Abstract
We prove non-archimedean analogue of Sendov’s conjecure. We also provide complete list of polynomials over an algebraically closed non-archimedean field \(K\) that satisfy the optimal bound in the Sendov’s conjecture.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We only consider characteristic 0 fields.
References
B. Bojanov, “Extremal problems for polynomials in the complex plane,” Approximation and Computation, pp. 61–85, Springer Optim. Appl. 42 (Springer, New York, 2011).
J. E. Brown and X. Guangping, “Proof of the Sendov conjecture for polynomials of degree at most eight,” J. Math. Anal. Appl. 232 (2), 272–292 (1999).
T. P. Chalebgwa, “Sendov’s Conjecture: A Note on a paper of Dégot,” Anal. Math. 46 (3), 447–463 (2020).
J. Dégot, “Sendov conjecture for high degree polynomials,” Proc. Amer. Math. Soc. 142 (4), 1337–1349 (2014).
I. G. Kasmalkar, “On the Sendov conjecture for a root close to the unit circle,” Aust. J. Math. Anal. Appl. 11 (1), art. 4, pp. 34 (2014).
M. Marden, “Conjectures on the critical points of a polynomial,” Amer. Math. Monthly 90, 267–276 (1983).
J. Neukirch, Algebraic Number Theory (Springer Science & Business Media, 2013).
T. Tao, “Sendov’s conjecture for sufficiently high degree polynomials,” arXiv:2012.04125 (2020).
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Choi, D., Lee, S. Non-Archimedean Sendov’s Conjecture. P-Adic Num Ultrametr Anal Appl 14, 77–80 (2022). https://doi.org/10.1134/S2070046622010058
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070046622010058