$f(x)=\sum\_{i=0}^\infty a\_ix^i$ with $a\_i\in\Bbb Q$. Let $S$ be finite set of places of $\Bbb Q$ such that:


**1.** $\forall p\notin S$, $|a\_i|\_p\leq1$ $\forall i\geq0$.


**2.** $\forall v\in S$, $f(x)$ extends to a meromorphic function on a disc $D\_v$ of radius $R\_v$ in $\Bbb C\_v$ and $\prod\_{v\in S}R\_v>1$.


Dwork says $f(x)\in\Bbb Q(x)$.


Is there a further natural constraint that will force $f(x)\in\Bbb Q[x]$?


Is there a multivariate version of this theorem?



Is there a multivariate analog of Dwork's theorem?
*Update* (Nov, 2019). The multivariate rationality criterion is true, and it follows from the reference to Andre's paper below. However, in my suggested proof scheme there was a point I had overlooked at the time of answering this question. When fully worked out, the straightforward outline with ``$Qf - P$'' below leads to $f \in \mathbb{Q}(\mathbf{x})$ only under the stronger hypothesis that 
$$
(\star) \quad \quad \quad \sum\_{v \in S} \log{R\_v} > (6 + 4\sqrt{2}) \sum\_{v \in S} \log^+{|1/\rho\_v|},
$$
 where $\rho\_v$ is the $v$-adic radius of convergence of $f(\mathbb{x})$. (Meaning: the largest $\rho\_v$ such that $f(\mathbf{x})$ converges on the polydisk $\max\_{i=1}^d |x\_i| < \rho\_v$ of radius $\rho\_v$.) The latter is a particular case of a result (Cor. VIII 1.6) of Andre in his book *G-Functions and Geometry*, which concerns arbitrary $f \in K[[\mathbf{x}]]$. 


As an up-date to this answer, I firstly give an outline of Andre's argument which suffices for rationality under the stronger condition ($\star$); this is the content of my original answer, explaining the Diophantine approximation proof scheme in a simplest case. Then I explain how to up-grade to a proof of the full multivariate generalization of the Borel-Dwork rationality criterion, valid under the best-possible condition $\sum\_{v \in S} \log{R\_v} > 0$ in place of $(\star)$.




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**Original answer.** 
Here is a slightly more detailed answer as requested. You have an obvious statement, the literal extension in which simply $x$ is replaced by a block of variables $\mathbf{x} = (x\_1,\ldots,x\_k)$ *and we assume meromorphy on the $v$-adic polydisk of radius $R\_v$*. This statement is true, and it is due originally to Yves Andre as Theoreme 5.4.3 of "Sur la conjecture des p-courbures de Grothendieck-Katz et un probleme de Dwork" in the volume *Geometric Aspects of Dwork Theory*, vol. 1. *[Added, Nov, 2019: Andre's theorem rather proves the algebraicity of $f(\mathbf{x})$, under a more general condition of simultaneous meromorphic uniformization of the pair $\mathbf{x}$ and $f(\mathbf{x})$. But it is not hard in the given situation to derive rationality from this. See below the line for the reduction argument.]*


[Added Nov, 2019: *The following direct ``$Qf - P$'' approximation scheme works only under the condition $(\star)$ which is stronger than $\sum\_{v \in S} \log{R\_v} > 0$. The general case is similar but one has to also add higher powers of $f$ into this argument. This is what Andre did in his paper in the Dwork volume.*]


The proof is by a familiar transcendence argument. With Siegel's lemma, a rational approximation $P/Q$ to $f$ is constructed having controlled height. This means $P,Q \in \mathbb{Z}[\mathbf{x}]$ are polynomials with small coefficients and such that $Qf - P$ vanishes at $\mathbf{x} = \mathbf{0}$ to an order $M$ comparable with $N := \max(\deg{P},\deg{Q})$. Using the product formula and the assumption [corrected!] ($\star$), one then proves that necessarily $f = P/Q$.


Cauchy's derivative estimates are what makes this work. If $f = p\_v/q\_v$ in the $v$-adic polydisk of radius $R\_v$, with $p\_v$ and $q\_v$ holomorphic in that polydisk and with the normalization $q\_v(\mathbf{0}) = 1$, then the $v$-adic valuation of the $\mathbf{x}^{\mathbf{i}}$ coefficient in $p\_v$ and $q\_v$ are each at least $|\mathbf{i}| \cdot \log{R\_v} + O(1)$, while a minimal order coefficient in $Qf-P$ is equal to the corresponding coefficient in $q\_v(Qf-P) = Qp\_v - Pq\_v$. This coefficient on the other hand is an $S$-integer, and now [corrected!], *when one makes this approach fully explicit, one finds that the condition $(\star)$ and the product formula yield its vanishing (for an appropriate choice of parameters)*, thus forcing $f = P/Q$. 


