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Maximal polynomials and the Ilieff-Sendov conjecture


Author: Michael J. Miller
Journal: Trans. Amer. Math. Soc. 321 (1990), 285-303
MSC: Primary 30C15; Secondary 26C10, 30C10
DOI: https://doi.org/10.1090/S0002-9947-1990-0965744-X
MathSciNet review: 965744
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Abstract: In this paper, we consider those complex polynomials which have all their roots in the unit disk, one fixed root, and all the roots of their first derivatives as far as possible from a fixed point. We conjecture that any such polynomial has all the roots of its derivative on a circle centered at the fixed point, and as many of its own roots as possible on the unit circle. We prove a part of this conjecture, and use it to define an algorithm for constructing some of these polynomials. With this algorithm, we investigate the 1962 conjecture of Sendov and the 1969 conjecture of Goodman, Rahman and Ratti and (independently) Schmeisser, obtaining counterexamples of degrees $ 6$, $ 8$, $ 10$, and $ 12$ for the latter.


References [Enhancements On Off] (What's this?)

  • [1] L. Ahlfors, Complex analysis, 2nd ed., McGraw-Hill, New York, 1966. MR 510197 (80c:30001)
  • [2] J. Farkas, Theorie der einfachen Ungleichungen, J. Reine Angew. Math. 124 (1901), 1-27.
  • [3] F. Gacs, On polynomials whose zeros are in the unit disk, J. Math. Anal. Appl. 36 (1971), 627-637. MR 0288237 (44:5435)
  • [4] A. W. Goodman, Q. I. Rahman, and J. S. Ratti, On the zeros of a polynomial and its derivative, Proc. Amer. Math. Soc. 21 (1969), 273-274. MR 0239062 (39:421)
  • [5] W. K. Hayman, Research problems in function theory, The Athelone Press, London, 1967. MR 0217268 (36:359)
  • [6] F. Lucas, Theoremes concernant les equations algebriques, C. R. Acad. Sci. Paris 78 (1874), 431-433.
  • [7] M. Marden, Geometry of polynomials, 2nd ed., Amer. Math. Soc., Providence, R. I., 1966. MR 0225972 (37:1562)
  • [8] -, Conjectures on the critical points of a polynomial, Amer. Math. Monthly 90 (1983), 267-276. MR 700266 (84e:30007)
  • [9] A. Meier and A. Sharma, On Ilyeff's conjecture, Pacific J. Math. 31 (1969), 459-467. MR 0249587 (40:2831)
  • [10] D. Phelps and R. Rodriguez, Some properties of extremal polynomials for the Ilieff conjecture, Kodai Math. Sem Rep. 24 (1972), 172-175. MR 0304618 (46:3753)
  • [11] C. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, Göttingen, 1975. MR 0507768 (58:22526)
  • [12] G. Schmeisser, Bemerkungen zu einer Vermutung von Ilieff, Math. Z. 111 (1969), 121-125 MR 0264040 (41:8637)
  • [13] -, On Ilieff's conjecture, Math. Z. 156 (1977), 165-173. MR 0486436 (58:6182)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0965744-X
Keywords: Ilieff, Sendov, geometry of polynomials, roots of polynomials, maximal polynomials
Article copyright: © Copyright 1990 American Mathematical Society

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