Vestnik КRAUNC. Fiz.-Mat. Nauki. 2026. vol. 54. no. 1. P. 56 – 63. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2026-54-1-56-63
Research Article
Full text in English
MSC 30C15, 46L05

Contents of this issue

Read Russian Version

C^{\ast}-algebraic Gauss-Lucas Theorem and C^{\ast}-algebraic Sendov’s Conjecture

K. M. Krishna^{\ast}

School of Mathematics and Natural Sciences, Chanakya University Global Campus, NH-648, Haraluru Village, Devanahalli Taluk, Bengaluru North District, Karnataka State, 562 110, India

Abstract. Building on a foundational result established by Robertson [Proc. Edinburgh Math. Soc., 1976], we develop a framework for differentiation of maps defined on specific classes of unital commutative C^{\ast}-algebras. This construction allows us to extend classical analytic ideas into the C^{\ast}-algebraic setting, thereby enriching the interplay between operator theory and complex analysis. Within this framework, we derive a C^{\ast}-algebraic analogue of the Gauss–Lucas theorem, which traditionally describes the geometric location of polynomial zeros in relation to their derivatives. Furthermore, we propose a C^{\ast}-algebraic version of Sendov’s conjecture, a long-standing problem in complex analysis concerning the proximity of critical points to polynomial zeros. Our formulation adapts this conjecture to the algebraic context and provides new insights into its structure. As a concrete verification, we establish the validity of the C^{\ast}-algebraic Sendov’s conjecture for all polynomials of degree two, thereby offering evidence for its broader applicability.

Key words: Sendov’s conjecture, C^{\ast}-algebra, Gauss-Lucas theorem.

Received: 01.12.2025; Revised: 02.02.2026; Accepted: 05.02.2026; First online: 29.03.2026

For citation. Krishna K.M. C^{\ast}-algebraic Gauss-Lucas theorem and C^{\ast}-algebraic Sendov’s conjecture. Vestnik KRAUNC. Fiz.-mat. nauki. 2026, 54: 1, 56-63. EDN: EILBHL. https://doi.org/10.26117/2079-6641-2026-54-1-56-63.

Funding.The study was conducted without the support of foundations.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for the submission of the final version of the article for publication.

^{\ast}Correspondence: E-mail: kmaheshak@gmail.com

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Krishna K. M., 2026

© Institute of Cosmophysical Research and Radio Wave Propagation, 2026 (original layout, design, compilation)

