Mystery of point charges
Andrei Gabrielov, Dmitry Novikov, Boris Shapiro
Abstract
We discuss the problem of finding an upper bound for the number of equilibrium points of a potential of several fixed point charges in Rn. This question goes back to J.C.Maxwell and M.Morse. Using fewnomial theory we show that for a given number of charges there exists an upper bound independent on the dimension, and show it to be 12 for three charges. We conjecture the exact upper bound for a given configuration of nonnegative charges in terms of its Voronoi diagram, and prove it asymptotically.
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