufficiently large $n$. [Tao2022-known-n-less-9] [Tao2022-high-degree]

• Near-unit-circle regime. Tao's proof in this regime refines earlier Miller arguments and invokes Chijiwa's results in an extreme subregime. [Tao2022-near-unit-circle-history]

• Wikipedia page on Sendov's conjecture

References (from the repository page)

• [Tao2022] Tao, Terence. Sendov’s conjecture for sufficiently-high-degree polynomials. Acta Mathematica 229 (2022), no. 2, 347-392 (December 2022). DOI: https://doi.org/10.4310/ACTA.2022.v229.n2.a3. Publisher page: https://projecteuclid.org/journals/acta-mathematica/volume-229/issue-2/Sendovs-conjecture-for-sufficiently-high-degree-polynomials/10.4310/ACTA.2022.v229.n2.a3.full. arXiv PDF: https://arxiv.org/pdf/2012.04125.pdf. Google Scholar

• [Tao2022-sendov-statement] loc: arXiv PDF p.1, Conjecture 1.1 quote: "Conjecture 1.1 (Sendov’s conjecture). Let $f:\mathbb{C}\to\mathbb{C}$ be a polynomial of degree $n\ge 2$ that has all zeroes in the closed unit disk $D(0,1)$. If $\lambda_0$ is one of these zeroes, then $f'$ has at least one zero in $D(\lambda_0,1)$."

• [Tao2022-surveys] loc: arXiv PDF p.1, Introduction paragraph after Conjecture 1.1 quote: "There is a long history of partial results towards this conjecture; see for instance [17], [23], [24], [20], [25] for some surveys of results."

• [Tao2022-known-n-less-9] loc: arXiv PDF p.1, Introduction paragraph after Conjecture 1.1 quote: "The conjecture is known for low degrees, and specifically for all $n<9$ [1]."

• [Tao2022-high-degree] loc: arXiv PDF p.2, Theorem 1.2 quote: "Sendov’s conjecture is true for all sufficiently large $n$. That is, there exists an absolute constant $n_0$ such that Sendov’s conjecture holds for $n\ge n_0$."

• [Tao2022-example-zn1] loc: arXiv PDF p.3, Example 1.4 quote: "For each $n\in\mathbb{N}$, set $f(z):=z^n-1$, and $a:=1$. Then all the zeroes of $f$ lie in $D(0,1)$, and $f'$ just barely has zeroes in $D(a,1)$ since the zeroes are all at the origin which lies on the boundary circle $\partial D(a,1)$."

• [Tao2022-near-unit-circle-history] loc: arXiv PDF p.1, Abstract quote: "for $\lambda_0$ near the unit circle we refine a previous argument of Miller (and also invoke results of Chijiwa when $\lambda_0$ is extremely close to the unit circle)"

• [BrownXiang1999] Brown, J. E.; Xiang, G. Proof of the Sendov conjecture for polynomials of degree at most eight. Journal of Mathematical Analysis and Applications 232 (1999), 272-292. Google Scholar

• [Miller1993] Miller, M. J. On Sendov's conjecture for roots near the unit circle. Journal of Mathematical Analysis and Applications 175 (1993), no. 2, 632-639. Google Scholar

• [Chijiwa2011] Chijiwa, T. A quantitative result on Sendov's conjecture for a zero near the unit circle. Hiroshima Mathematical Journal 41 (2011), no. 2, 23-273. DOI: https://doi.org/10.32917/hmj/1314204564. Publisher page: https://projecteuclid.org/journals/hiroshima-mathematical-journal/volume-41/issue-2/A-quantitative-result-on-Sendovs-conjecture-for-a-zero-near/10.32917/hmj/1314204564.full. Google Scholar

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1; best known upper bound 2. Values marked with an asterisk have minimal verification.

Known upper bounds

Known lower bounds