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

TL;DR

This paper demonstrates that for any finite set in R^n, there exists a polynomial with local minima exactly at those points and no other critical points, impacting the design of stable vector fields.

Contribution

It provides an explicit construction of polynomials with prescribed local minima and no other critical points, and explores implications for polynomial vector fields.

Findings

Existence of polynomials with specified local minima and no other critical points

Construction of polynomial vector fields with stable equilibria at given points

Implications for the behavior of gradient flows and omega-limit sets

Abstract

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.