Number of stationary areas for polynomial of many variables

Question

Suppose we have a polynomial $f$ of $k$ variables of degree $n$. $\mathcal S$ is a set of stationary points, on which equation $\nabla f=0$ holds. What is the maximum number of connected components of $\mathcal S$.

For example, for ($k=2$, $n=2$): $$ f(x,y)=x^2 $$ $\mathcal S$ has one connected component.

Answer 1

Let be $p$ a one variable polynomial with $r$ zeros (for example...). Consider the polynomial $$f(x,y) = \int p(x)\,dx.$$ How many connected components has your $S$?