The set of zeros in $\mathbb{C}$ for a polynomial of one variable and of any degree is the set of points, i.e. dimension 0.
I have seen a few examples, e.g. in the book Complex Analysis by Freitag or other books, shows that the set of zeros of a polynomial of two variables (domain $\mathbb{C^2}$) is an curve, i.e. dimension 1. But none proves if it valid in general.
Is it true and if so how to show that the set of zero of a polynomial of $n$ variables has dimension $n-1$?