Consider a $m+n$ times differentiable function $$f : \mathbb{R}^2 \to \mathbb{R} \\ (x,y) \mapsto f(x,y),$$
where two-dimensions was only chosen out of convenience for notation, but the question would also apply to higher dimensions. Consider the set $$S_{m,n,c} = \{ (x,y) \in \mathbb{R}^2 | \left(\frac{d^m}{dx^m} f(x,y) \right)\left(\frac{d^n}{dy^n} f(x,y)\right) = c \frac{d^{m+n}}{d x^m y^n} f(x,y) \}$$
Now one for example has that for polynomials of the form $$f(x,y) = a x^m y^n,$$
$ S_{m,n,c}$ is exactly the level set, where $f(x,y) = c$. Furthermore, for any polynomial
$$f(x,y) = \sum_{k} a x^{i_k} y^{j_k} ,$$
such that $i_k + j_k < m +n $, this will contain points where one of the partial derivatives is $0$ (i.e. for $m =n = 1$, extrema would be in the set).
Now I would be interested what one could say about this in the general case. I am sure this has been studied, so I would be glad to get indications where to look at.