References

  1. Hayman W. K. Research problems in function theory. London: The Athlone Press, 1967.
  2. Marden M. Conjectures on the critical points of a polynomial, Amer. Math. Monthly, 1983. vol. 90, no. 4, pp. 267–276 DOI: 10.1080/00029890.1983.11971207.
  3. Rahman Q. I., Schmeisser G. Analytic theory of polynomials, London Mathematical Society Monographs. New Series, vol. 26. Oxford: The Clarendon Press, Oxford University Press, 2002.
  4. Brannan D. A.On a conjecture of Ilieff, Proc. Cambridge Philos. Soc., 1968. vol. 64, pp. 83–85 DOI: 10.1017/S0305004100042596.
  5. Rubinstein Z.On a problem of Ilieff, Pacific J. Math., 1968. vol. 26, pp. 159–161 DOI: 10.2140/pjm.1968.26.159.
  6. Meir A., Sharma A.On Ilieff’s conjecture, Pacific J. Math., 1969. vol. 31, pp. 459–467 DOI: 10.2140/pjm.1969.31.459.
  7. Katsoprinakis E. S.On the Sendov–Ilieff conjecture, Bull. London Math. Soc., 1996. vol. 28, no. 6, pp. 605–612 DOI: 10.1112/blms/28.6.605.
  8. Borcea I.On the Sendov conjecture for polynomials with at most six distinct roots, J. Math. Anal. Appl., 1996. vol. 200, no. 1, pp. 182–206 DOI: 10.1006/jmaa.1996.0198.
  9. Brown J. E.A proof of the Sendov conjecture for polynomials of degree seven, Complex Variables Theory Appl., 1997. vol. 33, no. 1-4, pp. 75–95 DOI: 10.1080/17476939708815013.
  10. Brown J. E., Xiang G. Proof of the Sendov conjecture for polynomials of degree at most eight, J. Math. Anal. Appl., 1999. vol. 232, no. 2, pp. 272–292 DOI: 10.1006/jmaa.1999.6267.
  11. Tao T. Sendov’s conjecture for sufficiently high degree polynomials, Acta Math., 2022. vol. 229, no. 2, pp. 347–392 DOI: 10.4310/ACTA.2022.v229.n2.a3.
  12. Borcea I. The Sendov conjecture for polynomials with at most seven distinct zeros, Analysis, 1996. vol. 16, no. 2, pp. 137–159 DOI: 10.1524/anly.1996.16.2.137.
  13. Brown J. E.On the Sendov conjecture for sixth degree polynomials, Proc. Amer. Math. Soc., 1991. vol. 113, no. 4, pp. 939–946 DOI: 10.1090/S0002-9939-1991-1081693-X.
  14. Brown J. E.On the Ilieff–Sendov conjecture, Pacific J. Math., 1988. vol. 135, no. 2, pp. 223–232 DOI: 10.2140/pjm.1988.135.223.
  15. Bojanov B. Extremal problems for polynomials in the complex plane / Approximation and computation. In Honor of Gradimir V. Milovanović, vol. 42. N-Y, Springer, 2010, pp. 61–85.
  16. Schmeisser G. Bemerkungen zu einer Vermutung von Ilieff, Math. Z., 1969. vol. 111, pp. 121–125 DOI: 10.1007/BF01111192.
  17. Degot J. Sendov conjecture for high degree polynomials, Proc. Amer. Math. Soc., 2014. vol. 142, no. 4, pp. 1337–1349 DOI: 10.1090/S0002-9939-2014-11888-0.
  18. Chalebgwa T.P. Sendov’s conjecture: a note on a paper of Degot, Anal. Math., 2020. vol. 46, no. 3, pp. 447–463 DOI: 10.1007/s10476-020-0050-x.
  19. Goodman A. W., Rahman Q. I., Ratti J. S.On the zeros of a polynomial and its derivative, Proc. Amer. Math. Soc., 1969. vol. 21, pp. 273–274 DOI: 10.1090/S0002-9939-1969-0239062-6.
  20. Miller M. J.On Sendov’s conjecture for roots near the unit circle, J. Math. Anal. Appl., 1993. vol. 175, no. 2, pp. 632–639 DOI: 10.1006/jmaa.1993.1194.
  21. Chijwa T.A quantitative result on Sendov’s conjecture for a zero near the unit circle, Hiroshima Math. J., 2011. vol. 41, no. 2, pp. 235–273 DOI: 10.32917/hmj/1314204564.
  22. Kasmalkar I. G.On the Sendov conjecture for a root close to the unit circle, Aust. J. Math. Anal. Appl., 2014. vol. 11, no. 1, 34.
  23. Joyal A.On the zeros of a polynomial and its derivative, J. Math. Anal. Appl., 1969. vol. 26, pp. 315–317 DOI: 10.1016/0022-247X(69)90155-3.
  24. Vajaitu V., Zaharescu A. Ilyeff’s conjecture on a corona, Bull. London Math. Soc., 1993. vol. 25, no. 1, pp. 49–54 DOI: 10.1112/blms/25.1.49.
