Arnold's seminar, October, 19th

Mystery of point charges
B.Shapiro (joint with A.Gabrielov, D.Novikov)

  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 of the dimension, and 
show it to be at most 12 for three
charges. We conjecture an exact upper bound for a given configuration
of nonnegative charges in terms of its Voronoi diagram,  and prove it
asymptotically.
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