Background: Suppose first that $f \in \mathbb{C}[z]$ and consider the "analytic hypersurfaces" $V(f)$ and $V(df)$, which are of course just sets of points with multiplicities. (That is, we think of these hypersurfaces in a "scheme-theoretic sense.") It's easy to check that the center of mass of $V(f)$ is the same as the center of mass of $V(df)$ -- for the proof, see e.g. the first paragraph of
The average of the roots of a polynomial equals the average of the roots of its derivative
Question: Is there a notion of "center of mass" such that, for $f \in \mathbb{C}[z_1, \ldots, z_n]$, we can say that the "center of mass" of $V(f)$ is the same as that of $V(df)$? The usual definition won't work in general; consider e.g. $f(z_1, z_2) = z_1$.
Alternative question: Is there some general result about complex hypersurfaces or complex affine varieties for which the statement "$V(f)$ and $V(df)$ have he same center of mass" is the "$n = 1$ case"?