# Euler products associated to multivariate rational functions: maximal domain of meromorphy, zeroes and poles

Arnaud Chadozeau, Oswaldo Velásquez Castañón

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Arnaud Chadozeau, Oswaldo Velásquez Castañón. Euler products associated to multivariate rational functions: maximal domain of meromorphy, zeroes and poles. Dissertationes Mathematicae, In press. hal-05143650

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Distributed under a Creative Commons Attribution 4.0 International License
EULER PRODUCTS ASSOCIATED TO MULTIVARIATE
RATIONAL FUNCTIONS: MAXIMAL DOMAIN OF
MEROMORPHY, ZEROES AND POLES

by

Arnaud Chadozeau & Oswaldo Velásquez Castañón

*Abstract.* — We determine the maximal domain of meromorphy of any Euler product $f(s_1, \dots, s_k) = \prod_p h(p^{-s_1}, \dots, p^{-s_k})$, where $h$ is the quotient of two polynomials in $k$ variables, with integer coefficients and with constant term equal to 1. More precisely, we define a domain $\Gamma \subseteq \mathbb{C}^k$, on which $f$ admits a meromorphic extension, and such that for any $z$ in $\partial\Gamma$, there is no neighborhood of $z$ on which $f$ admits a meromorphic extension. This maximal domain $\Gamma$ is described as $\tilde{K} + i\mathbb{R}^k$, where $K \in \mathbb{R}^k$ is a rational cone, computed from the set of exponents of the two polynomials defining $h$.

We also describe the divisor of $f$ over $\Gamma$, which comes from the local factors $h(p^{-s_1}, \dots, p^{-s_k})$, and from the zeta factors $\zeta(\alpha_1 s_1 + \dots + \alpha_k s_k)^{-c_\alpha}$, where $\zeta$ denotes the Riemann zeta function, corresponding to terms in the expansion of $h$ as a formal infinite product $\prod_\alpha (1 - X_1^{\alpha_1} \dots X_k^{\alpha_k})^{c_\alpha}$. We focus our study on the hyperplanes in the divisor, allowing us to use tools developed for the single variable case.

We complete our study by giving a geometric and arithmetic description of the set of exponents occurring in the infinite product expansion of $h$, and by showing a new result on the geometric nature of the set of singular points of an holomorphic function defined over a tubular domain.

This is the version of the manuscript accepted for publication in *Dissertationes Mathematicae*. The final version will be published by the Institute of Mathematics of the Polish Academy of Sciences (IMPAN). This version may differ from the published version.

This work is licensed under a Creative Commons “Attribution
4.0 International” license.

<figure><img src="image_2.png" /></figure>

<footnote n="2020">2020 Mathematics Subject Classification. — 11M32 (52B20 13F25 32D15).</footnote>

<footnote>*Key words and phrases.* — Euler products; meromorphic continuation; rational power series; rational polyhedral cone.</footnote>
# 1. Introduction

Zeta functions are analytic objects associated to discrete phenomena that need to be described asymptotically, such as counting of objects or function values. They are named after the Riemann Zeta Function, the function of one complex variable defined for $\Re(e(s)) > 1$ by

$$\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$$

and extended meromorphically to all the complex plane, with an only pole at $s = 1$ of residue equal to 1. Some of these functions do have special representations related to arithmetic properties, such as Euler products, named after the representation found by Euler (1737)

$$\zeta(s) = \prod_{p} \left(1 - \frac{1}{p^s}\right)^{-1}$$

(valid for $\Re(s) > 1$) or other symmetries, like the so called functional equations.

Typically, the asymptotic information of the counting function is extracted through tauberian techniques, after two procedures are held. First, the analytic continuation of the zeta function is computed to the extent it *uncovers* the underlying date. Then, a growth analysis is done to the extended region. After that, the tauberian techniques extract the asymptotic behavior of the counting function from the poles of the zeta function.

While a lot of applications come from the use of one-variable functions, the use of several variables functions allow to use several parameters in the counting. As such, several variable functions appear in the work of Bhowmik, Essouabri and Lichtin [BEL07].

Our goal is to study the Euler product $f(s_1, \dots, s_k) = \prod_p h(p^{-s_1}, \dots, p^{-s_k})$ as a meromorphic function, where $h \in \mathbb{Z}[[X_1, \dots, X_k]]$ is any quotient of two polynomials in $k$ variables with constant term equal to 1. In particular, we determine its domain of meromorphy and its divisor over this domain.

Before giving an historical account on this problem and stating our results, we mention some obvious facts about Euler products.

1. The process associating a function $h$ to its (somewhere convergent) Euler product $f$ is multiplicative: if a function $h_1$ (resp. $h_2$) is associated to its Euler product $f_1$ (resp. $f_2$) which admits a meromorphic extension on a domain $D_1$ (resp. $D_2$), then the function $h_1h_2$ is associated to the Euler product $f_1f_2$ which admits a meromorphic extension on a connected component of $D_1 \cap D_2$. Of course $h_1/h_2$ is associated to $f_1/f_2$, which is meromorphic over a connected component of $D_1 \cap D_2$.

2. The polynomial $h = 1 - X^n$ is associated to the Euler product $\prod_p (1 - p^{-ns}) = \zeta(ns)^{-1}$ which is meromorphic over $\mathbb{C}$. More generally, the multivalued polynomial $1 - X_1^{\alpha_1} \dots X_k^{\alpha_k}$, which we denote $1 - X^\alpha$, with $\alpha = (\alpha_1, \dots, \alpha_k) \in \mathbb{N}^k$, is associated to the Euler product $\zeta(\alpha_1 s_1 + \dots + \alpha_k s_k)^{-1}$ which is meromorphic over $\mathbb{C}^k$.

These are facts from which we can derive some general consequences.
3. The $n^{\text{th}}$ cyclotomic polynomial $\Phi_n$ can be written as a quotient $\prod_{d|n}(1 - X^d)^{\mu(n/d)}$ where $\mu$ is the Möbius function. In consequence, its associated Euler product is $\prod_{d|n} \zeta(ds)^{-\mu(n/d)}$, which is meromorphic on $\mathbb{C}$. More generally, we call **cyclotomic polynomial in k variables** any polynomial of the form $\Phi(X^\alpha)$ with $\Phi$ an univariate cyclotomic polynomial, and $\alpha \in \mathbb{N}^k$ with $\alpha \neq (0, \dots, 0).$ (*) The Euler product associated to any cyclotomic polynomials in $k$ variables is meromorphic on $\mathbb{C}^k$.

4. The Euler products $f$ and $f'$ associated to the functions $h$ and $h' = \Phi(X^\alpha)h$ respectively, admit meromorphic extension on the same domains. If $h$ and $h'$ are polynomials, we say that $h'$ admits a **cyclotomic factor**. Without loss of generality, we can assume that a polynomial has no cyclotomic factor, or for a rational function, that its numerator and denominator have no cyclotomic factor. In particular, the meromorphic functions $f$ and $f'$ have the same singularities. (†)

**1.1. Previous works on the subject.** — The first study for a generic $h$ can be credited to Estermann [Est28], who treated the case where $h$ is a univariate polynomial with integer coefficients. Here is an overview of his argument.

If $h \in \mathbb{Z}[X]$ is a product of cyclotomic polynomials, it can be written as a finite product $\prod_{n<N}(1 - X^n)^{c_n}$ with $c_n \in \mathbb{Z}$ for some $N \in \mathbb{N}$. The associated Euler product equals $\prod_{n<N} \zeta(ns)^{-c_n}$, which is meromorphic on $\mathbb{C}$.

If $h$ is not such a product, then, for any $N \in \mathbb{N}$, $h$ can still be approximated by a finite product $\prod_{n<N}(1 - X^n)^{c_n}$ with $c_n \in \mathbb{Z}$ with an arbitrary small error term:

$$h(z) = (1 - c_N z^N + o(z^N)) \prod_{n<N} (1 - z^n)^{c_n} \quad \text{when } z \text{ tends to 0}$$

and $f$ can be meromorphically extended to the half-plane $\Re(e) > \frac{1}{N}$. This provides us with the meromorphic continuation of $f$ up to the half-plane $\Re(e) > 0$. More exactly, for any $N \in \mathbb{N}$ there is some large $P$ such that

$$f(s) = \prod_{p<P} h(p^{-s}) \prod_{n<N} \zeta(ns)^{-c_n} G_{N,P}(s)$$

where $G_{N,P}$ is holomorphic and without zero on $\Re(e) > 1/N$. This proves that zeros and poles of $f$ on $\Re(e) > 0$ come either from a factor $h(p^{-s})$ or from a factor $\zeta(ns)^{-c_n}$. Since $h$ is not a product of cyclotomic polynomials, it has at least one root $\rho$ with $|\rho| < 1$. For each prime number $p$, this root creates an arithmetic progression of zeros of $s \mapsto h(p^{-s})$ on the half space $\Re(e) > 0$, namely $\frac{1}{\log p}(-\log \rho + 2\pi i \mathbb{Z})$. As $p$ grows, these arithmetic progressions cluster at all points of the imaginary line, proving that this line is made up of singularities of $f$. More precisely, these zeros may cancel with poles coming from $\zeta$ factors, but one root of $h$ creates so many zeros of $f$ that they cannot be significantly cancelled by the poles coming from the $\zeta$ factors.

<footnote>(*) Montgomery [Mon76, Theorem 1] has shown that this family is a reasonable generalization of the family of univariate cyclotomic polynomials.</footnote>

<footnote>(†) Note that in general, we cannot derive the set of singularities of $f_1 f_2$ from the singularities of $f_1$ and those of $f_2$. But if a singularity of $f_1$ is a point of meromorphy of $f_2$, then it is a singularity of $f_1 f_2$.</footnote>
Notice that the result and the proof of Estermann can be easily adapted to a
rational fraction $h \in \mathbb{Z}(X)$ with $h(0) = 1$ and without cyclotomic factor. By Fatou's
lemma ([Fat04, pp. 369–370], [Pól16, Hilfssatz III]), $h$ is the quotient of two coprime
polynomials $q^+$ and $q^-$ in $\mathbb{Z}[X]$ with $q^+(0) = q^-(0) = 1$. To each polynomial $q^\pm$ is
associated its Euler product $g^\pm$ meromorphic on the half space $\Re e(s) > 0$, therefore
$f = g^+/g^-$ is meromorphic on the half space. If they are not constant, the polynomials
$q^+$ and $q^-$ have at least a root inside the unit disc, giving rise to a large collection
of zeros of $g^+$ or $g^-$ that cluster at all points of the imaginary line. The collection
associated with $g^+$ might cancel the collection associated with $g^-$, but only when
these collections are generated by the same root for $q^+$ and for $q^-$, which contradicts
the fact that $q^+$ and $q^-$ don't share a common factor.

Dahlquist [Dah52] extended the result for $h \in \mathbb{Z}\{X\}$ (i.e. when $h$ is a power
series with integer coefficients that converges on a neighborhood of 0), obviously with
$h(0) = 1$. His convergence theorem states that if $h$ is meromorphic on the open
unit disc, then $f$ is meromorphic on the half-plane $\Re e(s) > 0$. The proof roughly
follows the lines of Estermann's, with a more careful treatment of the zeta factors.
The maximality of this domain of meromorphy is proved when $h$ admits a finite
number of zeros and poles in the open unit disc, again with an argument similar to
Estermann's. The main novelty is an elaborate argument to prove the maximality
of the half-plane when $h$ has no zero nor pole inside the unit disc. In this case, the
zeros and poles of $f$ are all coming from the $\zeta$ factors and Estermann's argument is
no longer useful. (‡)

Dahlquist's treatment of the case of $h$ without zero nor pole has some drawbacks:

- The argument is convoluted and somewhat unconventional. Up to our knowledge, this argument has never been used in another setting.

