It is well known that if $k$ is an infinite field, then polynomials in $k[X_1,\dots,X_n]$ can be identified with polynomial functions from $k^n$ to $k$. I am wondering if this generalizes in a suitable sense to algebraic functions, at least for $k=\mathbb{C}$.
To be more precise, is it true that for any finitely many elements $f_1,\dots,f_m$ in the algebraic closure of $\mathbb{C}(X_1,\dots,X_n)$, there exists a dense open subset $U\subseteq\mathbb{C}^n$, such that there is an embedding of $\mathbb{C}[f_1,\dots,f_m]$ into $\mathcal{O}(U)$? Here "embedding" means embedding of $\mathbb{C}$-algebra, and it's reasonable to also require it to extend the obvious embedding of $\mathbb{C}[X_1,\dots,X_n]$; $\mathcal{O}(U)$ is the ring of holomorphic functions on $U$.
Remarks: This is certainly possible if $f_1,\dots,f_m\in\mathbb{C}(X_1,\dots,X_n)$; in that case we can even let $U$ be a nonempty Zariski open set. But we cannot hope for that in general, e.g., $\sqrt{X}$ cannot be defined on a nonempty Zariski open (aka cofinite) subset of $\mathbb{C}$.
Motivation: If this is possible, then by picking any point $\bar{a}\in U$ and evaluating at this point, we get a $\mathbb{C}$-homomorphism from $\mathbb{C}[f_1,\dots,f_m]$ into $\mathbb{C}$. As mentioned in my previous question this can be used to prove Nullstellensatz.
