On a conjecture of Sendov about the critical points of a polynomial

Bojanov, Borislav; Rahman, Q. I.; Szynal, Jan

doi:10.1007/bf01160464

Cited by 27 publications

(16 citation statements)

References 9 publications

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“…Tracing back, we must verify several preliminary results. 1 We set R s and A s 0.562. Since p is extremal and G 1, we 9 9 j 2 assert that it must have the form < To verify this we set r s a y z and note that since z is repeated 0 0 Ž .…”

Section: žmentioning

confidence: 99%

Proof of The Sendov Conjecture for Polynomials of Degree at Most Eight

Brown

Xiang

1999

Journal of Mathematical Analysis and Applications

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Section: žmentioning

confidence: 99%

Proof of The Sendov Conjecture for Polynomials of Degree at Most Eight

Brown

Xiang

1999

Journal of Mathematical Analysis and Applications

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“…We are now in a position to show how the existence of a suitable number p > 1 such that p 5^ p implies that p x < 1. Proof R x and R 2 are the roots of x 2 -5x + [2 + (\/2p)] 2 , and the condition R x < S<R 2 means that S 2 -5S+[2 + (l/2p)] 2 < 0. That is,…”

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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“…As n tends to infinity, u 1 We now estimate the size of the coefficients of P with Proposition 9. Suppose that P ∈ S(n + 1, β) with P monic and …”

Section: Preliminariesmentioning

confidence: 99%

A quadratic approximation to the Sendov radius near the unit circle

Miller

2004

Trans. Amer. Math. Soc.

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Abstract. Define S(n, β) to be the set of complex polynomials of degree n ≥ 2 with all roots in the unit disk and at least one root at β. For a polynomial P , define |P | β to be the distance between β and the closest root of the derivative P . Finally, define rn(β) = sup{|P | β : P ∈ S(n, β)}. In this notation, a conjecture of Bl. Sendov claims that rn(β) ≤ 1.In this paper we investigate Sendov's conjecture near the unit circle, by computing constants C 1 and C 2 (depending only on n) such that rn(β) ∼ 1 + C 1 (1 − |β|) + C 2 (1 − |β|) 2 for |β| near 1. We also consider some consequences of this approximation, including a hint of where one might look for a counterexample to Sendov's conjecture.

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On a conjecture of Sendov about the critical points of a polynomial

Cited by 27 publications

References 9 publications

Proof of The Sendov Conjecture for Polynomials of Degree at Most Eight

Proof of The Sendov Conjecture for Polynomials of Degree at Most Eight

On the Sendov-Ilyeff Conjecture

A quadratic approximation to the Sendov radius near the unit circle

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