- This argument is no longer needed if the Riemann Hypothesis is assumed. On the plus side, only a small amount of knowledge on the zeros of the Riemann zeta function is required for that proof. This allows Dahlquist to generalize his results to different kinds of Euler products, which can be expressed using Dirichlet $L$-functions instead of the Riemann $\zeta$ function.

- The functions $h$ concerned by this part of the proof (power series with integer coefficients, with radius of convergence equal to 1, and which are not rational fractions) are pathological: by Pólya-Carlson Theorem [Rem98, p.265], any point of the unit circle is a singular point for such functions.

However, this article is our main source of inspiration for focusing on the terms
of the divisor of $f$ which comes from the $\zeta$ factors instead of those coming from the
local factors (even if, for cancellation reasons, both are to be considered). We will
give a new look on Dahlquist's argument, and we will show that the absence of zeros
or poles is indeed common in the multivariate case, even for elementary examples.

<footnote>(‡) Notice that Estermann's argument also fails when there are infinitely many zeros and poles of $h$ inside the unit disc. Dahlquist [Dah52, Section 4.2] remarked that if the number of zeros and poles of $h$ in the disc of radius $r < 1$ does not grow too fast when $r$ tends to 1, it is still possible to conclude with a version of Estermann's argument.</footnote>
Regarding the multivariate case, Bhomwik, Essouabri and Lichtin [BEL07] gave the first general convergence result when $h$ is a multivariate polynomial. We call **support** of a polynomial $h \in \mathbb{Z}[X]$, the set of exponents $\alpha$ such that the term $X^\alpha$ has a non-zero coefficient in $h$. When $h$ does not have any cyclotomic factor, Bhomwik et al. proved the meromorphy of $f$ on the tubular polyhedral open convex cone $\Gamma$ of $\mathbb{C}^k$ defined by the inequalities $\Re\langle\alpha, s\rangle > 0$ for every $\alpha \neq 0$ in the support of $h$, where $\langle\cdot, \cdot\rangle$ is the canonical hermitian inner product of $\mathbb{C}^k$. From a geometric point of view, the closure $\bar{\Gamma}$ is the dual cone of the support of $h$. On the other hand, they proved a weak maximality result: if $h$ has a non cyclotomic factor, then the origin $0$ is a singular point for $f$ [BEL07, Theorem 2].

In his PhD thesis, Delabarre [Del13] precised the maximality result in a remarkable way. By duality, the facets of $\Gamma$ are in relation with the extreme rays of its dual cone $\Gamma^\vee$, which is the conical hull of the support of $h$ in this case. To an extreme ray $\nu\mathbb{R}_+$ of $\Gamma^\vee$ (where $\nu$ can be chosen so that its coordinates are coprime integers), we associate the facet $\mathcal{F}_\nu = \bar{\Gamma} \cap \{s : \Re\langle\nu, s\rangle = 0\}$ and the subpolynomial $h_\nu$ of $h$ constituted of terms of $h$ whose exponents belong to the ray $\nu\mathbb{R}_+$. The polynomial $h_\nu$ is an element of $\mathbb{Z}[X^\nu]$ and can be considered as a non-constant polynomial in one single variable $T = X^\nu$ with constant coefficient equal to 1. Delabarre proved that [Del10, Théorème principal] if $h$ has no cyclotomic factor, then any point of the facet $\mathcal{F}_\nu$ is a singular point for $f$ as long as $h_\nu$ is square-free. The proof follows the work of Estermann, in the way that Delabarre constructs zeros of $h$ in the neighborhood of zeros of $h_\nu$ and those zeros create a large collection of zeros in $f$ near the facet $\mathcal{F}_\nu$. The study of possible cancellations between zeros is complicated in this setting, but more tractable $h_\nu$ is square-free.

The results of Bhowmik et al. and Delabarre can also be applied to a rational function $h = q^+/q^-$ where $q^+, q^- \in \mathbb{Z}[X]$ have no cyclotomic factor and $q^+(0) = q^-(0) = 1$. The polynomials $q^+$ and $q^-$ are associated to their respective Euler products $g^+$ and $g^-$ which are meromorphic over $\Gamma^+$ and $\Gamma^-$ respectively. Their quotient $f = g^+/g^-$ is a meromorphic extension of the Euler product associated to $h$ over $\Gamma^+ \cap \Gamma^-$, which is the open dual cone of $\text{supp}(q^+) \cup \text{supp}(q^-)$.

On the maximality of this domain, we may conclude that $h$ has a singularity at any points of $\Gamma^+ \cap \partial\Gamma^-$ and of $\partial\Gamma^+ \cap \Gamma^-$, at least if the corresponding $h_\nu$ is square-free, but we cannot conclude for a point of $\partial\Gamma^+ \cap \partial\Gamma^-$. In particular, it is not possible to get a maximality result for $h(X_1, X_2) = \frac{1+X_1+X_2-X_1X_2}{1+X_1+X_2}$ directly from the previous theorems since the domains $\Gamma^+$ and $\Gamma^-$ are the same.

## 1.2. Statement of results.
In this article, we extend and complete previous results in several ways.

First, we produce a tubular convex cone $\Gamma$ on which the Euler product $f$ is meromorphic, provided the function $h$ verifies the following conditions:

$$ - h(0) = 1, $$

- on a neighborhood of $0$, $h$ is the sum of a converging power series of $\mathbb{Z}[[X]]$,

- $h$ is meromorphic on the Reinhardt domain $\{(e^{-s_1}, \dots, e^{-s_k}) \mid s \in \Gamma\}$.
This domain $\Gamma$ is not described from the support of the power series defining $h$, but from the support of the infinite product representing $h$. Indeed, we prove that any function $h$ verifying the two first assumptions can be written as $\prod_{\alpha \in \mathbb{N}^k} (1 - X^\alpha)^{c_\alpha}$, where $c_\alpha \in \mathbb{Z}$ for any $\alpha \in \mathbb{N}^k$, with $c_0 = 0$, and where the product converges on a small neighborhood around 0. We introduce the notion of **exponential support** of $h$, denoted $\text{xsupp}(h)$ as the set of exponents $\alpha \in \mathbb{N}^k$ for which $c_\alpha \neq 0$. The exponential support of $h$ is useful to describe not only the domain $\Gamma$ (as the interior of the inferior limit of the halfspaces $\{s \in \mathbb{C}^k | \Re\langle\alpha, s\rangle > 0\}$ for $\alpha \in \text{xsupp}(h)$) but also the divisor of $h$ and especially the divisor of $f$ over $\Gamma$.

**Theorem 1.1.** — Let $h$ be a power series of $\mathbb{Z}[[X]]$ with $h(0) = 1$. Set $C$ as its exponential support and $(c_\alpha)_{\alpha \in C}$ as the family of exponents in its product representation $\prod_{\alpha \in C} (1 - X^\alpha)^{c_\alpha}$. Define $\Gamma$ as the interior of the set of $s \in \mathbb{C}^k$ such that $\Re\langle\alpha, s\rangle \ge 0$ for all but finitely many $\alpha \in C$. If the function $s \mapsto h(e^{-s_1}, \dots, e^{-s_k})$ is meromorphic on $\Gamma$, then $f$ is meromorphic on $\Gamma$ and the divisor of $f$ over $\Gamma$ is

$$ (f) = \sum_{p \text{ prime}} \left[ (h_p) - \sum_{\alpha \in C} c_{\alpha} \sum_{\kappa \in \mathbb{Z}} H_{\alpha}\left(\frac{2\kappa\pi i}{\log p}\right) - \sum_{\alpha \in C} \sum_{\gamma \in G} c_{\alpha} m_{\gamma} H_{\alpha}(\gamma) \right], $$

where $h_p$ is the meromorphic function $s \mapsto h(p^{-s_1}, \dots, p^{-s_k})$ defined on $\Gamma$, $G$ and $m_\gamma$ are the support and coefficients of the divisor of the Riemann zeta function on $\mathbb{C}$, i.e. $(\zeta) = \sum_{\gamma \in G} m_\gamma \gamma$, and

$$ H_{\lambda}(\gamma) = \{s \in \Gamma | \langle \lambda, s \rangle = \gamma\}. $$

Notice that if all the real parts of $s_1, \dots, s_k$ are large enough, the double product $\prod_p \prod_{\alpha \in C} (1 - p^{-\langle\alpha, s\rangle})^{c_\alpha}$ is convergent, and we derive three expressions for $f$ as an infinite product

$$ f(s) = \prod_p h_p(s) = \prod_p \prod_{\alpha \in C} (1 - p^{-\langle\alpha, s\rangle})^{c_\alpha} = \prod_{\alpha \in C} \zeta(\langle\alpha, s\rangle)^{-c_\alpha}. $$

The divisors of the terms of these three products correspond exactly to the terms in the expression of the divisor ($f$) over $\Gamma$ of our Theorem 1.1. More exactly, the divisor ($f$) is the formal sum of the divisors of the terms of the first and third products, corrected by the divisors of the terms of the second one.

Notice also that $\Gamma$ contains the dual cone of $\text{supp}(h)$. Of course, in the case of a rational function without cyclotomic factors, our domain of meromorphy $\Gamma$ is exactly the open dual cone of $\text{supp}(q^+) \cup \text{supp}(q^-)$ obtained by Bhowmik et al. and Delabarre, since this domain turns up to be maximal as we will see. In that particular case, the main novelty of Theorem 1.1 is the complete description of the divisor of $f$ on $\Gamma$. However, there is no assumption about cyclotomic factors in our statement. This is taken care of by the construction of $\Gamma$: the multiplication or division of $h$ by a cyclotomic factor affects its exponential support only by adding or removing finitely many elements, it has no effect on $\Gamma$ described as a limit inferior.

We now state our theorem of maximality in the case where $h \in \mathbb{Z}[[X]]$ is a rational power series. In that case, the domain of meromorphy $\Gamma$ is a rational polyhedral convex cone.
**Theorem 1.2.** — Let $h \in \mathbb{Z}[[\mathbf{X}]]$ a rational power series with constant term 1. There exist two coprime polynomials $q^+$ and $q^-$ of $\mathbb{Z}[\mathbf{X}]$ such that $q^-h = q^+$. We assume that neither $q^+$ nor $q^-$ have a cyclotomic factor. Then, the Euler product $f(s) = \prod_p h(p^{-s_1}, \dots, p^{-s_k})$ admits an meromorphic extension on the interior of the dual cone of $\text{supp}(q^+) \cup \text{supp}(q^-)$, that is on

$$ \Gamma = \bigcap_{\substack{\alpha \in \text{supp}(q^+) \cup \text{supp}(q^-) \\ \alpha \neq 0}} \{s \in \mathbb{C}^k \mid \Re e(\alpha, s) > 0\}. $$

This domain is the domain of meromorphy of $f$, meaning that any point on the boundary of $\Gamma$ is a singular point of $f$.

This result completely determines the maximal domain of meromorphy in the polynomial case, but also in the rational case. The methods used may also apply to more general functions $h$, but some steps in the argumentation are only valid for rational functions. We have not tried to determine the broadest class of functions $h$ for which our methods can prove the maximality of $\Gamma$, but it seems to us that there is no simple and practical description of a class of functions $h$ for which our methods are sufficient to conclude.

Our strategy to prove Theorem 1.2 is to use the divisor of $f$ described in Theorem 1.1 and to find points on the different faces of the polyhedral cone $\Gamma$ where the local finiteness of $(f)$ is not verified. Due to the shape of the expected result (a polyhedral cone) and the terms in the second sum of the divisor $f$ (which all are hyperplanes), we have restricted our study to hyperplanes in the divisor.

For each face $\mathcal{F}_\nu$ of $\Gamma$, we split the argument into three cases.

