This question: "Can a real quartic polynomial in two variables have at most 4 isolated local minima?" came up in this post:

https://math.stackexchange.com/questions/4620663/how-many-strict-local-minima-a-quartic-polynomial-in-two-variables-might-have/4653751

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 while searching for quite a while.

The only relevant results I can find are in the papers: "Counting Critical Points of Real Polynomials in Two Variables" http://www.jstor.org/stable/2324459 and "The Index of grad f(x, y)", Alan H. Durfee, Topology Vol. 37, No. 6, pp. 1339Ð1361, 1998

Durfee shows that the number of critial points is bounded by the index i of the gradient field of the polynomial. 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.