For a positively charged insulated d-dimensional sphere we investigate how
the distribution of this charge is affected by proximity to a nearby positive
or negative point charge when the system is governed by a Riesz s-potential
1/r^s, s>0, where r denotes Euclidean distance between point charges. Of
particular interest are those distances from the point charge to the sphere for
which the equilibrium charge distribution is no longer supported on the whole
of the sphere (i.e. spherical caps of negative charge appear). Arising from
this problem attributed to A. A. Gonchar are sequences of polynomials of a
complex variable that have some fascinating properties regarding their zeros.Comment: 44 pages, 9 figure
