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Corrigendum: ‘on The Sendov‐Ilyeff Conjecture’

Corrigendum: ‘on The Sendov‐Ilyeff Conjecture’ ERRATUM TO 'ON THE SENDOV-ILYEFF CONJECTURE' EMMANUEL S. KATSOPRINAKIS 1. Introduction The conjecture of Sendov (also known as 'Ilyeff's conjecture') asserts that if p(z) = (z — z j (z — z ).. . (z — z ), n ^ 2, is a complex polynomial with zeros \z \ ^ 1, 2 n k 1 ^ k < n, then each disk \z — z \ ^ 1 contains a zero of p'{z). In [6] we claimed to have proved this conjecture for n = 6 using some auxiliary results, but in a letter to us (dated 1 February 1994), Arne Meurman noticed that the conclusion of Lemma B among these auxiliary results is false. In view of this, our Theorem 2 in [6] (see Theorem 1 below) still needs to be proved, and we shall do this in Section 2. We also mention that for n ^ 5, F. Gacs proved in [4] the following stronger conjecture proposed by Goodman, Rahman and Ratti [5] and independently by Schmeisser [10]: 'Under the hypotheses of the conjecture of Sendov, each disk |z —(z /2)| ^ 1— (|z |/2) contains a zero of p'(z)\ fc fc (For n ^ 4, see http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Corrigendum: ‘on The Sendov‐Ilyeff Conjecture’

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/28.6.605
Publisher site
See Article on Publisher Site

Abstract

ERRATUM TO 'ON THE SENDOV-ILYEFF CONJECTURE' EMMANUEL S. KATSOPRINAKIS 1. Introduction The conjecture of Sendov (also known as 'Ilyeff's conjecture') asserts that if p(z) = (z — z j (z — z ).. . (z — z ), n ^ 2, is a complex polynomial with zeros \z \ ^ 1, 2 n k 1 ^ k < n, then each disk \z — z \ ^ 1 contains a zero of p'{z). In [6] we claimed to have proved this conjecture for n = 6 using some auxiliary results, but in a letter to us (dated 1 February 1994), Arne Meurman noticed that the conclusion of Lemma B among these auxiliary results is false. In view of this, our Theorem 2 in [6] (see Theorem 1 below) still needs to be proved, and we shall do this in Section 2. We also mention that for n ^ 5, F. Gacs proved in [4] the following stronger conjecture proposed by Goodman, Rahman and Ratti [5] and independently by Schmeisser [10]: 'Under the hypotheses of the conjecture of Sendov, each disk |z —(z /2)| ^ 1— (|z |/2) contains a zero of p'(z)\ fc fc (For n ^ 4, see

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Nov 1, 1996

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