Ilyeff's Conjecture on a Corona

Vâjâitu, Viorel; Zaharescu, Alexandru

doi:10.1112/blms/25.1.49

Cited by 12 publications

(22 citation statements)

References 0 publications

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“…Our first consequence improves on estimates in [7] and [12] (by providing a value for the coefficient of the linear term) with Recall that r n (0) = (1/n) 1/(n−1) . This quantity is increasing in n, so it is tempting to conjecture that for all fixed β the quantity r n (β) is increasing in n. Indeed, the graphs in [6, figure 4.8] provide some evidence of this for n = 4, 6, 8, 10, and 12.…”

Section: Introductionsupporting

confidence: 67%

“…It is already known that r 2 (β) = (1 + β)/2 and that Since r n (1) = 1, an obvious place to look for counterexamples to Sendov's conjecture is in a neighborhood of β = 1. This has already been done in [7,Theorem 3] and [12], where a linear upper bound on r n (β) suffices to verify the Sendov conjecture if β is sufficiently close to 1. Unfortunately, having only an upper bound leaves many interesting questions about the conjecture unanswered.…”

Section: Introductionmentioning

confidence: 83%

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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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Section: Introductionsupporting

confidence: 67%

Section: Introductionmentioning

confidence: 83%

A quadratic approximation to the Sendov radius near the unit circle

Miller

2004

Trans. Amer. Math. Soc.

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“…[9] and [20]). Moreover, Sendov's conjecture and related problems have been studied almost exclusively by means of analytical or variational methods and all the results obtained until very recently seemed to suggest a negative answer to the following question: Problem 3.…”

Section: Second Order Variations and Locally Maximal Polynomialsmentioning

confidence: 98%

“…[15]). Moreover, this polynomial was shown to be a local maximum for d ( [9], [20]). These are actually all the examples of α-maximal polynomials known so far, as no such polynomials were found explicitly for |α| < 1.…”

Section: Introductionmentioning

confidence: 95%

Maximal and linearly inextensible polynomials

Borcea¹

2006

MATH. SCAND.

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Abstract. Let S(n, 0) be the set of monic complex polynomials of degree n ≥ 2 having all their zeros in the closed unit disk and vanishing at 0. For p ∈ S(n, 0) denote by |p| 0 the distance from the origin to the zero set of p ′ . We determine all 0-maximal polynomials of degree n, that is, all polynomials p ∈ S(n, 0) such that |p| 0 ≥ |q| 0 for any q ∈ S(n, 0). Using a second order variational method we then show that although some of these polynomials are linearly inextensible, they are not locally maximal for Sendov's conjecture.

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“…A variety of special cases Ž w x . have been dealt with over the years see 3, 11, 19 for references , among w x which we mention that of polynomials with at most five distinct roots 9 , w x as well as Miller's qualitative result 13 according to which those roots of p lying sufficiently close to the unit circle satisfy an even stronger condi-Ž w x. tion than the one stated in Sendov's conjecture see also 20 . w x A recent paper by E. S. Katsoprinakis 7 claims that both conjectures w x are true if n s 6.…”

Section: May Then Be Viewed As An Extremal Problemmentioning

confidence: 96%