A note on the structure of the zeros of a polynomial and Sendov's conjecture





In this note we prove a result that highlights an interesting connection between the structure of the zeros of a polynomial \(p(z)\) and Sendov's conjecture.


Polynomials , zeros , critical points

Mathematics Subject Classification:

30A10 , 30C15
  • Pages: 515–521
  • Date Published: 2023-12-31
  • Vol. 25 No. 3 (2023)

B. D. Bojanov, Q. I. Rahman, and J. Szynal, “On a conjecture of Sendov about the critical points of a polynomial,” Math. Z., vol. 190, no. 2, pp. 281–285, 1985, doi: 10.1007/BF01160464.

I. Borcea, “On the Sendov conjecture for polynomials with at most six distinct roots,” J. Math. Anal. Appl., vol. 200, no. 1, pp. 182–206, 1996, doi: 10.1006/jmaa.1996.0198.

J. E. Brown and G. Xiang, “Proof of the Sendov conjecture for polynomials of degree at most eight,” J. Math. Anal. Appl., vol. 232, no. 2, pp. 272–292, 1999, doi: 10.1006/jmaa.1999.6267.

T. P. Chalebgwa, “Sendov’s conjecture: a note on a paper of Dégot,” Anal. Math., vol. 46, no. 3, pp. 447–463, 2020, doi: 10.1007/s10476-020-0050-x.

J. Dégot, “Sendov conjecture for high degree polynomials,” Proc. Amer. Math. Soc., vol. 142, no. 4, pp. 1337–1349, 2014, doi: 10.1090/S0002-9939-2014-11888-0.

W. K. Hayman, Research problems in function theory. The Athlone Press [University of London], London, 1967.

P. Kumar, “A remark on Sendov conjecture,” C. R. Acad. Bulgare Sci., vol. 71, no. 6, pp. 731–734, 2018.

M. J. Miller, “Maximal polynomials and the Ilieff-Sendov conjecture,” Trans. Amer. Math. Soc., vol. 321, no. 1, pp. 285–303, 1990, doi: 10.2307/2001603.

Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, ser. London Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, Oxford, 2002, vol. 26.

Z. Rubinstein, “On a problem of Ilyeff,” Pacific J. Math., vol. 26, pp. 159–161, 1968.

G. Schmeisser, “Bemerkungen zu einer Vermutung von Ilieff.” Math Z., vol. 111, pp. 121–125, 1969, doi: 10.1007/BF01111192.

G. Schmeisser, “Zur Lage der kritischen Punkte eines Polynoms,” Rendiconti del Seminario Matematico della Università di Padova, vol. 46, pp. 405–415, 1971.

G. M. Sofi and W. M. Shah, “On Sendov’s conjecture,” Rend. Circ. Mat. Palermo (2), vol. 72, no. 1, pp. 493–497, 2023, doi: 10.1007/s12215-021-00690-y.

G. M. Sofi, S. A. Ahanger, and R. B. Gardner, “Some classes of polynomials satisfying Sendov’s conjecture,” Studia Sci. Math. Hungar., vol. 57, no. 4, pp. 436–443, 2020, doi: 10.1556/012.2020.57.4.1475.

T. Tao, “Sendov’s conjecture for sufficiently-high-degree polynomials,” Acta Math., vol. 229, no. 2, pp. 347–392, 2022.


Download data is not yet available.



How to Cite

G. M. Sofi and W. M. Shah, “A note on the structure of the zeros of a polynomial and Sendov’s conjecture”, CUBO, vol. 25, no. 3, pp. 515–521, Dec. 2023.