  The following observation must surely be "well-known", but it seems worth
giving a simple and quite explicit proof. Take any finite subset X of Rn, n>1.
Then, there is a polynomial function P:Rn -> R which has local minima on the
set X, and has no other critical points. Applied to the negative gradient flow
of P, this implies that there is a polynomial vector field with asymptotically
stable equilibria on X and no other equilibria. Some trajectories of this
vector field are not pre-compact; a complementary observation says that, again
for arbitrary X, one can find a vector field with asymptotically stable
equilibria on X, no other equilibria except saddles, and all omega-limit sets
consisting of singletons.
