Acta Mathematica Hungarica

, Volume 137, Issue 1–2, pp 10–26 | Cite as

A fascinating polynomial sequence arising from an electrostatics problem on the sphere

  • J. S. Brauchart
  • P. D. Dragnev
  • E. B. Saff
  • C. E. van de Woestijne
Article

Abstract

A positive unit point charge approaching from infinity a perfectly spherical isolated conductor carrying a total charge of +1 will eventually cause a negatively charged spherical cap to appear. The determination of the smallest distance ρ(d) (d is the dimension of the unit sphere) from the point charge to the sphere where still all of the sphere is positively charged is known as Gonchar’s problem. Using classical potential theory for the harmonic case, we show that 1+ρ(d) is equal to the largest positive zero of a certain sequence of monic polynomials of degree 2d−1 with integer coefficients which we call Gonchar polynomials. Rather surprisingly, ρ(2) is the Golden ratio and ρ(4) the lesser known Plastic number. But Gonchar polynomials have other interesting properties. We discuss their factorizations, investigate their zeros and present some challenging conjectures.

Key words and phrases

Coulomb potential electrostatics problem Golden ratio Gonchar problem Gonchar polynomial Plastic number signed equilibrium sphere 

2010 Mathematics Subject Classification

30C10 31B10 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2012

Authors and Affiliations

  • J. S. Brauchart
    • 1
  • P. D. Dragnev
    • 2
  • E. B. Saff
    • 3
  • C. E. van de Woestijne
    • 4
  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  2. 2.Department of Mathematical SciencesIndiana-Purdue UniversityFort WayneUSA
  3. 3.Center for Constructive Approximation, Department of MathematicsVanderbilt UniversityNashvilleUSA
  4. 4.Lehrstuhl für Mathematik und StatistikMontanuniversität LeobenLeobenAustria

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