Suppose $f:\mathbb{C}^n\to \mathbb{C}$ is a polynomial of degree $k$. Let $S=\{(z_1,...,z_n)\in \mathbb{C}^n: Df=0\}$, where $Df=(f'_{z_1},...,f'_{z_n})$ is the gradient. My question is: when $S$ has only isolated points?
Maybe this question is related to the Bézout's theorem. Then do we have the following fact: if $gcd(f'_{z_1},...,f'_{z_n})$=constant, then $S$ has only isolated points? Here $gcd$ means the greatest common divisor of these multivariate polynomials. I don't know whether I understand the theorem correctly. Any help will be appreciated.
P.S. As pointed out by Mohan in the comments, the $gcd$ condition does not lead to the fact that $S$ has only isolated points. What if we add an assumption that $deg_{z_1}(f)=...=deg_{z_n}(f)=deg(f)=k$? Here $deg_{z_j}(f)$ means the highest degree of $z_j$ in $f$.
P.S. I realized that the answer is still no with the assumption above. An example is $f(x,y,z)=\frac29x^3+3y^3+z^3-x^2y$. $S$ contains a line $x=3y,z=0$
P.S. But what if we assume that $gcd(f'_{z_i},f'_{z_j})=constant$, for each pair of $(i,j), i\ne j$?
P.S. I realized that the answer is still no. An example is $f(x,y,z)=x^2+y^2+xyz$. $S$ contains a line $x=y=0$.