TITLE: Conjecture: Finitely many points where gravitational field due to N masses vanishes
QUESTION [13 upvotes]: Given a configuration $C$ of $N$ distinct fixed points of equal mass in the plane (eventually in space), let $f_C(N)$ denote the number of points $P$ for which the gravitational field at $P$ vanishes. The gravitational force is Newtonian, i.e $1/r^2$.
For example $f_C(2)=1$ for all $C$ and for a configuration $C$ of $3$ unequally spaced collinear points, $f_C(3)=2$.
Conjecture: $f_C(N)$ is always finite and nonzero.
Is this problem known? Can we establish an upper bound on $f_C(N)$?

REPLY [14 votes]: Since you are talking about gravitation (rather than electrostatics) I assume
that all charges are positive. (With charges of different sign it is easy to arrange a whole curve of equilibrium points).
It is certainly non-zero if the number of points is at least 2. (From very general consideration it must be at least $N-1$ if you count multiplicitis properly. Without counting multiplicities it can be $1$ for any $N$).
But an exact upper estimate is not known. J. C. Maxwell conjectured that is is 
at most $(N-1)^2$, but this is wide open. Even finiteness is not known. I know only  computer-assisted proof for
$N=3$ but I have not checked it. If one assumes finiteness, then there are
upper estimates but they are much worse than expected.  
Reference: http://www.math.purdue.edu/~eremenko/dvi/equil2.pdf