Suppose $M$ is a complex affine algebraic manifold in ${\mathbb C}^n$ (I mean, a set of common zeroes of a system of polynomials on ${\mathbb C}^n$, which is at the same time a smooth manifold). Consider the space ${\mathcal O}({\mathbb C}^n)$ of all holomoprphic functions on ${\mathbb C}^n$ with the usual topology of uniform convergence on compact sets. Alexei Pirkovskii in his paper Arens–Michael envelopes... (page 42, example 3.6) shows that
If a function $f\in {\mathcal O}({\mathbb C}^n)$ vanishes on $M$, $$ f\Big|_M=0, $$ then it can be approximated in ${\mathcal O}({\mathbb C}^n)$ by a sequence of polynomials $g_i\in {\mathbb C}[x_1,...,x_n])$ which also vanish on $M$: $$ g_i\Big|_M=0,\qquad g_i\overset{{\mathcal O}({\mathbb C}^n)}{\underset{i\to\infty}{\longrightarrow}}f $$
Pirkovskii uses Cartan's theorems for this, but I have a feeling that this is too complicated. Is it possible that there is a more or less simple way to prove this, without Cartan?
