Title

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

Document Type

Presentation

Presentation Date

Spring 5-19-2011

Conference Name

International Symposium in Approximation Theory

Conference Location

Nashville, TN

Peer Review

Invited

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 r(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+r(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, r(2) is the Golden ratio and r(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.

Keywords

minimal energy problems, polynomials, zeros, factorization

Disciplines

Mathematics

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