This question: "Can a real quartic polynomial in two variables have at most 4 isolated local minima?" came up in this post on Math SE but with no answer so far.
Finding examples of 4 isolated minima to a degree 4 polynomial is trivial (see posting) but proving that you cannot have 5 isolated minima seems hard. I cannot find any proof after searching for quite a while.
The only relevant results I can find are in the papers:
Alan Durfee, Nathan Kronefeld, Heidi Munson, Jeff Roy, Ina Westby, Counting critical points of real polynomials in two variables, The American Mathematical Monthly, Volume 100, Number 3, March 1993.
Alan H. Durfee, The index of grad f(x, y), Topology, Volume 37, Issue 6, November 1998.
Durfee shows that the number of critical points is bounded by the index of the polynomial’s gradient field. A consequence of this seems to be that you can have at most have 5 minima. However, there seems to be no known example with 5 minima, only 4, leading me to believe 4 is the most you can have.
