Sendov's Conjecture

Made by Blagovest Sendov circa 1958, this conjecture has eluded proof despite a heated interest among many mathematicians. It states simply that for a polynomial with and each root located inside the closed unit disk in the complex plane, it must be the case that every closed disk of radius 1 centered at a root will contain a critical point of . Since the Lucas–Gauss theorem implies that the critical points of must themselves lie in the unit disk, it seems completely implausible that the conjecture could be false. Yet, at present, it has not been proven for polynomials with real coefficients or for any polynomial whose degree exceeds 8.
Set the degree of the polynomial (i.e., the number of roots) using the popup menu. Initially, the polynomial is used, so that the roots are the roots of unity. The roots of are blue locators; simply drag a root to change its value. The critical points of (the roots of the derivative) are shown in orange. Sendov's conjecture will be disproved if you can manipulate things in such a way that there is a disk that does not contain an orange point.



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Further reading:
Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford: Oxford University Press, 2002.
G. Schmeisser, "The Conjectures of Sendov and Smale," Approximation Theory: A Volume Dedicated to Blagovest Sendov (B. Bojoanov, ed.), Sofia: DARBA, 2002 pp. 353-369.
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