First, if $q^+$ or $q^-$ admits a factor which creates an hyperplane orthogonal to $\nu$ in the divisor $(f)$ (this factor has to appear in $h_\nu$ as well), then, similarly to the argument of Estermann in [Est28], this factor creates an arithmetic progression of hyperplanes for each factor $p$, and this double family of hyperplanes clusters everywhere along $\mathcal{F}_\nu$.

Second, if no such factor is to be found, we are in a situation very similar to the one detailed by Dahlquist in [Dah52], where only zeros and poles from the $\zeta$ factors appear. Actually, his argument works as is in this situation, unless $h_\nu$ is a finite quotient of cyclotomic polynomials, a case in which the hyperplanes in $(f)$ that are parallel to the face $\mathcal{F}_\nu$ form a discrete set, and therefore produce no accumulation.

Taking care of this third case has required a completely new set of arguments. To begin with, the fact that the domain $\Gamma$ described in Theorem 1.1 coincides with expected dual cone of $\text{supp}(q^+) \cup \text{supp}(q^-)$ is no longer obvious. This fact relies on the following geometric description of the support of a rational power series.

**Proposition 1.3.** — Let $g$ be a rational power series of $\mathbb{Z}[[\mathbf{X}]]$, and $p^+$ and $p^-$ be the two coprime polynomials in $\mathbb{Z}[\mathbf{X}]$ with $p^-(0) = 1$ and $p^-g = p^+$. We have

$$ \overline{\text{conv}}(\text{supp}(g)) = \text{conv}(\text{supp}(p^+)) + \text{cone}(\text{supp}(p^-)). $$

Moreover, there exists $\alpha \in \mathbb{Z}^k$, such that the convex hull $\text{conv}(\text{supp}(g))$ can be spanned only by elements of $\text{supp}(g)$ which do not belong to the affine cone $\alpha + \text{cone}(\text{supp}(p^-))$.
To prove this result, we establish a multivariate generalization of a result known for Fatou rings. We will provide a proof of this result only for a unique factorization domain $R$.

**Proposition 1.4.** — Let $R$ be a Fatou ring with fraction field $F$. For any rational power series $g \in R[[\mathbf{X}]]$, there exists an unique pair of polynomials $p^+, p^- \in R[[\mathbf{X}]]$, coprime in $F[[\mathbf{X}]]$, with $p^-(0) = 1$, such that $p^-g = p^+$.

These results help to understand the support of a rational power series, but the objects in Theorem 1.1 are defined in terms of the exponential support. If their geometric properties are related, a proper study of $\mathrm{xsupp}(h)$ is required. A striking result is the following.

**Proposition 1.5.** — Let $h \in \mathbb{Z}[[\mathbf{X}]]$ be a rational power series with $h(0) = 1$. Set $(c_\alpha)_{\alpha \in \mathbb{N}^k}$ as the family of exponents in its product representation $\prod_{\alpha \in \mathbb{N}^k} (1 - \mathbf{X}^\alpha)^{c_\alpha}$, and $K$ as the cone generated by $\mathrm{xsupp}(h)$. For any $\nu \in \mathbb{N}^k$ directing an extremal ray of $K$ and any $\mu \in \mathbb{N}^k$ not collinear with $\nu$, the power series $\sum_{n \in \mathbb{N}} c_{\mu+n\nu} T^n \in \mathbb{Z}[[T]]$ is rational.

With this proposition, we can ensure an accumulation of hyperplanes along any face of $\Gamma$, but only in a tubular neighborhood of $0$. To conclude in this case, we establish the following general result about tubular domains of holomorphy:

**Proposition 1.6.** — Let $\xi$ be a function holomorphic on a tubular domain $K + i\mathbb{R}^k$, with $K \subseteq \mathbb{R}^k$ convex, and let $F$ be a proper face of $\bar{K}$ and set $W$ as the direction of the affine hull of $F$ in $\mathbb{R}^k$. The function $\xi$ is holomorphic at $\mathbf{x} + i\mathbf{y} \in F + i\mathbb{R}^k$ if, and only if, $\xi$ is holomorphic at any $\mathbf{x}' + i\mathbf{y}' \in F + i(\mathbf{y} + W)$.

This result, based on a theorem of Siciak [Sic69], can be seen as a qualitative local version of Bochner Theorem for tubular domains. It exhibits an all-or-nothing property which completes perfectly our findings, since we have a polyhedral tubular convex cone as domain of meromorphy for $f$ with singular points along its faces over a tubular neighborhood of $0$.

Although we use tools of different areas of mathematics, we make particular use of the geometry of convex cones, a subject that pops up at different points in this article, within various settings. We spend the next part of this section introducing notations and conventions that will be used throughout this paper, and the most basic definitions and results about convex cones that we will need.

**1.3. Notations, definitions and basic results.** — Out of force of habit, we will use the Bourbaki brackets notation for open intervals, and we will denote the natural logarithm (of basis e) by log. The set of non-negative integers is denoted by $\mathbb{N}$.

In this article, $k \ge 1$ always refers to the dimension of the parameter space, or equivalently, the number of variables of its power series expansion. Any bold letter or symbol denotes a $k$-tuple, whose coordinates are denoted by the same letter or symbol with a regular fontweight, and with an index running from 1 to $k$. For example, $\mathbf{s}$ stands for $(s_1, \dots, s_k)$ and $\mathbf{0}$ (resp. **1**) denotes the $k$-tuple $(0, \dots, 0)$ (resp. $(1, \dots, 1)$). The family $\{\epsilon_1, \dots, \epsilon_k\}$ denotes the canonical basis of $\mathbb{R}^k$.

The ring of formal power series in $k$ variables with integer coefficients is denoted by $\mathbb{Z}[[\mathbf{X}]]$, where $\mathbf{X}$ stands for $X_1, \dots, X_k$. Members of the monomial basis are denoted
$\mathbf{X}^{\alpha}$, which stands for the power product $X_1^{\alpha_1} \cdots X_k^{\alpha_k}$. Here, $\alpha$ runs through $\mathbb{N}^k$. An exponent $\alpha \in \mathbb{N}^k$ with coprime coordinates is said to be **primitive**.

Algebraically, a formal series $h \in \mathbb{Z}[[\mathbf{X}]]$ is identified to its **coefficient function** $b \in \mathbb{Z}^{\mathbb{N}^k}$, related by the equation $h = \sum_{\beta \in \mathbb{N}^k} b(\beta)\mathbf{X}^{\beta}$. The **support** of $h$, denoted $\text{supp } h$, is defined as $\{\beta \in \mathbb{N}^k \mid b(\beta) \neq 0\}$. For any given primitive $\nu \in \mathbb{N}^k$, we set $h_\nu$ as the formal series $\sum_{n \in \mathbb{N}} b(n\nu)\mathbf{X}^{n\nu}$, which is a power series in the single variable $T = \mathbf{X}^\nu$. The map $h \mapsto h_\nu$ is generally not a ring homomorphism.

The ring of polynomials in $k$ variables with integer coefficients $\mathbb{Z}[\mathbf{X}]$ is seen as a subring of $\mathbb{Z}[[\mathbf{X}]]$. It is a unique factorization domain. For any UFD $R$, the content and the primitive part of a polynomial of $R[\mathbf{X}]$ is defined up to an invertible element of $R$. If the only common factors between two polynomials are invertible, we say that these polynomials are **coprime** in $R[\mathbf{X}]$. They are coprime in $F[\mathbf{X}]$, where $F$ is the fraction field of $R$, if and only if their primitive parts are coprime. A power series $h \in \mathbb{Z}[[\mathbf{X}]]$ is **rational** if there is a nonzero polynomial $q^{-}$ such that $q^{-}h$ is a polynomial, denoted by $q^{+}$. Moreover, there exists a unique choice of $q^{-}$ up to the sign, such that $q^{+}$ and $q^{-}$ are coprime. In that case, the polynomials $q^{+}$ and $q^{-}$ are referred to respectively as *the numerator* and *the denominator* of the rational power series.

Analytically, the power series $h$ is identified with the holomorphic function it defines on its **domain of convergence** in $\mathbb{C}^k$. By convergence, we mean absolute convergence and by domain of convergence we mean the largest open set of $z \in \mathbb{C}^k$ for which $\sum_{\beta \in \mathbb{N}^k} |b(\beta)z^\beta| < +\infty$. Once again, $z^\beta$ denotes the product $z_1^{\beta_1} \cdots z_k^{\beta_k} \in \mathbb{C}$ where as usual $z^0 = 1$ for any value of $z \in \mathbb{C}$. It is the only sensible mode of (pointwise) convergence for this series, since the set of indices (here: $\mathbb{N}^k$) is not well-ordered. By Abel's Lemma (see [*Sch05*, Lemma 1.5.8]), if the family $(b(\beta)z^\beta)_{\beta \in \mathbb{N}^k}$ is bounded for some $z \in \mathbb{C}^k$, then the power series $h$ converges absolutely on the **open polydisc** $D(0, r)$ of **polyradius** $r = (|z_1|, \dots, |z_k|)$, that is on the cardinal product of complex open disks $D(0, |z_1|) \times \dots \times D(0, |z_k|) \subseteq \mathbb{C}^k$. Therefore, for any weaker mode of convergence requiring the boundedness of the terms, the possible new points of convergence would be in the closure of the *domain of convergence* or have a zero coordinate.

Actually, we will be more interested in the convergence of the **associated Dirichlet** series, obtained after an exponential change of variables: in that case, there is no chance to get a zero coordinate. We set $h_p$ as the function $s \mapsto h(p^{-s_1}, \dots, p^{-s_k})$ where $p \in ]1, +\infty[$ (generally, $p$ is a prime number). Notice that for any $\lambda \in \mathbb{R}$, we have $h_p \circ S_\lambda = h_{p^\lambda}$, where $S_\lambda$ is the scaling operator $s \mapsto \lambda s$.

The same letter will denote the power series, the holomorphic function defined by it on its domain of convergence, and often, the meromorphic continuation of this function on any admissible domain. To a meromorphic function $f$ defined on a domain $\Gamma$ of $\mathbb{C}^k$ is associated its **divisor** $(f)$ (the domain is always specified in the context) which is a formal sum $\sum_S n_S S$ where $S$ run through irreducible analytic subsets of $\Gamma$ of codimension 1, and where its coefficient $n_S \in \mathbb{Z}$ is called the **order** of $S$. An analytic set of nonzero order is said to **appear in** the divisor $(f)$. A critical property is that any compact subset of $\Gamma$ intersects finitely many analytic sets appearing in
(f). To simplify the expression of (f), we will allow some of its terms to be analytic sets not intersecting Γ, and which are not considered as appearing in (f).

The only analytic surfaces that we care about are affine hyperplanes. The canonical hermitian inner product on $\mathbb{C}^k$ is defined as $\langle \mathbf{x}, \mathbf{y} \rangle = \sum_{i=1}^k \bar{x}_i y_i$, with linearity on the second variable. Any hyperplane can be described as $\{ \mathbf{s} \in \mathbb{C}^k \mid \langle \mathbf{v}, \mathbf{s} \rangle = \lambda \}$, with $\mathbf{v} \in \mathbb{C}^k \setminus \{0\}$ and $\lambda \in \mathbb{C}$. We denote this hyperplane by $H_\mathbf{v}(\lambda)$. Two hyperplanes $H_\mathbf{v}(\lambda)$ and $H_{\mathbf{v}'}(\lambda')$ are parallel if and only if the vectors $\mathbf{v}$ and $\mathbf{v}'$ are proportional; they denote the same hyperplane if and only if the vectors $(\mathbf{v}, \lambda)$ and $(\mathbf{v}', \lambda')$ are proportional. If $\Gamma$ is a convex domain of $\mathbb{C}^k$ and $f$ a meromorphic function on $\Gamma$, the intersection $H_\mathbf{v}(\lambda) \cap \Gamma$ is either empty or an irreducible analytic subset of $\Gamma$. In the latter case, we denote this analytic subset $H_\mathbf{v}(\lambda)$ as well.

