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Mystery of point charges

Andrei Gabrielov·Dmitry Novikov·Boris Shapiro
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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 R^n. 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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Andrei Gabrielov, Dmitry Novikov, Boris Shapiro. 2004-09-03. Mystery of point charges. https://arxiv.org/abs/math-ph/0409009

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