[*Added*, Nov 2019: This is contained in Corollary VIII 1.6 in Andre's book, where the coefficient of $6 + 4\sqrt{2}$ arises from the optimal choice of $\tau := 1 + \sqrt{2}$ (see the remark in Andre's proof). This theorem of Andre applies to arbitrary power series $f \in K[[\mathbf{x}]]$, with not necessarily $S$-integral coefficients, upon replacing the $\sum\_{v \in S}$ in the right-hand side of condition $(\star)$ by a $K$-relative height of the power series $f$. ]




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Let me now explain how to complete the proof of rationality from Andre's algebraicity theorem to which I referred with my original answer (see also the question at the end for the full statement of this theorem of Andre). The proof of the latter is by the same Diophantine approximation scheme sketched above in the simplest setting of a $\mathbb{Q}(\mathbf{x})$-linear dependency among $1$ and $f$, only extended to a proof of $\mathbb{Q}(\mathbf{x})$-linear dependency among the powers $1, f, f^2, \ldots$ of our power series $f \in O\_{K,S}[[\mathbf{x}]]$. 
The addition of higher powers $f^j$ is what gives the new degree of freedom needed to relax $(\star)$ to the best-possible condition $\sum\_{v \in S} \log{R\_v} > 0$. We refer to *Theoreme 5.4.3* in Andre's paper (also see below for its formulation), for the details of this Diophantine approximation argument. 


To answer now the OP's question, it remains to see that in the particular situation of meromorphy on the $v$-adic polydisks of radii $R\_v$ (in Andre's language, this is the case of a "trivial simultaneous meromorphic uniformization"), an algebraic such power series is *a fortiori* rational. 


This is directly seen as follows. Let $q(\mathbf{x}) \in \mathbb{Q}[\mathbf{x}] \setminus \{0\}$ be the leading coefficient in a minimal algebraic equation for $f(\mathbf{x})$ over $\mathbb{Q}[\mathbb{x}]$. Then the power series $q \cdot f$ satisfies a monic algebraic equation over $\mathbb{Q}[\mathbb{x}]$, and we are also given that $q \cdot f$ is meromorphic on the $v$-adic polydisk of radius $R\_v$, for every $v \in S$. But then $q \cdot f$ is in fact *holomorphic* on the said polydisks. (Consider the valuation attached to a hypothetical prime polar divisor of this $v$-adic meromorphic function $q \cdot f$, and derive a contradiction with the existence of a monic polynomial relation.) 


At this point we have reduced the problem to the case of power series that are in fact *holomorphic* on the polydisk of radius $R\_v$, for each $v \in S$. For such power series, as previously noted, it is trivial from the Cauchy derivative estimates and the product formula in $K$ that $\sum\_{v \in S} \log{R\_v} > 0$ yields that the power series is a polynomial. Applying this to the power series $q \cdot f$ of the previous paragraph, we conclude that $p := q \cdot f \in K[\mathbf{x}]$ is a polynomial, and hence that $f = p/q \in K(\mathbf{x}) \cap K[[\mathbf{x}]]$ is a rational function, as requested by the OP.




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Let me end with a question about generalizing this multivariate rationality criterion in the style of Carlson and Polya, who proved this in the univariate case $d = 1$.


**Question.** *Consider again $f \in O\_{K,S}[[x\_1, \ldots, x\_d]]$, but suppose now that for each $v \in S$ there exists a meromorphic mapping* $G\_v : D(\mathbf{0}, R\_v) \dashrightarrow \mathbb{C}\_v^d$, *where $D(\mathbf{0}, R\_v)$ is the $v$-adic polydisk $\max\_{i=1}^d |x\_i|\_v < R\_v$ of radius $R\_v$, such that $G\_v(\mathbf{0}) = \mathbf{0}$, $(DG\_v(\mathbf{z}) / D\mathbf{z} )\_{\mathbf{z} = \mathbf{0}} = I\_d$ (the identity $d \times d$-matrix), and the power series $f(G\_v(\mathbf{z}))$ is also the $\mathbf{z} = \mathbf{0}$ germ of a meromorphic function on the $v$-adic polydisk $D(\mathbf{0}, R\_v)$, for each $v \in S$. Andre's theorem states the algebraicity of $f(\mathbf{x})$ under the condition $\sum\_{v \in S} \log{R\_v} > 0$ on the radii.* But suppose now additionally that our mappings $G\_v : D(\mathbf{0}, R\_v) \hookrightarrow \mathbb{C}\_v$ are injective (''univalence''). *Does it follow then that the algebraic function $f(\mathbf{x})$ is in fact a rational function?*


In the setting of formal functions on an algebraic curve, such results were obtained by J.-B. Bost and A. Chambert-Loir in the paper *Analytic curves in algebraic varieties over number fields* (Progress in Math. vol. **269**, 2009), indeed in a finer form containing also F. Bertrandias's and D. Cantor's extension of the Borel-Dwork-Carlson-Polya theorem to adelic domains in $\mathbb{P}^1$ that are not necessarily of the above ("simply connected $\mathbb{C} \supset U \ni 0$") form. As in the above outline, they firstly prove that the formal function is algebraic. They then derive its rationality by an argument using the Hodge index theorem on the arithmetic surface, with ideas inspired from works of Harbater and Ihara (cf. also Bost's earlier paper *Potential theory and Lefschetz theorems for arithmetic surfaces*, 1999). Their results are for the case of dimension one. 