Since convexity is considered, $\mathbb{C}^k$ is also regarded as a real vector space. The canonical inner product is the real part of the hermitian inner product and is denoted by $\Re\langle \mathbf{x}, \mathbf{y} \rangle = \sum_{i=1}^k (\Re e(x_i) \Re e(y_i) + \Im m(x_i) \Im m(y_i))$. The hyperplane for this structure will also be considered and has to be distinguished from the previous type of hyperplanes (which have real codimension 2), especially since one of our major argument is that a given family of (complex) hyperplanes clusters along a (real) hyperplane.

The vector space $\mathbb{R}^k$ is canonically regarded as the subspace of $\mathbb{C}^k$ made up of the $k$-tuples with real coordinates, and $i\mathbb{R}^k$ the set made up of the $k$-tuples with imaginary coordinates. As a real vector space, we have $\mathbb{C}^k = \mathbb{R}^k \oplus i\mathbb{R}^k$. The canonical inner product of $\mathbb{R}^k$ is the restriction of the Hermitian inner product, and it is denoted in the same way. A subset of $\mathbb{C}^k$ is said to be **tubular** when it can be written as $K + i\mathbb{R}^k$ where $K \subseteq \mathbb{R}^k$ and $+$ denotes the Minkowski sum.

The **Minkowski sum** (or simply the sum) of two subsets $A, B$ of $\mathbb{C}^k$, denoted $A + B$, is defined as $\{a + b \mid a \in A, b \in B\}$. The sum of a subset $A$ by a singleton $\{b\}$ is simply denoted $A + b$. We extend this formalism to scalar multiplication: For $A$ a subset of $\mathbb{C}^k$ and $L$ a subset of scalars, we set $L \cdot A$, or simply $LA$, as $\{\lambda a \mid \lambda \in L, a \in A\}$.

Finally, the norm generally used on $\mathbb{C}^k$, or $\mathbb{R}^k$, is the one associated with the inner product (hermitian or not), and is denoted by $\| \cdot \|$.

**1.4. Classical definitions and results in convex cones.** — We state some basic facts about convex sets in a $\mathbb{R}$-vector space of finite dimension.

The **convex hull** of a set $S$, denoted by $\text{conv}(S)$, is the set of all linear combinations $\sum_{i \in I} \lambda_i s_i$ of elements $s_i \in S$ with the condition $\forall i \in I, \lambda_i \ge 0$ and $\sum_{i \in I} \lambda_i = 1$. Obviously, a set is convex if, and only if, it is its own convex hull.

The set of linear combinations of elements of $S$ with the condition $\sum_{i \in I} \lambda_i = 1$ only, forms an affine space, named the **affine hull** of $S$. The condition $\sum_{i \in I} \lambda_i = 0$ would instead define a vector space, which is the direction of the affine hull.

Finally, the set of linear combinations of elements of $S$ with the condition $\forall i \in I, \lambda_i \ge 0$ only, form a convex cone, named the **conical hull** of $S$ and denoted by $\text{cone}(S)$. As convex cones are often easier to deal with if they are topologically closed, we denote by $\overline{\text{cone}}(S)$ the closed conical hull of $S$, as a short-hand for $\text{cone}(S)$.
The **dimension** of a convex set is the dimension of its affine hull. The interior of a convex set as a subset of its affine hull is never empty, and is always convex, this is the **relative interior** of the convex set. The closure of a convex set is also convex. The relative interior and the closure have the same affine hull as the original set.

The intersection of any family of convex sets is convex, the union of a non-decreasing sequence of convex sets is convex. The **limit inferior** of a countably infinite family of sets is the set of elements belonging to all but finitely many sets of this family. In particular, for any enumeration $(A_n)_{n \in \mathbb{N}}$ of this family, the limit inferior can be written as

$$ \liminf A_n = \bigcup_{n \ge 0} \bigcap_{m > n} A_m. $$

In particular, the limit inferior of a countably infinite family of convex sets is convex.

The **boundary** of a convex set $K$, denoted $\partial K$, is $K \setminus \text{ri} K$. For any $x \in \partial K$, there is a nontrivial linear form $\phi$ such that $\forall x' \in \bar{K}, \phi(x') \ge \phi(x)$. The affine hyperplane $\phi^{-1}(\phi(x))$ is a **supporting hyperplane** of $K$ at $x$.

A (proper) **face** of a convex $K$ is a (proper) subset $F$ of $K$ such that for any $x, y \in K$, if $]x, y[$ intersects $F$ then $[x, y] \subseteq F$. A proper face $F$ of $K$ is also a convex set and its dimension is strictly inferior to the dimension of $K$. Every proper face of $F$ is also a proper face of $K$. The union of all the proper faces of $\bar{K}$ is $\partial K$. The open convex $\text{ri} K$ has no proper face.

Faces of dimension 0 of a convex $K$ are called **extreme points** of $K$, and 1-codimensional faces are called **facets**. The only possible extreme point of a convex cone is the origin. A face of dimension 1 of a convex cone are either a ray (called **extreme ray**) or a line (in that case, there is no extreme point). We will be using different characterizations of extreme rays. For a ray $R$ in a convex cone $K$, the following assertions are equivalent: $R$ is extreme in $R$; the set $K \setminus R$ is convex; for any $x \in R$, if $x = y_1 + y_2$ with $y_1, y_2 \in K$, then $y_1, y_2 \in R$; any set generating the convex cone $K$ contains a non zero element of $R$.

For any subset $S \subseteq \mathbb{R}^k$, the set of linear forms $\phi$ such that $\forall s \in \bar{S}, \phi(s) \ge \phi(0)$ is a closed convex cone of the dual space, called the **dual cone** of $S$ and denoted $S^\vee$. The dual cone of the dual cone can be viewed as the closure of the conical hull of $S$. We will often identify $\mathbb{R}^n$ with its dual using the euclidean inner product; in particular, the dual cone will be often thought as a subset of $\mathbb{R}^k$.

By definition, we expect a convex cone to contain the origin. The relative interior of a convex cone $K$ usually does not meet this requirement. However, we will still refer to it as an **open convex cone**. Notice that $\text{ri}(K) + \bar{K} = \text{ri}(K)$ by [Roc97, Theorem 6.1].

Any convex cone $K$ is stable under addition. It has a natural structure of **com-mutative monoid**. The group of invertible elements of this monoid $K$ is the vector space $K \cap (-K)$. The **preordering** $\prec$ on $K$ defined by: $\forall x, y \in K, x \prec y \Leftrightarrow y - x \in K$ is antisymmetric when $K \cap (-K) = \{0\}$. In that case, the preordering $\prec$ is a **partial order** and the origin is the lower bound of $K$ for this order.
A convex cone which is the conical hull of a finite set is said to be **polyhedral**. Polyhedral cones are all closed sets. If moreover all the points of this generating finite set have rational coordinates, the convex cone is said to be **rational**.

**1.5. Examples.** — In this section we give a handful of examples to illustrate some significant points of our claims.

The first example aims to present the algorithmic efficiency of determining $\Gamma$ from a polynomial $h \in \mathbb{Z}[\mathbf{X}]$ with $h(0) = 1$, by removing the obstructing cyclotomic factors. This can be adapted for any quotient of such polynomials that are relatively coprime: basically, the support $B$ has to be replaced by the union of the supports of the two polynomials.

A possible algorithm is the following one:

1. Find the support $B$ of $h$

2. Find the extreme rays of the conical hull of $B$

3. For each extreme ray $\nu\mathbb{R}_+$

(a) Compute the associated polynomial $h_\nu \in \mathbb{Z}[\mathbf{X}^\nu]$.

(b) Check if $h_\nu$ is a product of a cyclotomic polynomials

(c) Check if $h_\nu$ divides $h$ (and compute the quotient if it does)

(d) If both tests (b) and (c) are verified, run the algorithm again for $h/h_\nu$
Otherwise, mark the ray $\nu\mathbb{R}_+$ as an extreme ray of $\Gamma^\vee$

4. Construct $\Gamma$ for the obtained extreme rays of $\Gamma^\vee$

From a computational point of view, there are fast algorithms to perform the item 2, based on repeated linear programming problems (see [Cha96]). If this point is repeated for the quotient $h/h_\nu$, it is even faster since all extreme rays computed for $h$ other than $\nu\mathbb{R}_+$ are extreme rays for $h/h_\nu$. This remark is also valid for the point 3, and for an extreme ray different from $\nu\mathbb{R}_+$, the points 3a.-3d. give the same answer for the polynomial $h$ or the quotient $h/h_\nu$.

The test 3b) is also fast, by obtaining a fixed point after at most $O(\log \deg h_\nu)$ iterations of the Graeffe's transform (see [BD89]).

Finally, in 3c), the divisibility and the quotient of $h$ by $h_\nu$ can also be determined by a fast algorithm (as the Fast Fourier Transform) since the divisor is essentially univariate: if we write $h$ as a polynomial in $k-1$ variables with coefficients in $\mathbb{Z}[\mathbf{X}^\nu]$, the computation is equivalent to the divisibility and the quotient of every coefficients in $\mathbb{Z}[\mathbf{X}^\nu]$ by $h_\nu$.

We give here the example of a simple polynomial where the different steps of the algorithms are to be checked by hand.

Consider the polynomial in three variables $h = 1+X_1+X_2+X_3 - X_1X_3^2+X_2X_3$.

1. The support of this polynomial is a finite set of $\mathbb{N}^3$ which elements are $0, \epsilon_1 = (1,0,0), \epsilon_2 = (0,1,0), \epsilon_3 = (0,0,1), \epsilon_1 + 2\epsilon_3$ and $\epsilon_2 + \epsilon_3$.

2. $\epsilon_1, \epsilon_2, \epsilon_3$ are the directions of the three extreme rays.

3. for $i \in \{1,2,3\}$, we have $h_{\epsilon_i} = 1 + \mathbf{X}^{\epsilon_i} = 1 + X_i$, which is a cyclotomic polynomial. We check the divisibility by $h_{\epsilon_3} = 1 + X_3$ first. We write $h$ as a polynomial with coefficients in $\mathbb{Z}[X_3]: h = (1+X_3)+(1-X_3^2)X_1+(1+X_3)X_2$. All the coefficients are divisible by $h_{\epsilon_3} = 1 + X_3$ and the quotient is
1 + $X_1 + X_2 - X_1X_3$. We rerun the algorithm for this polynomial, but this new polynomial is clearly irreducible and not a cyclotomic polynomial. The extreme rays are directed by $\epsilon_1$, $\epsilon_2$ and $\epsilon_1 + \epsilon_3$.

4. By a direct application of Theorem 1.2, we have the following description of the maximal domain of meromorphy:

$$ \Gamma = \{(s_1, s_2, s_3) \in \mathbb{C}^3 \mid \mathrm{Re}(s_1) > 0, \mathrm{Re}(s_2) > 0, \mathrm{Re}(s_1 + s_3) > 0\}. $$

We now give other examples to describe and discuss the three different cases we will encounter during the proof of Theorem 1.2, and the arguments we will use.

First consider $h = 1 - 2X_2 + 4X_1X_2^2 - 4X_1^2X_2^2$. The extreme rays are directed by $(1, 1)$ and $(0, 1)$, and are associated to polynomials $h_{(1,1)} = 1 - 4(X_1X_2)^2$ and $h_{(0,1)} = 1 - 2X_2$: no risk of a relevant cyclotomic factor, and the maximal domain of meromorphy is determined by these two vectors.

