This question: "Can a real quartic polynomial in two variables have at most 4 isolated local minima?" came up in [this post][1] 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][2], The American Mathematical Monthly, Volume 100, Number 3, March 1993.

- Alan H. Durfee, [The index of grad f(x, y)][3], 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.


  [1]: https://math.stackexchange.com/q/4620663/4653751
  [2]: https://doi.org/10.2307/2324459
  [3]: https://doi.org/10.1016/S0040-9383(97)00089-X