It has been more than fifty years for famous Sendov's conjecture which states that if $p(z)$ is a polynomial of degree $n$ having all its zeros in the unit disc $|z|\leq 1$ then each of the n roots is at a distance no more than 1 from at least one critical point. It has also been more than twenty years for its proof for polynomials of degree $n=8$. I believe, after that no progress on greater values of $n.$ Some claims are there on arxiv platform, which are not yet validated by the experts I feel. There is a review article http://www.indianmathsociety.org.in/mathstudent-part-1-2019.pdf#page=101

It is very sad and unfortunate to hear that Sendov passed away on January 19, 2020 and a nice tribute article is published in Journal of Approximation Theory, see https://www.sciencedirect.com/science/article/pii/S0021904520300423?via%3Dihub

There is another paper on the conjecture for high degree of polynomials, see https://www.ams.org/journals/proc/2014-142-04/S0002-9939-2014-11888-0/home.html

A new approach of solving this problem using Grace Theorem by constructing apolar polynomials is given in the article 'A remark on Sendov Conjecture' published in Comptes rendus de l’Acade'mie bulgare des Sciences, Vol 71 (6), 2018, pp.731-734.

Let me know if anything more significant latest contributions?


A 2010 status report is given by D. Khavinson et al. in Borcea's variance conjectures on the critical points of polynomials. A 2019 update is in A note on a recent attempt to prove Sendov's conjecture, by N.A. Rather and Suhail Gulzar.

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