Let us nonetheless continue the process of factorization to understand why the face $\mathcal{F}_{(1,1)}$ is made of singular points. Writing $h$ as a polynomial with coefficients in $\mathbb{Z}[X_1X_2]$, we get a new factorization

$$ h = (1 - 4(X_1X_2)^2) + (-2 + 4X_1X_2)X_2 = (1 - 2X_1X_2)(1 + 2X_1X_2 - 2X_2). $$

Each factor is associated to an Euler product which is meromorphic over $\Gamma$ (at least). This gives a factorization of $f$ over $\Gamma$:

$$ f = \prod_p (1 - 2p^{-(s_1+s_2)}) \prod_p (1 + 2p^{-(s_1+s_2)} - 2p^{-s_2}). $$

Now consider the hyperplanes normal to $(1, 1)$ (that is: lines defined by $s_1 + s_2 = \gamma$ with $\gamma \in \mathbb{C}$) which may appear in the divisor of $f$. They may come from a zeta factor (in that case $\gamma$ is a zero of $\zeta$ divided by a positive integer) or from a local factor. No term $1 + 2p^{-(s_1+s_2)} - 2p^{-s_2}$ has such hyperplane in its divisor, but each term $1 - 2p^{-(s_1+s_2)}$ does (in that case $\gamma$ is a zero of $1 - 2p^{-s}$). Indeed, the divisor of $\prod_p(1 - 2p^{-(s_1+s_2)})$ is only composed of such hyperplanes, whose constants $\gamma$ are the zeros and poles of the meromorphic extension of the univariate Euler product $\prod_p(1 - 2p^{-s})$.

This boils down to the study of the zeros and poles of $\prod_p(1 - 2p^{-s})$, which was done by Estermann [Est28], who proved that the zeros coming from the local factors accumulate along the imaginary line and are more plentiful than the poles coming from the zeta factors. This proves the accumulation of hyperplanes in the divisor of $f$ along the face $\mathcal{F}_{(1,1)}$ supported by the (real) hyperplane $\mathrm{Re}(s_1 + s_2) = 0$.

The argument we have just given (when $h$ can be factorized by a polynomial whose exponents belong to one extreme ray) will be referred to as *Estermann's argument*.

We now consider the other face $\mathcal{F}_{(0,1)}$ of $\Gamma$, and the family of hyperplanes normal to $(0, 1)$ (defined by an equation of the form $s_2 = \gamma$ with $\gamma$ in $\mathbb{C}$) that appears in the divisor of $f$. It is obvious that no such hyperplane appears in the divisor of $\prod_p(1 - 2p^{-(s_1+s_2)})$, nor in the zero set of a local factor $1 + 2p^{-(s_1+s_2)} - 2p^{-s_2}$. There all come from the zeta factors of $\prod_p(1 + 2p^{-(s_1+s_2)} - 2p^{-s_2})$.
These zeta factors are completely described by the expression of $h$ as a infinite product $\prod_\alpha (1 - \mathbf{X}^\alpha)^{c_\alpha}$, where $\alpha = (\alpha_1, \alpha_2)$ goes through $\mathbb{N}^2 \setminus \{(0,0)\}$, and $c_\alpha \in \mathbb{Z}$. Such formal expansion of $h$ is always possible, and is uniquely determined. In particular, a factor $(1 - \mathbf{X}^\alpha)^{c_\alpha}$ with $c_\alpha \neq 0$ generates an Euler product $\zeta((\alpha, s))^{-c_\alpha}$, whose divisor is made of hyperplanes normal to $\alpha$ and appears in the divisor of $f$ as in Theorem 1.1. In particular, the hyperplanes of $(f)$ normal to $(0,1)$ are completely described by the infinite subproduct $\prod_{\alpha_2>0} (1 - 2X_2^{\alpha_2})^{c(0,\alpha_2)}$, which is equal to the whole product evaluated at $X_1 = 0$, that is $h(0, X_2) = h_{(0,1)} = 1 - 2X_2$. Therefore, the hyperplane defined by $s_2 = \gamma$ is in the divisor of $f$ if and only $\gamma$ is a zero or pole coming from the zeta factors of $\prod_p (1 - 2p^{-s_2})$.

Once again, the problem boils down to the study of an univariate Euler product. And in this case, Dahlquist [Dahl52] has essentially proven the accumulation along the imaginary axis of zeros and poles coming from the zeta factors of such Euler product, provided that there are infinitely many zeta factors (that is as long as $h_{(0,1)}$ is not a product of cyclotomic polynomials). This proves the accumulation of hyperplanes in the divisor of $(f)$ that are orthogonal to $(0,1)$ along the face $F_{(0,1)}$ supported by the (real) hyperplane $\mathfrak{Re}(s_2) = 0$.

This kind of argument (when $h_\nu$ has no common factor with $h$ and $h_\nu$ is not a finite product of cyclotomic factors) will be referred to as *Dahlquist's argument*.

Finally, it remains the case for which:

- There is no common factor between $h_\nu$ and $h$

- The polynomial (resp. a rational function) $h_\nu$ is a finite product (resp. a quotient of finite products) of cyclotomic polynomials.

This is precisely the case of our previous example $1 + X_1 + X_2 - X_1 X_3$ for its three extreme rays of respective directions $\epsilon_1, \epsilon_2$ and $\epsilon_1 + \epsilon_3$. The corresponding polynomials are $1 + X_i$ for the two first directions and $1 - X_1 X_3$ for the third. Since we are allowed to multiply or divide $h$ by a cyclotomic polynomial, we may as well consider the rational fraction $h^\circ = \frac{1+X_1+X_2-X_1 X_3}{(1+X_1)(1+X_2)(1-X_1 X_3)}$, for which the Euler product has the same domain of meromorphy.

For that function $h^\circ$, we have $h_{\epsilon_1}^\circ = h_{\epsilon_2}^\circ = h_{(1,0,1)}^\circ = 1$. This third case then boils down to the case $h_\nu = 1$, thus the divisor of the Euler product $f$ doesn't have any hyperplane orthogonal to $\nu$. Our previous strategy of accumulation of parallel hyperplanes (as in Estermann's or Dahlquist's arguments) is bound to fail in that case.

We illustrate this case with a simpler expression $h = \frac{1+X_1+X_2-X_1 X_2}{1+X_1+X_2}$, to which we have already referred earlier. Both numerator and denominator are irreducible polynomials, therefore there is no hyperplane in the divisor of $f$ which comes from the local factors. The only hyperplanes in the divisor $(f)$ are then described by the exponents in the infinite product $h = \prod_\alpha (1 - \mathbf{X}^\alpha)^{c_\alpha}$. The two extreme rays are clearly directed by $(1,0)$ and $(0,1)$; by symmetry, we will only consider the direction $(0,1)$. We have $h_{(0,1)} = h(0, X_2) = \prod_\alpha (1 - X_2^\alpha)^{c(0,\alpha)} = 1$, therefore $c_{(0,\alpha)} = 0$ for any $\alpha \ge 1$, and there are no hyperplanes in $(f)$ orthogonal to $(0,1)$.
<figure><img src="image_3.png" /><figcaption>FIGURE 1. Real part projections on $\mathbb{R}^2$ of the domain of meromorphy $\Gamma$ and of an accumulating family of hyperplanes of the divisor $(f)$ for the functions $h = \frac{1+X_1+X_2-X_1X_2}{1+X_1+X_2}$ (left) and $h = 1+2X_1X_2-2X_2$ (right).</figcaption></figure>

We look for other hyperplanes in $(f)$ by computing more coefficients $c_\alpha$. In this case, even if computing $c(1, \alpha)$ is straightforward for this particular polynomial, it is still a fair description of the argument we use for any polynomial.

Consider the value of the infinite product modulo $X_1^2$ (setting $X_1 = 0$ in the product is equivalent to computing its residue modulo $X_1$). Taking advantage of the fact that $c(0, \alpha) = 0$ for any $\alpha \ge 1$, we have:

$$ \prod_{\alpha} (1 - X^{\alpha})^{c_{\alpha}} \equiv \prod_{\alpha \ge 1} (1 - X_1 X_2^{\alpha})^{c_{(1,\alpha)}} \equiv 1 - \sum_{\alpha \ge 1} c_{(1,\alpha)} X_1 X_2^{\alpha} \pmod{X_1^2}. $$

But computing the MacLaurin series of $h$ with coefficient in $\mathbb{Q}(X_2)$ is fairly straightforward:

$$ h = 1 - X_1 X_2 \frac{1}{1 + X_1 + X_2} = 1 + \sum_{n \ge 1} (-1)^n \frac{X_2}{(1 + X_2)^n} X_1^n. $$

Therefore, we have:

$$ 1 - \sum_{\alpha \ge 1} c_{(1,\alpha)} X_1 X_2^{\alpha} \equiv h \equiv 1 - \frac{X_2}{1 + X_2} X_1 \equiv 1 - \sum_{\alpha \ge 1} (-1)^{\alpha-1} X_1 X_2^{\alpha} \pmod{X_1^2}, $$

and finally $c_{(1,\alpha)} = (-1)^{\alpha-1}$ for any $\alpha \ge 1$.

Therefore, for any $\alpha \ge 1$, and any non trivial zero $\gamma$ of the Riemann $\zeta$ function with $\Re(\gamma) \ge 1/2$, the hyperplane $s_1 + \alpha s_2 = \gamma$ is a simple zero or a pole of $f$, which could only be cancelled with another zeta factor, if $2\gamma$ (or $k\gamma$ with $k \ge 2$) were a zero of $\zeta$. By Prime Number Theorem, $\zeta$ has no zero within the half-plane $\Re(s) \ge 1$.

The set of clustering points in $\bar{\Gamma} = \{s \in \mathbb{C} \mid \Re e(s) \ge 0\}^2$ of this family of hyperplanes contains the set $[0, 1/2] \times \{0\} + i\mathbb{R}^2$ as shown in Figure 1. Notice the difference with the previous cases where a family of parallel hyperplanes were exhibited. Therefore, only considering hyperplanes in the divisor is not enough to determine the singularity of $f$ at any point of the face $\mathbb{R}_+ \times \{0\} + i\mathbb{R}^2$. We will reach this goal thanks to Proposition 1.6.
## 2. The convergence theorem

In this section, we provide a proof of our main theorem of convergence, which gives an explicit description of a domain on which the Euler product $\prod_p h(p^{-s_1}, \dots, p^{-s_k})$ can be meromorphically extended, as well as an expression of the divisor of this meromorphic function on this domain.

This theorem applies to any power series $h$ with some simple conditions:

- it is convergent in a neighborhood of the origin.

- its constant term is equal to 1.

- its coefficient are integers.

The two first conditions are necessary to ensure the existence of the Euler product. As a matter of fact, these assumptions imply that $h(z) = 1 + O(\max_i |z_i|)$ as $z = (z_1, \dots, z_k)$ tends to $0$, therefore the Euler product $\prod_p h(p^{-s_1}, \dots, p^{-s_k})$ converges to a limit $f(s_1, \dots, s_k)$ if all the real parts of $s_1, \dots, s_k$ are larger than 1. The convergence being locally uniform, the limit $f$ is meromorphic at $s$ whenever every term of the product is meromorphic at $s$.

The primary reason for the third condition is to simplify the treatment. The presence of a non-integer coefficient roughly corresponds to the factorization of $f$ by a non-integer power of the Riemann $\zeta$ function, which accounts for a family of singularities of $f$. In specific examples, these singularities are usually easy to keep track of, for example by considering terms with non-integer multiplicities in the expression of the divisor of $f$.

Actually, we will ask for a fourth assumption on $h$, which is difficult to state at this point. Basically, any default of meromorphy of the function $s \to h(e^{-s_1}, \dots, e^{-s_k})$ is passed down to every term $h_p$ of the product defining $f$. Since we want to avoid every foreseeable singularity for $f$ from local factors $h_p$, we will ask the function $s \to h(e^{-s_1}, \dots, e^{-s_k})$ to be meromorphic on the domain where $f$ is meromorphic. However, we will suggest a less precise statement for functions which do not satisfy this assumption.

