Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A remark about polynomials with specified local minima and no other
  critical points   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.

32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th