Suppose we have $f \in \mathbf{C}[x_1,...,x_n]$ and that $f(z_1,...,z_n) = 0$.
Is it true that $f \in (x_1-z_1,...,x_n-z_n)$. Where this is the ideal generated by $x_i - z_i$.
For a single variable polynomial this clearly holds due to the fundamental theorem of algebra.
Yes it is true. For instance, if you consider the subvariety $X=V(x_1-z_1, \cdots , x_n-z_n)$ (which is the point $\{ (z_1, \cdots, z_n) \})$ of $\mathbb{C}^n$, then $f$ belongs to the ideal $I(X)$ of polynomials that vanish on $X$. But by the Nullstellensatz, $I(X)=(x_1-z_1, \cdots , x_n-z_n)$.
Maybe there is a simpler way to see this.