**2.1. Formal development as an infinite product.** — We show that any formal series

$$ (1) \qquad \sum_{\beta \in \mathbb{N}^k} b(\beta) \mathbf{X}^{\beta} \in \mathbb{Z}[[\mathbf{X}]] \quad \text{with } b(\mathbf{0}) = 1 $$

has a unique representation as a formal infinite product

$$ (2) \qquad \prod_{\alpha \in \mathbb{N}^k} (1 - \mathbf{X}^{\alpha})^{c(\alpha)} \quad \text{with } c(\mathbf{0}) = 0, $$

and conversely. Moreover we give formulas to compute the exponent function $c : \mathbb{N}^k \to \mathbb{Z}$ of (2) from the coefficient function $b : \mathbb{N}^k \to \mathbb{Z}$ of (1), and the other way around. We will always assume that $c(\mathbf{0}) = 0$ so that the term with $\alpha = \mathbf{0}$ is interpreted as 1, and that $b(\mathbf{0}) = 1$.
Our results generalize the argument of Dahlquist [Dah52, Equation (2.1)] for power series in a single variable and the argument of Delabarre [Del13, Corollary 2.2] for polynomials in multiple variables.

We first show that these infinite products (2) describe exactly the set of formal power series in $k$ variables with constant term equal to 1.

**Lemma 2.1.** — For any integer exponent function $c : \mathbb{N}^k \to \mathbb{Z}$ with $c(\mathbf{0}) = 0$, there exists an integer coefficient function $b : \mathbb{N}^k \to \mathbb{Z}$ with $b(\mathbf{0}) = 1$ such that

$$ (3) \qquad \prod_{\alpha \in \mathbb{N}^k} (1 - \mathbf{X}^\alpha)^{c(\alpha)} = \sum_{\beta \in \mathbb{N}^k} b(\beta) \mathbf{X}^\beta $$

is an identity in $\mathbb{Z}[[\mathbf{X}]]$. Moreover, for any $\beta \in \mathbb{N}^k$, we have

$$ (4) \qquad b(\beta) = \sum_{\substack{m:\mathbb{N}^k \to \mathbb{N} \\ \sum_{\alpha \in \mathbb{N}^k} m(\alpha)\alpha = \beta}} \prod_{\alpha \in \mathbb{N}^k} (-1)^{m(\alpha)} \binom{c(\alpha)}{m(\alpha)}. $$

Note that the outer sum runs over the partitions of $\beta$ with parts in $\mathbb{N}^k$, which are described through the function $m : \mathbb{N}^k \to \mathbb{N}$. Since $m$ is of finite support, almost every term of the inner product is equal to 1. Thus, this product is well defined. Since $c(\mathbf{0}) = 0$, the term with $\alpha = \mathbf{0}$ of this product is zero, unless $m(\mathbf{0}) = 0$. Since there are finitely many partitions of $\beta$ in non zero parts, only finitely many terms of the outer sum are different from zero; thus, the sum is well defined as well.

*Proof.* — Recall that $\binom{c}{m} = \frac{1}{m!} \prod_{i=0}^{m-1} (c-i)$ and that $(1-X)^c = \sum_{m \ge 0} (-1)^m \binom{c}{m} X^m$ for any $c \in \mathbb{Z}$. Therefore, the product $\prod_{\alpha \in \mathbb{N}^k} (1 - \mathbf{X}^\alpha)^{c(\alpha)}$ is formally equal to

$$ (5) \qquad \sum_{\beta \in \mathbb{N}^k} \left( \sum_{\substack{m:\mathbb{N}^k \to \mathbb{N} \\ \sum_{\alpha} m(\alpha)\alpha = \beta}} \prod_{\alpha \in \mathbb{N}^k} (-1)^{m(\alpha)} \binom{c(\alpha)}{m(\alpha)} \right) \mathbf{X}^\beta \in \mathbb{Z}[[\mathbf{X}]], $$

where the inner sum runs over the partitions of $\beta$ with parts in $\mathbb{N}^k$. We have already noticed that this expression is well defined. For the constant term of this series (with $\beta = \mathbf{0}$) the inner sum has only one non-zero term — the empty partition with $m(\alpha) = 0$ for all $\alpha$ — and its value is equal to 1. $\square$

We now prove the converse of Lemma 2.1, generalizing a formula of Delabarre [Del10, Lemma 25] obtained when the coefficient function $b : \mathbb{N}^k \to \mathbb{Z}$ has finite support.

**Lemma 2.2.** — For any integer coefficient function $b : \mathbb{N}^k \to \mathbb{Z}$ with $b(\mathbf{0}) = 1$, there exists an integer exponent function $c : \mathbb{N}^k \to \mathbb{Z}$ with $c(\mathbf{0}) = 0$ such that (3) is an identity in $\mathbb{Z}[[\mathbf{X}]]$. Moreover, for any $\alpha \in \mathbb{N}^k$, we have

$$ (6) \qquad c(\alpha) = \sum_{r \ge 1} \sum_{d \ge 1} \frac{(-1)^r \mu(d)}{dr} \sum_{\substack{\beta_1, \dots, \beta_r \ne \mathbf{0} \\ d \sum_{i=1}^r \beta_i = \alpha}} \left( \prod_{i=1}^r b(\beta_i) \right). $$
The inner sum is over $r$-tuples of non zero elements $\beta_i$ of $\mathbb{N}^k$ whose sum is equal to $\frac{1}{d}\alpha$. In particular, the integers $d$ and $r$ and the sum of the coordinates of any $\beta_i$ are all less than or equal to the sum of coordinates of $\alpha$. Thus, every sum is finite in this formula.

*Proof.* — We first show that there exists an exponent function $c$ verifying the first identity, then give a necessary expression for its values. For the existence, we use inductive reasoning thanks to a non-decreasing(§) sequence $(\lambda_n)_{n \in \mathbb{N}}$ enumerating $\mathbb{N}^k \setminus \{0\}$.

We recursively define the sequence $(c_n)_{n \in \mathbb{N}}$ of integers such that for any $n \in \mathbb{N}$, $c_n$ is the opposite of the coefficient of $X^{\lambda_n}$ in the finite product of power series

$$P_n = \prod_{i=0}^{n-1} (1 - X^{\lambda_i})^{-c_i} \left( 1 + \sum_{\beta \neq \mathbf{0}} b(\beta) X^{\beta} \right) \in \mathbb{Z}[[X]]$$

where the sum is defined over non zero $\beta \in \mathbb{N}^k$.

We prove by induction over $n \in \mathbb{N}$ that for any $m < n$, the coefficient of $X^{\lambda_m}$ in $P_n$ is zero. For $m = n - 1$, it comes from the definition of $c_{n-1}$, for $m < n - 1$, it comes from the induction hypothesis for $P_{n-1}$ and the fact that the sequence $(\lambda_n)_{n \in \mathbb{N}}$ is non-decreasing.

Since the sequence $(\lambda_n)_{n \in \mathbb{N}}$ enumerates $\mathbb{N}^k \setminus \{0\}$, we have the first formal identity by setting $c(\mathbf{0}) = 0$ and $c(\alpha)$ as $c_n$ where $n$ is such that $\lambda_n = \alpha$. We derive

$$(7) \quad \log \left( 1 + \sum_{\beta \neq \mathbf{0}} b(\beta) X^{\beta} \right) = \log \left( \prod_{\alpha \in \mathbb{N}^k} (1 - X^{\alpha})^{c(\alpha)} \right) = \sum_{\alpha \in \mathbb{N}^k} c(\alpha) \log (1 - X^{\alpha}).$$

The left-hand-side term may be rewritten as

$$\sum_{r \ge 1} \frac{(-1)^{r-1}}{r} \left( \sum_{\beta \neq \mathbf{0}} b(\beta) X^{\beta} \right)^r = \sum_{r \ge 1} \frac{(-1)^{r-1}}{r} \sum_{\beta_1, \dots, \beta_r \neq \mathbf{0}} \left( \prod_{i=1}^{r} b(\beta_i) \right) X_{i=1}^{\sum_{\beta_i}}.$$

By Möbius inversion of the formal identity $-\log(1-X) = \sum_{m \ge 1} X^m/m$, we get $X = -\sum_{d \ge 1} \frac{\mu(d)}{d} \log(1-X^d)$, and we derive the formal identity

$$(8) \quad \log \left( 1 + \sum_{\beta \neq \mathbf{0}} b(\beta) X^{\beta} \right) = \sum_{r \ge 1} \sum_{d \ge 1} \frac{(-1)^r \mu(d)}{dr} \sum_{\beta_1, \dots, \beta_r \neq \mathbf{0}} \left( \prod_{i=1}^{r} b(\beta_i) \right) \log \left( 1 - X^d \sum_{i=1}^{r} \beta_i \right).$$

In (7) and (8), we get two expressions of the same series as an infinite expansion with respect to the family of $(\log(1 - X^\alpha))_{\alpha \in \mathbb{N}^k}$. Since $\log(1 - X^\alpha)$ is the sum

<footnote>(§)for the usual semi-order of $\mathbb{N}^k$, that is $\lambda_n - \lambda_m \in \mathbb{N}^k \Rightarrow n > m$. Any sequence such that the sum of coordinates is not decreasing verifies this condition.</footnote>
of $-\mathbf{X}^{\alpha}$ and of monomials of larger exponent vector, this expansion is unique. By identifying coefficients, we have a necessary expression for $c(\alpha)$ for every $\alpha \in \mathbb{N}^k$

$$c(\alpha) = \sum_{r \ge 1} \sum_{d \ge 1} \frac{(-1)^r \mu(d)}{dr} \sum_{\substack{\beta_1, \dots, \beta_r \ne 0 \\ d \sum_{i=1}^r \beta_i = \alpha}} \left( \prod_{i=1}^r b(\beta_i) \right). \quad \square$$

Incidentally, this proves that this complicated formula always spills out an integer (cfr. [Est28, p. 448]).

**2.2. Analytic convergence.** — In this section, we give an analytic meaning of the formal identity in $\mathbb{Z}[\mathbf{X}]$ of Lemmata 2.1 and 2.2. This will allow us to replace the three conditions on $h$ by the equivalent assumption:

For $z$ in the neighborhood of $0$, $h(z)$ is defined by an infinite product $\prod_{\alpha \in \mathbb{N}^k} (1 - z^{\alpha})^{c(\alpha)}$ with $c(\alpha) \in \mathbb{Z}^*$ and $c(0) = 0$.

**Lemma 2.3.** — Let $b: \mathbb{N}^k \to \mathbb{Z}$ and $c: \mathbb{N}^k \to \mathbb{Z}$ be two functions with $b(0) = 1$ and $c(0) = 0$ verifying the formal relation

$$\sum_{\beta \in \mathbb{N}^k} b(\beta) \mathbf{X}^{\beta} = \prod_{\alpha \in \mathbb{N}^k} (1 - \mathbf{X}^{\alpha})^{c(\alpha)} \in \mathbb{Z}[[\mathbf{X}]].$$

The LHS-term converges in a neighborhood of 0 if and only if the RHS-term converges in a neighborhood of 0.

We recall that we refer here to the absolute convergence of the series $\sum_{\beta \in \mathbb{N}^k} b(\beta)z^{\beta}$, and of the absolute convergence of the infinite product $\prod_{\alpha \in \mathbb{N}^k} (1 - z^{\alpha})^{c(\alpha)}$ for some $z \in \mathbb{C}^k$.

