On Ilyeff’s conjecture

Meir, A.; Sharma, A.

doi:10.2140/pjm.1969.31.459

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Proof Of P! < 1 When 0 < a <1

Proof Of Theorem 111

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Cited by 26 publications

(11 citation statements)

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“…For a first application of Lemma 1 we need another lemma, which follows by Lemma B (see Section 5) as in [1,8,9,12,14]. If we set (in Lemma B)…”

Section: Proof Of P! < 1 When 0 < a <mentioning

confidence: 99%

On the Sendov-Ilyeff Conjecture

Katsoprinakis

1992

Bulletin of the London Mathematical Society

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“…For a first application of Lemma 1 we need another lemma, which follows by Lemma B (see Section 5) as in [1,8,9,12,14]. If we set (in Lemma B)…”

Section: Proof Of P! < 1 When 0 < a <mentioning

confidence: 99%

On the Sendov-Ilyeff Conjecture

Katsoprinakis

1992

Bulletin of the London Mathematical Society

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“…It is easily checked that q{a) is positive when a £ [0.5,0.8] (in fact, one can show that q{a) > g(0.8) > 175), which implies that r is a strictly increasing function on contradicting (10). Hence /(a) < 1 for 0.5 < a < 0.8.…”

Section: Proof Of Theorem 11mentioning

confidence: 96%

The Sendov Conjecture for Polynomials With at Most Seven Distinct Zeros

Borcea¹

1996

Analysis

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We prove that Sendov's conjecture is true for polynomials of degree N = 7 and determine the so-called extremal polynomials in this case. As a consequence of a general result concerning polynomials with a prescribed number of distinct zeros (Theorem 2.4), we also check the validity of the conjecture for polynomials with at most seven distinct roots. Some properties of the extremal polynomials are discussed as well. 1991 MATH. SUBJ.

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“…Suppose we focus attention on any arbitrarily fixed zero of p(z) and ask for the location of a zero of p'(z) relative to it. This leads to the well-known conjecture of Ilieff and Sendov [3;Problem 4.5] which asserts that if p(z) has the form conjecture is nearly a quarter of a century old and has been verified in some special cases, most notably if p(z) has the form (1) and if (A)2<n<5 [1,6,8]…”

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confidence: 92%