  25. Miller M. J.A quadratic approximation to the Sendov radius near the unit circle, Trans. Amer. Math. Soc., 2005. vol. 357, no. 3, pp. 851–873 DOI: 10.1090/S0002-9947-04-03766-3.
  26. Miller M. J. Unexpected local extrema for the Sendov conjecture, J. Math. Anal. Appl., 2008. vol. 348, no. 1, pp. 461–468 DOI: 10.1016/j.jmaa.2008.07.049.
  27. Saff E. B., Twomey J. B.A note on the location of critical points of polynomials, Proc. Amer. Math. Soc., 1971. vol. 27, pp. 303–308 DOI: 10.1090/S0002-9939-1971-0271312-1.
  28. Miller M. J. Maximal polynomials and the Ilieff–Sendov conjecture, Trans. Amer. Math. Soc., 1990. vol. 321, no. 1, pp. 285–303 DOI: 10.1090/S0002-9947-1990-0965744-X.
  29. Borcea J.Two approaches to Sendov’s conjecture, Arch. Math. (Basel), 1998. vol. 71, no. 1, pp. 46–54 DOI: 10.1007/s000130050232.
  30. Rahman Q. I., Tariq Q. M.On a problem related to the conjecture of Sendov about the critical points of a polynomial, Canad. Math. Bull., 1987. vol. 30, no. 4, pp. 476–480 DOI: 10.4153/CMB-1987-070-9.
  31. Goodman A. W.On the derivative with respect to a point, Proc. Amer. Math. Soc., 1987. vol. 101, no. 2, pp. 327–330 DOI: 10.1090/S0002-9939-1987-0902551-3.
  32. Pawlowski P.On the zeros of a polynomial and its derivatives, Trans. Amer. Math. Soc., 1998. vol. 350, no. 11, pp. 4461–4472 DOI: 10.1090/S0002-9947-98-02291-0.
  33. Miller M. J. Continuous independence and the Ilieff–Sendov conjecture, Proc. Amer. Math. Soc., 1992. vol. 115, no. 1, pp. 79–83 DOI: 10.1090/S0002-9939-1992-1113647-X.
  34. Chijwa T.A quantitative result on polynomials with zeros in the unit disk, Proc. Japan Acad. Ser. A Math. Sci., 2010. vol. 86, no. 10, pp. 165–168 DOI: 10.3792/pjaa.86.165.
  35. Schmieder G., Szynal J.On the distribution of the derivative zeros of a complex polynomial, Complex Var. Theory Appl., 2002. vol. 47, no. 3, pp. 239–241 DOI: 10.1080/02781070290001373.
  36. Daepp U., Gorkin P., Voss K.Poncelet’s theorem, Sendov’s conjecture, and Blaschke products, J. Math. Anal. Appl., 2010. vol. 365, no. 1, pp. 93–102 DOI: 10.1016/j.jmaa.2009.09.058.
  37. Bojanov B. D., Rahman Q. I., Szynal J.On a conjecture of Sendov about the critical points of a polynomial, Math. Z., 1985. vol. 190, no. 2, pp. 281–285 DOI: 10.1007/BF01160464.
  38. Byrne A. Some results for the Sendov conjecture, J. Math. Anal. Appl., 1996. vol. 199, no. 3, pp. 754– 768 DOI: 10.1006/jmaa.1996.0173.
  39. Byrne A. Results pertaining to the Sendov conjecture, J. Math. Anal. Appl., 1997. vol. 212, no. 2, pp. 333–342 DOI: 10.1006/jmaa.1997.5500.
  40. Choi D., Lee S. Non-Archimedean Sendov’s Conjecture, p-Adic Numbers Ultrametric Anal. Appl., 2022. vol. 14, no. 1, pp. 77–80 DOI: 10.1134/S2070046622010058.
  41. Krishna K. M. C*-algebraic Schur product theorem, Polya–Szegő–Rudin question and Novak’s conjecture, J. Korean Math. Soc., 2022. vol. 59, no. 4, pp. 789–804.
  42. Bratteli O., Jorgensen P. E. T., Kishimoto A., Robinson D. W.A C*-algebraic Schoenberg theorem, Ann. Inst. Fourier, 1984. vol. 34, no. 3, pp. 155–187 DOI: 10.5802/aif.981.
  43. Belavkin V.P., Staszewski P. C*-algebraic generalization of relative entropy and entropy, Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A, 1982. vol. 37, pp. 51–58.
  44. Berzi V. Remarks on the Goldstone theorem in the C*-algebraic approach, Lett. Math. Phys., 1981. vol. 5, pp. 373–377 DOI: 10.1007/BF02285308.
  45. Robertson G.On the density of the invertible group in C*-algebras, Proc. Edinburgh Math. Soc., 1976. vol. 20, no. 2, pp. 153–157 DOI: 10.1017/S001309150001066X.

Information about the author

Krishna Mahesh – PhD (Phys. & Math.), Assistant Professor, School of Mathematics and Natural Sciences, Chanakya University Global Campus, Haraluru, India, ORCID 0000-0003-4872-8634.

Read articleDownload article