It is standard (see [Kno90, Chapter VII]) that the absolute convergence of an infinite product $\prod_{n \in \mathbb{N}} (1 + a_n)$ is equivalent to the absolute convergence of the series $\sum_{n \in \mathbb{N}} a_n$. Taking account of the multiplicities among the terms, we derive that the absolute convergence of an infinite product $\prod_{n \in \mathbb{N}} (1 + a_n)^{c_n}$ is equivalent to the absolute convergence of the series $\sum_{n \in \mathbb{N}} c_n a_n$, where $c_n$ are positive integers. Treating positive exponents $c_n$ and negative ones separately, this equivalence also holds for any sequence $(c_n)_{n \in \mathbb{N}}$ of integers, as long as $a_n \ne -1$ for every $n \in \mathbb{N}$.

Therefore, if $z$ is in the open unit polydisc $D(0, 1)$, the absolute convergence of this latter product $\prod_{\alpha \in \mathbb{N}^k} (1 - z^{\alpha})^{c(\alpha)}$ is equivalent to the absolute convergence of the series $\sum_{\alpha \in \mathbb{N}^k} c(\alpha)z^{\alpha}$, which is easier to deal with.

*Proof.* — If the series $\sum_{\alpha \in \mathbb{N}^k} c(\alpha)z^{\alpha}$ converges absolutely at a non-zero $z \in D(0, 1)$, then it converges normally on the open polydisc of polyradius ($|z_1|, \dots, |z_k|$), and the product defines a holomorphic function on this polydisc, which is the sum of its power series expansion, that is $\sum_{\beta \in \mathbb{N}^k} b(\beta)X^{\beta}$, on this polydisc. More generally, the domain of convergence of $\sum_{\beta \in \mathbb{N}^k} b(\beta)X^{\beta}$ contains the intersection of the domain of convergence of $\sum_{\alpha \in \mathbb{N}^k} c(\alpha)X^{\alpha}$ with the open unit polydisc $D(0, 1)$.

We now consider the converse. Assume that $\sum_{\beta \in \mathbb{N}^k} b(\beta)X^{\beta}$ converges on a compact polydisc of polyradius $r \in ]0, 1[^k$, then there exists $M \ge 1$ such that $|b(\beta)| \le$
$Mr^{-\beta}$ for any $\beta \in \mathbb{N}^k$. By Lemma 2.2 we deduce that for any $\alpha$

$$|c(\boldsymbol{\alpha})| \le \sum_{d \ge 1} \sum_{r \ge 1} \frac{1}{dr} M^r \boldsymbol{r}^{-\boldsymbol{\alpha}/d} \left| \left\{ (\boldsymbol{\beta}_1, \dots, \boldsymbol{\beta}_r) \in (\mathbb{N}^k \setminus \{\boldsymbol{0}\})^r \mid d \sum_{i=1}^r \boldsymbol{\beta}_i = \boldsymbol{\alpha} \right\} \right|;$$

therefore

$$|c(\boldsymbol{\alpha})| \le M^{\alpha_1 + \dots + \alpha_k} \boldsymbol{r}^{-\boldsymbol{\alpha}} \frac{\sigma(\alpha_1 + \dots + \alpha_k)}{\alpha_1 + \dots + \alpha_k} P(\boldsymbol{\alpha}),$$

where $P(\boldsymbol{\lambda})$ is the number of representations of $\boldsymbol{\lambda}$ as a sum of non zero elements of $\mathbb{N}^k$, and where $\sigma(n)$ is the sum of the divisors of $n$. It is well-known that for any $\varepsilon > 0$ we have that $\sigma(n)/n \ll_{\varepsilon} (1 + \varepsilon)^n$ for $n \in \mathbb{N}$.

The series $\sum_{\boldsymbol{\lambda} \in \mathbb{N}^k} P(\boldsymbol{\lambda}) \boldsymbol{X}^{\boldsymbol{\lambda}}$ is formally equal to the product $\prod_{\mu \neq \boldsymbol{0}} (1 + \boldsymbol{X}^{\boldsymbol{\mu}})$. This product converges absolutely wherever the series $\sum_{\boldsymbol{\mu} \in \mathbb{N}^k} \boldsymbol{X}^{\boldsymbol{\mu}}$ does. But this sum has a formal expression $\prod_{i=1}^k (1 - X_i)^{-1}$ which corresponds to a holomorphic function on the open unit polydisc $\mathcal{D}(\boldsymbol{0}, 1)$. For $\boldsymbol{z}$ in this polydisc, the power series $\sum_{\boldsymbol{\mu} \in \mathbb{N}^k} \boldsymbol{z}^{\boldsymbol{\mu}}$ converges, the infinite product $\prod_{\boldsymbol{\mu} \neq \boldsymbol{0}} (1 + \boldsymbol{z}^{\boldsymbol{\mu}})$ does also, and so does the power series $\sum_{\boldsymbol{\lambda} \in \mathbb{N}^k} P(\boldsymbol{\lambda}) \boldsymbol{z}^{\boldsymbol{\lambda}}$. Therefore, for any $\rho \in ]0, 1[$ we have that

$$P(\boldsymbol{\lambda}) \ll_{\rho} \rho^{-(\lambda_1 + \dots + \lambda_k)}, \quad \text{for } \boldsymbol{\lambda} \in \mathbb{N}^k.$$

We now set $\boldsymbol{r}' = \frac{\rho}{M(1+\varepsilon)}\boldsymbol{r}$ and we get $|c(\boldsymbol{\alpha})| \ll \boldsymbol{r}'^{-\boldsymbol{\alpha}}$ for any $\boldsymbol{\alpha} \in \mathbb{N}^k$. Therefore both the power series $\sum_{\boldsymbol{\alpha} \in \mathbb{N}^k} c(\boldsymbol{\alpha}) \boldsymbol{z}^{\boldsymbol{\alpha}}$ and the infinite product $\prod_{\boldsymbol{\alpha} \in \mathbb{N}^k} (1 - \boldsymbol{X}^{\boldsymbol{\alpha}})^{c(\boldsymbol{\alpha})}$ converge on the open polydisc of polyradius $\boldsymbol{r}' \in ]0, 1]^k$. $\square$

**2.3. Uniform convergence of a power series.** — In this section, we precise a classical result on the convergence of power series in several variables.

Let $\sum_{\boldsymbol{\alpha} \in \mathbb{N}^k} c(\boldsymbol{\alpha}) \boldsymbol{z}^{\boldsymbol{\alpha}}$ be a power series and define $E$ as the set of $\boldsymbol{z} \in \mathbb{C}^k$ for which there exists a real number $\gamma$ such that $|c(\boldsymbol{\alpha}) \boldsymbol{z}^{\boldsymbol{\alpha}}| \le \gamma$ for all $\boldsymbol{\alpha} \in \mathbb{N}^k$.

A direct generalization of Abel's Lemma (e.g. [Sch05, Lemma 1.5.8]) gives that the interior $\tilde{E}$ is the domain of convergence of the series and, if $\boldsymbol{z} \in \tilde{E}$, then the power series converges normally on the closed polydisc of polyradius ($|z_1|, \dots, |z_k|$).

This result is sufficient to deduce that the sum of a power series is an holomorphic function on its domain of convergence. We will show that if the support of the coefficient function $c$ is sufficiently small compared to $\mathbb{N}^k$, then we can replace the closed polydiscs in Abel's Lemma by larger, unbounded closed sets.

In the framework of Dirichlet series, namely substituting (¶) $\boldsymbol{z}$ by $(e^{-s_1}, \dots, e^{-s_k})$ with $\boldsymbol{s} \in \mathbb{C}^k$, Abel's Lemma is translated as follows.

**Lemma 2.4.** — The domain of convergence of the Dirichlet series $\sum_{\boldsymbol{\alpha} \in \mathbb{N}^k} c(\boldsymbol{\alpha}) e^{-\langle \boldsymbol{\alpha}, \boldsymbol{s} \rangle}$ is the interior of the set $D$ of $\boldsymbol{s} \in \mathbb{C}^k$ for which there exists $\theta \in \mathbb{R}$ such that $\Re\langle \boldsymbol{\alpha}, \boldsymbol{s} \rangle \ge \theta + \log |c(\boldsymbol{\alpha})|$ for every $\boldsymbol{\alpha} \in \mathbb{N}^k$.

For any $\boldsymbol{s}_0 \in \tilde{D}$, the Dirichlet series converges normally on $\boldsymbol{s}_0 + \{\boldsymbol{s} \in \mathbb{C}^k \mid \forall i \in [1, k], \Re s_i \ge 0\}$.

(¶)Only $\boldsymbol{z}$ with a zero coordinate cannot be represented this way.
Notice that $D$ is a tubular convex set($\mathcal{H}$), and if $c(\boldsymbol{\alpha}) = 0$, the condition $\Re\langle\boldsymbol{\alpha}, \boldsymbol{s}\rangle \ge \log|c(\boldsymbol{\alpha})| + \theta$ has to be interpreted as verified for all $\boldsymbol{s} \in \mathbb{C}^k$. The parameter $\theta$ in $D$ is related to the parameter $\gamma$ in $E$ by $\gamma = e^{-\theta}$.

**Lemma 2.5.** — Let $s_0 \in \mathbb{C}^k$ in the domain of convergence of the Dirichlet series $\sum_{\boldsymbol{\alpha} \in \mathbb{N}^k} c(\boldsymbol{\alpha}) e^{-\langle\boldsymbol{\alpha}, \boldsymbol{s}\rangle}$. Let $C \subseteq \mathbb{N}^k$ the support of the coefficient function $c$ and $C^\vee$ its dual cone in $\mathbb{C}^k$. Then, the Dirichlet series $\sum_{\boldsymbol{\alpha} \in \mathbb{N}^k} c(\boldsymbol{\alpha}) e^{-\langle\boldsymbol{\alpha}, \boldsymbol{s}\rangle}$ is normally convergent on $s_0 + C^\vee$.

*Proof.* — Let $\boldsymbol{s} = s_0 + \boldsymbol{s}'$ with $\boldsymbol{s}' \in C^\vee$. By definition of $C^\vee$, for any $\boldsymbol{\alpha} \in \mathbb{N}^k$ such that $c(\boldsymbol{\alpha}) \ne 0$, we have $\Re\langle\boldsymbol{\alpha}, \boldsymbol{s}'\rangle \ge 0$, thus

$$|c(\boldsymbol{\alpha}) e^{-\langle\boldsymbol{\alpha}, \boldsymbol{s}\rangle}| = |c(\boldsymbol{\alpha}) e^{-\langle\boldsymbol{\alpha}, s_0\rangle}| e^{-\Re\langle\boldsymbol{\alpha}, \boldsymbol{s}'\rangle} \le |c(\boldsymbol{\alpha}) e^{-\langle\boldsymbol{\alpha}, s_0\rangle}|.$$

Finally, for any $\boldsymbol{\alpha} \in \mathbb{N}^k$, we have $\sup_{\boldsymbol{s} \in s_0 + C^\vee} |c(\boldsymbol{\alpha}) e^{-\langle\boldsymbol{\alpha}, \boldsymbol{s}\rangle}| = |c(\boldsymbol{\alpha}) e^{-\langle\boldsymbol{\alpha}, s_0\rangle}|$, which is the term of a convergent series by assumption. $\square$

This result is optimal, in the sense that for any cone $K$ larger than $C^\vee$, there is an $\boldsymbol{\alpha} \in C$ such that the function $c(\boldsymbol{\alpha}) e^{-\langle\boldsymbol{\alpha}, \boldsymbol{s}\rangle}$ is not bounded on $s_0 + K$.

However, if we remove finitely many terms from the series, corresponding to a finite set $C_0 \subseteq C$ of indices, the lemma applies to the remaining sum but with normal convergence on the larger cone $(C \setminus C_0)^\vee$. The whole series may not converge normally on $s_0 + (C \setminus C_0)^\vee$, since some terms may not be bounded, but it still converges uniformly on this set.

