On the location of critical points of some complex polynomials

• Autorzy: Dronka J. , Sokół J.
• Języki publikacji: EN

Abstrakty

EN

Let P(a,n) be the set of all complex polynomials of degree n which have all their roots in the closed unit disk and one fixed root at a, 0 < a < 1. In this paper we show thiat for n > 3 all critical points of the polynomial f(z)= (zn-l+ l)(z - a) lie outside the set K(a,n) consisting of all b such that for some c the polynomial p(z)= (z - b) n - c belongs to P(a,n). Hence we infer that minimal sets satisfying the Sendov property (i.e. containing at least one critical point of each pzawiera P(a, n) ) exist but they are not unique.

Słowa kluczowe

EN

polynomials; zeros of the derivative; Sendov; minimal sets

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 1999
• Tom: Vol. 32, nr 2
• Strony: 297--302
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Dronka J.

• Department of Mathematics, Technical University of Rzeszów, ul. W. Pola 2, 35-959 Rzeszów, Poland

autor

Sokół J.

• Department of Mathematics, Technical University of Rzeszów, ul. W. Pola 2, 35-959 Rzeszów, Poland

Bibliografia

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• [6] M. J. Miller, Maximal polynomials and the Ilieff-Sendov conjecture, Trans. Amer. Math. Soc. 321 (1990), 285-303.
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• [8] M. J. Miller, On Sendov's conjecture for roots near the unit circle, J. Math. Anal. Appl. 175 (1993), 632-639.
• [9] E. B. Saff and J. B. Twomey, A note on the location of critical points of polynomials, Proc. Amer. Math. Soc. 27 (1971), 303-308.