KTH/SU Mathematics Colloquium
2005-01-26
Boris Shapiro, Stockholm University
Mystery of point charges
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.
Main references:
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J. C. Maxwell,
A Treatise on Electricity and Magnetism, vol. 1,
Republication of the 3rd revised edition, Dover Publ. Inc., 1954.
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M. Morse and S. Cairns,
Critical Point Theory in Global Analysis and Differential
Topology, Acad. Press, 1969.
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A. Gabrielov, D. Novikov, and B. Shapiro,
Mystery of point charges, math-ph/0409009.