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Sendov Conjecture


The Sendov conjecture, proposed by Blagovest Sendov circa 1958, that for a polynomial f(z)=(z-r_1)(z-r_2)...(z-r_n) with n>=2 and each root r_k located inside the closed unit disk |z|<=1 in the complex plane, it must be the case that every closed disk of radius 1 centered at a root r_k will contain a critical point of f. Since the Lucas-Gauss theorem implies that the critical points (i.e., the roots of the derivative) of f must themselves lie in the unit disk, it seems completely implausible that the conjecture could be false. Yet at present it has not been proved even for polynomials with real coefficients, nor for any polynomials whose degree exceeds eight.


This entry contributed by Bruce Torrence

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References

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

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Sendov Conjecture

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Torrence, Bruce. "Sendov Conjecture." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SendovConjecture.html

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