I repost here, after I tried here.
Lately I have been wondering on this problem: if $U \subset \mathbb C^n$ is an open Stein and I denote by $\mathcal R(U)$ the set of rational functions on $\mathbb C^n$ whose restriction to $U$ is holomorphic, is it true that the natural inclusion $\mathcal R(U) \to \mathcal H(U)$ is dense? Here $\mathcal H(U)$ denotes the algebra of holomorphic functions on $U$ endowed with the topology of uniform convergence over compacts.
I know that when $n = 1$ this is essentially the content of Runge's approximation theorem, but I didn't find any reference for the several variables case in the literature I know, and I do have problems in extending the proof of the $1$-dimensional case. Namely, the difficulty I encountered is the following: given an open Stein $U$, a compact $K \subset U$, I should provide a uniform approximation of any $f \in \mathcal H(K)$ with rational functions which are holomorphic inside $U$, and all what I can do is to approximate $f$ with rational functions which are holomorphic in a sufficiently small neighbourhood of $K$. In fact, if I am not mistaken, if one can prove this, then one can also prove the density of $\mathcal R(U)$ inside $\mathcal H(U)$ (and in this implication the hypothesis of $U$ being Stein is heavily used, because Stein exhaustions are necessary).
Remark. I would like to stress that $\mathcal R(U)$ is the set of rational functions and not polynomial functions (I am in fact well aware that the problem of understanding Runge domains in higher dimensions is way more complicated than the one-dimensional case).
