A remark on Maxwell's conjecture for planar charges
Kenneth Killian · Complex Variables and Elliptic Equations · 2009
In this article, we will approximate an upper bound for the number of critical points of a potential field produced by co-planar point charges. This is a simplified form of a problem posed by Maxwell [J.C. Maxwell, A Treatise on Electricity and Magnetism, Vol. 1 (Republication of the 3rd revised edition), Dover Publications, Inc., New York, 1954.], who conjectured an upper bound on the number of critical points of spatial configurations of the point charges. Using algebraic techniques, we will give an upper bound for planar charges that is sharper than the bound given in [A. Gabrielov, D. Novikov, and B. Shapiro, Mystery of point charges. Proc. Lond. Math. Soc. (3) 95(2) (2007), pp. 443–472.], but in a much more restricted case.