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The following observation must surely be "well-known", but it seems worthgiving 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 theset X, and has no other critical points. Applied to the negative gradient flowof P, this implies that there is a polynomial vector field with asymptoticallystable equilibria on X and no other equilibria. Some trajectories of thisvector field are not pre-compact; a complementary observation says that, againfor arbitrary X, one can find a vector field with asymptotically stableequilibria on X, no other equilibria except saddles, and all omega-limit setsconsisting of singletons.

A remark about polynomials with specified local minima and no other critical points