Sendov radius constant

Description of constant

Let $f:\mathbb{C}\to \mathbb{C}$ be a polynomial of degree $n\ge 2$ whose zeroes all lie in the closed unit disk $$\begin{gathered} D(0,1)= \\ z:|z|\le 1 \end{gathered}$$. Sendov's conjecture states that if ${\lambda }_0$ is one of these zeroes, then $f^{\prime }$ has at least one zero in $D({\lambda }_0,1)$.

$$\text{every zero }{\lambda }_0\text{ of }f\text{ has a critical point in }D({\lambda }_0,1).$$

<a href="#Tao2022-sendov-statement">[Tao2022-sendov-statement]</a>

We define the Sendov radius constant by

$$C_{69}:=\inf \left\{R>0:\begin{array}{l}\text{for all }n\ge 2,\text{all degree-}n\text{ polynomials }f\text{ with zeroes in }D(0,1),\\ \text{and all zeroes }{\lambda }_0\text{ of }f,\text{there exists a zero of }{f}^{\prime }\text{ in }D({\lambda }_0,R)\end{array}\right\}.$$

With this definition, Sendov's conjecture is equivalent to $C_{69}\le 1$. <a href="#Tao2022-sendov-statement">[Tao2022-sendov-statement]</a>

A standard example gives $C_{69}\ge 1$: take $f(z)=z^n-1$ and ${\lambda }_0=1$, for which the zeroes of $f^{\prime }$ are at the origin and lie on $\partial D({\lambda }_0,1)$. <a href="#Tao2022-example-zn1">[Tao2022-example-zn1]</a>

A trivial geometric bound gives $C_{69}\le 2$ (all zeroes of $f$ and $f^{\prime }$ lie in $D(0,1)$).

Hence the best established range currently is

$$1\le C_{69}\le 2.$$

Known upper bounds

Known lower bounds

Additional comments and links

• History surveys. Tao notes that there is a long history of partial results and points to several surveys. <a href="#Tao2022-surveys">[Tao2022-surveys]</a>

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

• Near-unit-circle regime. Tao's proof in this regime refines earlier Miller arguments and invokes Chijiwa's results in an extreme subregime. <a href="#Tao2022-near-unit-circle-history">[Tao2022-near-unit-circle-history]</a>

• Wikipedia page on Sendov's conjecture

References

• <a id="Tao2022"></a>[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
• <a id="Tao2022-sendov-statement"></a>[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 $f^{\prime }$ has at least one zero in $D({\lambda }_0,1)$."
• <a id="Tao2022-surveys"></a>[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."
• <a id="Tao2022-known-n-less-9"></a>[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]."
• <a id="Tao2022-high-degree"></a>[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 $n\ge n_0$."
• <a id="Tao2022-example-zn1"></a>[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 $f^{\prime }$ just barely has zeroes in $D(a,1)$ since the zeroes are all at the origin which lies on the boundary circle $\partial D(a,1)$."
• <a id="Tao2022-near-unit-circle-history"></a>[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)"

• <a id="BrownXiang1999"></a>[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

• <a id="Miller1993"></a>[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

• <a id="Chijiwa2011"></a>[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

What counts as progress

• A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
• A formal proof (Lean) of a known bound, or a precise error in a claimed one.
• New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.