Upper bounds for the number of isolated critical points via the Thom-Milnor theorem

We apply the Thom-Milnor theorem to obtain the upper bounds on the amount of isolated (1) critical points of a potential generated by several fixed point charges(Maxwell's problem on point charges), (2) critical points of SINR, (3) critical points of a potential generated by several fixed Newtonian point masses augmented with a quadratic term, (4) central configurations in the n-body problem. In particular, we get an exponential bound for Maxwell's problem and the polynomial bound for the case of an even dimensional potential in Maxwell's problem.

Automated summary

We apply the Thom-Milnor theorem to obtain upper bounds on the amount of critical points in various problems, including Maxwell's problem on point charges, SINR, potential generated by fixed Newtonian point masses with a quadratic term, and central configurations in the n-body problem.