A Fascinating Polynomial Sequence arising from an Electrostatics Problem on the Sphere

TL;DR

This paper explores Gonchar's problem on the sphere, revealing a polynomial sequence whose zeros relate to electrostatics, and uncovers surprising connections to special numbers like the Golden ratio and Plastic number.

Contribution

It introduces Gonchar polynomials, links their zeros to electrostatics on spheres, and investigates their properties and conjectures, revealing unexpected mathematical connections.

Findings

$ ho(2)$ equals the Golden ratio

$ ho(4)$ equals the Plastic number

Gonchar polynomials have intriguing factorizations and zeros

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 $\rho(d)$ ($d$ 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 $1+\rho(d)$ 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, $\rho(2)$ is the Golden ratio and $\rho(4)$ the lesser known Plastic number. But Gonchar polynomials have other interesting properties. We discuss their factorizations, investigate their zeros and present some challenging conjectures.