Analogue to Osgood's lemma for rational functions

Question

Osgood's Lemma says that if a function of several complex variables is continuous on an open set $D$ and holomorphic on each variable then it is holomorphic.

Is there a similar result but for rational functions? That is, suppose that $f(z_1,\dots,z_n)$ is holomorphic on an open set $D$ and every function of the form $$ \zeta \mapsto f(z_1,\dots,\zeta,\dots,z_n) $$ is a rational function, then $f$ is a rational function.

This looks intuitively true, but I cannot prove it. Could you please help me?

Answer 1

Introduction and motivation for a new edit.

I added this section and reworked a bit the whole answer in order to deal with a point that seems to have been left out by Solomon Bochner and William Ted Martin in their analysis of this problem: its consideration leads to a perfect analogue of Hartogs' theorem for separately rational functions, but let's see how.
The sought for result is based on an old theorem, whose statement is given below and whose full proof can be found in reference [1], chapter IX, §5, pp. 201-202, as theorem 5. I do not know if this is a result of the Authors.
We see that for this theorem to be applicable to our problem it is required that the function $f(z,w)$ is analytic on the product domain $A\times B$. However Hartogs' theorem guarantees this only for separately analytic functions while our datum is only separately rational and thus, more generally, only separately meromorphic: how can we proceed?
Well, we can simply use Renato Caccioppoli's extension of Hartogs' theorem ([2]. proved for dimension $n=2$), i.e.: "if a function of $n$ complex variables $f$ everywhere defined in a product domain, except at most for a finite number of points, is meromorphic respect to each one of its $i$ variables i.e. the map $$ \zeta \mapsto f(z_1,\dots,\underbrace{\zeta}_i,\dots,z_n) \quad \forall i= 1,\ldots, n, $$ is meromorphic, then it is meromorphic respect to all of its variables.
Said that, let's proceed and prove the "separate rationality theorem".

Bochner and Martin global meromorphy result.

Theorem. If $A\subseteq\Bbb C^N$ and $B\subseteq\Bbb C^M$ are domains in two sets of variables $z\triangleq (z_1, \ldots, z_N)$ and $w\triangleq (w_1, \ldots, w_M)$ and if an analytic function $f(z, w)$ in the product space $A\times B$ is a rational function in $z$ for each value of $w$ and a rational function of $w$ for each value of $z$ then $f(z, w)$ is rational in $(z, w)$.
The theorem is based on the following lemma, which in turn is a consequence of the vanishing of the determinant of linearly dependent system of functions:
Lemma ([1], chapter IX, §5, pp. 199-200, lemma 6). If $A\subseteq\Bbb C^N$ and $B\subseteq\Bbb C^M$ are domains in two sets of variables $z\triangleq (z_1, \ldots, z_n)$, if $F_1(z,w), \ldots, F_n(z,w)$ are analytic functions in the product domain $A\times B$, not all $\equiv 0$, if every $F_k(z,w)$ is a rational function in $w$ for every point in $z$, and if they satisfy a relation $$ c_1(w) F_1(z,w)+\ldots+c_N(w)F_N(z,w)\equiv 0 $$ with arbitrary (not necessarily analytic) functions $c_k(w)$, $k= 1, \ldots, N$ for which $$ \lvert c_1(w)\rvert^2 +\ldots +\lvert c_N(w)\rvert^2>0 $$ then there exist polynomials $p_1(w),\ldots, p_N(w)$ such that $$ p_1(w) F_1(z,w)+\ldots+p_N(w)F_N(z,w)\equiv 0. $$

Putting all together: separately rational functions are globally rational.

• By using Caccioppoli's extension of Hartogs' theorem we know that the function $f(z)$, which is rational and thus meromorphic respect to each variable $z_i$, $i=1, \ldots, n$, is globally meromorphic and thus analytic on the product domain $A\times B$. This implies that we can apply freely Bochner and Martin's theorem above even if the only thing we know about $f$ is its "separate rationality" structure.

• By using the said theorem, in order to get a complete analogue of the theorem of Hartogs now it is only necessary to proceed by induction:

  1. The function $f(z_1,\ldots,z_n)$ is holomorphic respect to all the variables and rational respect each one of them, then the can prove that $f$ is analytic respect to all the variables and rational with respect to the couple $(z_1, z_2)$ and each of the remaining ones, i.e. $z_k$, $k= 3,\ldots, n$.
  2. Now if $f$ is analytic with respect to all the variables and rational with respect to the $k$-tuple (with $k\le n-1$) $(z_1, \ldots, z_k)$ and each of the remaining variables, we can again apply the above theorem and prove $f$ is rational with respect to the $(k+1)$-tuple $(z_1, \ldots, z_{k+1})$ and to the remains ones.

The rational function version of the Hartogs'theorem is thus finally completely proved.

Notes

• For the sake of simplicity, I've not adhered strictly to Bochner and Martin's notation.

• I am quoting below Bochner and Martin's expert opinion on the above theorem:

  On the surface, this theorem strongly resembles the theorem of Hartogs (for complex variables) that analyticity in each variable implies analyticity in all variables. However, in substance the present theorem is totally different, being preponderantly algebraic with only a thin layer of analysis. ([1], chapter IX, §5, p. 201)
  

  However, this cannot be said of the result proved in this Q&A as it in turn relies on Hartogs' theorem (Caccioppoli's extension [2] relies on it) which is definitively an analytic tour de force whose proof is substantially the one given by Hartogs more than a hundred years ago.

References

[1] Salomon Bochner, William Ted Martin, Several complex variables (English), Princeton Mathematical Series, 10. Princeton: Princeton University Press, pp.X+216 (1948), MR27863, Zbl 0041.05205.

[2] Renato Caccioppoli, "Un teorema generale sulle funzioni di due variabili complesse" (Italian), Atti della Accademia Nazionale dei Lincei, Rendiconti, VI Serie, 19, pp. 699-703 (1934), JFM 60.0274.03, Zbl 0009.36301.