# Sendov's Conjecture

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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 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.

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Contributed by: Bruce Torrence (March 2011)

Additional contributions by: Paul Abbott

Open content licensed under CC BY-NC-SA

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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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