Remarks on the Gauss-Lucas Theorem in Higher Dimensional Space

Goodman, A. W.

doi:10.2307/2041850

Cited by 2 publications

(2 citation statements)

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“…First, we remark that since all the charges are positive, it is a direct consequence of the generalized Lucas theorem, proved in [3] (see also [6]), that all the critical points lie in a convex hall of the point charges. Therefore, it is clear that an equilibrium position r (n, β) cannot exceed (the length of) an apothem of a regular polygon, for all non-negative β and all n ≥ 3.…”

Section: Asymptotic Behavior Of Critical Pointsmentioning

confidence: 88%

Equilibria of Riesz Potentials Generated by Point Charges at the Roots of Unity

Bilogliadov

2015

Comput. Methods Funct. Theory

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We consider a special case of Maxwell's problem on the number of equilibrium points of the Riesz potential 1/r 2β (where r is the Euclidean distance and β is the Riesz parameter) for positive unit point charges placed at the vertices of a regular polygon. We show that the equilibrium points are located on the perpendicular bisectors to the sides of the regular polygon, and study the asymptotic behavior of the equilibrium points with regard to the number of charges n and the Riesz parameter β. Finally, we prove that for values of β in a small left-hand side neighborhood of β = 1, the Riesz potential has only one equilibrium point different from the origin on each perpendicular bisector, and one equilibrium point at the origin.

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Section: Asymptotic Behavior Of Critical Pointsmentioning

confidence: 88%

Equilibria of Riesz Potentials Generated by Point Charges at the Roots of Unity

Bilogliadov

2015

Comput. Methods Funct. Theory

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“…for n ≥ 3. Some further generalizations of the Gauss-Lucas theorem to higher dimensions can be found in [27], also cf. [20].…”

Section: Maxwell's Conjecturementioning

confidence: 99%

Borcea’s Variance Conjectures on the Critical Points of Polynomials

Khavinson

Pereira

Putinar

et al. 2011

Notions of Positivity and the Geometry of Polynomials

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Abstract. Closely following recent ideas of J. Borcea, we discuss various modifications and relaxations of Sendov's conjecture about the location of critical points of a polynomial with complex coefficients. The resulting open problems are formulated in terms of matrix theory, mathematical statistics or potential theory. Quite a few links between classical works in the geometry of polynomials and recent advances in the location of spectra of small rank perturbations of structured matrices are established. A couple of simple examples provide natural and sometimes sharp bounds for the proposed conjectures. Mathematics Subject Classification (2000). Primary 12D10; Secondary 26C10, 30C10, 15A42, 15B05.

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Remarks on the Gauss-Lucas Theorem in Higher Dimensional Space

Cited by 2 publications

References 1 publication

Equilibria of Riesz Potentials Generated by Point Charges at the Roots of Unity

Equilibria of Riesz Potentials Generated by Point Charges at the Roots of Unity

Borcea’s Variance Conjectures on the Critical Points of Polynomials

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