We set

$$D_\infty = \bigcup_{\substack{C_0 \subseteq C \\ C_0 \text{ finite}}} (C \setminus C_0)^\vee = \bigcup_{C_0 \subseteq C} \bigcap_{C_0 \text{ finite}} \{\boldsymbol{s} \in \mathbb{C}^k \mid \Re\langle\boldsymbol{\alpha}, \boldsymbol{s}\rangle \ge 0\}.$$

Notice that $D_\infty$ is the lower limit of the family of sets $\{\boldsymbol{s} \in \mathbb{C}^k \mid \Re\langle\boldsymbol{\alpha}, \boldsymbol{s}\rangle \ge 0\}$ indexed by $\boldsymbol{\alpha} \in C$, which means that $\boldsymbol{s} \in D_\infty$ if and only if $\Re\langle\boldsymbol{\alpha}, \boldsymbol{s}\rangle \ge 0$ for all but finitely many $\boldsymbol{\alpha} \in C$. This cone will play a central role in the following. For now, it can be seen as the maximal cone on which the series uniformly converges.

**Lemma 2.6.** — Let $s_0 \in \mathbb{C}^k$ in the domain of convergence of the Dirichlet series $\sum_{\boldsymbol{\alpha} \in \mathbb{N}^k} c(\boldsymbol{\alpha}) e^{-\langle\boldsymbol{\alpha}, \boldsymbol{s}\rangle}$. Let $K$ be a polyhedral cone. Then, the Dirichlet series converges uniformly on $s_0 + K$ if and only if $K \subseteq D_\infty$.

For this reason, we will call $D_\infty$ the **cone of uniform convergence** of the Dirichlet series $\sum_{\boldsymbol{\alpha} \in \mathbb{N}^k} c(\boldsymbol{\alpha}) e^{-\langle\boldsymbol{\alpha}, \boldsymbol{s}\rangle}$.

*Proof.* — Let $s_0 \in \mathbb{C}^k$ such that the Dirichlet series $\sum_{\boldsymbol{\alpha} \in \mathbb{N}^k} c(\boldsymbol{\alpha}) e^{-\langle\boldsymbol{\alpha}, s_0\rangle}$ converges. Let $K$ be a polyhedral cone, spanned by finitely many vectors $v_1, \dots, v_m$.

Assume $K \subseteq D_\infty$. For every $j \in [1, m]$, the vector $v_j$ is in some dual cone $(C \setminus C_j)^\vee$ where $C_j$ is a finite subset of $C$ which may depend on the vector $v_j$. The union $C_0 = \bigcup_{j=1}^m C_j$ is also a finite subset of $C$, and every generating vector $v_j$ belongs to $(C \setminus C_0)^\vee$, i.e. $K \subseteq (C \setminus C_0)^\vee$. By Lemma 2.5, the Dirichlet series converges uniformly on $s_0 + (C \setminus C_0)^\vee$, therefore on $s_0 + K$.

Assume $K \not\subseteq D_\infty$. Then, there exists a generating vector $v_j$ outside of $D_\infty$. In

<footnote>($\mathcal{H}$)which corresponds to the fact that $\hat{E}$ is a logarithmically convex Reinhardt domain.</footnote>
particular, there exist infinitely many $\alpha \in C$ for which $\Re\langle\alpha, v_j\rangle < 0$. For all these $\alpha$, the term $s \to c(\alpha) e^{-\langle\alpha, s\rangle}$ is not bounded on $s_0 + K$, which contradicts Cauchy's uniform convergence test. $\square$

**Corollary 2.7.** — Let $\sum_{\alpha \in \mathbb{N}^k} c(\alpha) e^{-\langle\alpha, s\rangle}$ be a Dirichlet series with integer coefficients, with $c(0) = 0$, and which converges at $s_0 \in \mathbb{C}^k$. Let $D_\infty$ be the cone of uniform convergence of this series. Then, the function $g$ defined by the product

$$g(s) = \prod_{\alpha \in \mathbb{N}^k} (1 - e^{\langle\alpha, s\rangle})^{c(\alpha)}$$

is meromorphic on $s_0 + \mathring{D}_\infty$, and its divisor on this domain is

$$(g) = \sum_{\kappa \in \mathbb{Z}} \sum_{\alpha \in C} c(\alpha) H_{\alpha}(2\kappa\pi i),$$

where $H_\lambda(\gamma)$ is the hyperplane of elements $s \in s_0 + \mathring{D}_\infty$ such that $\langle\lambda, s\rangle = \gamma$.

*Proof.* — Let $K$ be a polyhedral cone inside $D_\infty$. By uniform convergence, the Dirichlet series has at most finitely many terms taking values larger than $1/2$ on $s_0 + K$; we set $C_0$ as the finite set of indices of those terms. Then, the subseries $\sum_{\alpha \in C \setminus C_0} c(\alpha) \log(1 - e^{-\langle\alpha, s\rangle})$ of analytic functions converges uniformly on $s_0 + K$, the subproduct $\prod_{\alpha \in C \setminus C_0} (1 - e^{\langle\alpha, s\rangle})^{c(\alpha)}$ is uniformly convergent on $s_0 + K$ (see [Kno90, §51] for a detailed argument), and its logarithm is bounded over $s_0 + K$. Therefore, the subproduct is an analytic function of $s_0 + \mathring{K}$, with no zero on $s_0 + \mathring{K}$.

The whole product $\prod_{\alpha \in \mathbb{N}^k} (1 - e^{\langle\alpha, s\rangle})^{c(\alpha)}$ is therefore meromorphic on $s_0 + K$, and its divisor on $s_0 + \mathring{K}$ comes only from the terms of the infinite product : $(g) = \sum_{\kappa \in \mathbb{Z}} \sum_{\alpha \in C_0} c(\alpha) H_\alpha(2\kappa\pi i)$. Note that for any $\alpha \in C \setminus C_0$, we have $|e^{-\langle\alpha, s\rangle}| \le \frac{1}{2}$ for every $s \in s_0 + K$, therefore $\alpha \in K^\vee$, which means that $\Re\langle\alpha, s'\rangle \ge 0$ for any $s' \in K$, and $|e^{-\langle\alpha, s_0\rangle}| \le \frac{1}{2}$, which means that $\Re\langle\alpha, s_0\rangle \ge \log 2 > 0$. We deduce that for any $s \in s_0 + K$, we have $\Re\langle\alpha, s\rangle > 0$. Thus, $H_\alpha(2\kappa\pi i)$ does not intersect $s \in s_0 + K$ if $\alpha \in C \setminus C_0$. We may as well write the divisor of $g$ on $s_0 + \mathring{K}$ as $(g) = \sum_{\kappa \in \mathbb{Z}} \sum_{\alpha \in C} c(\alpha) H_\alpha(2\kappa\pi i)$, even if infinitely many terms of this sum are empty.

Since $g$ is meromorphic on $s_0 + \mathring{K}$ with divisor $(g) = \sum_{\kappa \in \mathbb{Z}} \sum_{\alpha \in C} c(\alpha) H_\alpha(2\kappa\pi i)$ for any polyhedral cone inside $D_\infty$, we deduce that $g$ is meromorphic on $s_0 + \mathring{D}_\infty$ with divisor $(g) = \sum_{\kappa \in \mathbb{Z}} \sum_{\alpha \in C} c(\alpha) H_\alpha(2\kappa\pi i)$. $\square$

This result gives some legitimacy to the study of the infinite product expansion of a power series: (some of) the terms of the product correspond directly to (some of) the terms of the divisor of the meromorphic extension, which is intrinsic to the power series. The parentheses in the previous statement come from the fact that this correspondence only concerns the terms of the divisor which intersect the domain $s_0 + \mathring{D}_\infty$.

It is natural to ask under which conditions the hyperplane of equation $\langle\alpha, s\rangle = \gamma$ meets the domain $s_0 + \mathring{D}_\infty$. Since $\mathring{D}_\infty$ is tubular, we can only consider the real parts. Recall that $\alpha \in \mathbb{N}^k \setminus \{0\}$ and $\mathbb{R}_+^k \subseteq D_\infty$. Therefore, the image of $\mathring{D}_\infty$ by the real linear form $s \mapsto \Re\langle\alpha, s\rangle$ is either $]0, +\infty[$ if $\alpha \in D_\infty^\vee$ or $]-\infty, +\infty[$ if $\alpha \notin D_\infty^\vee$.
We deduce that the hyperplane of equation $\langle \alpha, s \rangle = \gamma$ meets the domain $s_0 + D^\circ_\infty$ if and only if $\Re\langle \alpha, s_0 \rangle < \Re e(\gamma)$ or $\alpha \notin D^\vee_\infty$.

With this result in hands, we give two examples. The first one shows how the product representation can help to study the meromorphic expansion.

*First example.* — Consider the power series $\sum_{k,n} p_k(n) X^k Y^n \in \mathbb{Z}[[X, Y]]$, where $p_k(n)$ counts the partitions of $n$ in $k$ parts, and let $g$ be the associated Dirichlet series. The domain of convergence of $g$ is $\mathbb{C}^2$, where $\mathbb{C}_r$ denotes $\{s \in \mathbb{C} | \Re e(s) > 0\}$. This power series has the well-known product expansion $\prod_{m \ge 1} (1 - XY^m)^{-1}$. Therefore $\text{xsupp}(g) = \{(1, m) | m \ge 1\}$ and $D^\circ_\infty = \mathbb{C} \times \mathbb{C}_r$. The dual $D^\vee_\infty = \{0\} \times \mathbb{R}_+$ does not contain any element of $\text{xsupp}(g)$.

Using Corollary 2.7, we deduce that $g$ is meromorphic on $D^\circ_\infty$, and its divisor is exactly $(g) = \sum_{\kappa \in \mathbb{Z}} \sum_{m \ge 1} (-1)^m H_{(1,m)}(2\kappa\pi i)$, where every hyperplane actually intersects the domain of meromorphy, which is easy to verify in this case.

As an exercise, the reader could prove that the family of poles clusters at any point $(s_1, s_2) \in \partial D_\infty$ with $\Re e(s_1) \le 0$, and with the help of Theorem 4.14, that every point of $\partial D_\infty$ is a singular point for the meromorphic function $g$. This means that $\mathbb{C} \times \mathbb{C}_r$ is the domain of meromorphy of $g$.

In this second example, we use our knowledge on the meromorphic extension to deduce the property of the exponential support.

*Second example.* — Consider an algebraic power series $\sum_{\beta \in \mathbb{N}^k} b(\beta) X^\beta \in \mathbb{Z}[[X]]$ with minimal polynomial

$$Q(\mathbf{X}, Y) = P_0(\mathbf{X}) + Y P_1(\mathbf{X}) + \dots + Y^d P_d(\mathbf{X}) \in \mathbb{Z}[\mathbf{X}, Y],$$

and let $g$ be the associated Dirichlet series. With $d=1$, we recover the case of rational power series. We assume that $b(0) = 1$, and that the series converges around $0$.(***) If neither $P_0$ nor $P_d$ have a cyclotomic factor, no meromorphic extension of $h$ can have a zero or a pole written as $H_\alpha(2\kappa\pi i)$ (see Lemma 4.1). Therefore, by Corollary 2.7, the divisor of $h$ is empty on $s_0 + D^\circ_\infty$. For example, for any $\alpha \in \mathbb{C}$, the hyperplane $H_\alpha(2\kappa\pi i)$ does not meet $s_0 + D^\circ_\infty$, which implies that $\alpha \in D^\vee_\infty$ and $\Re\langle \alpha, s_0 \rangle \ge 0$. The first fact, namely $C \subseteq D^\vee_\infty$, can be also written as $(C \setminus C_0)^\vee = C^\vee = \overline{D^\vee_\infty}$, for all finite subset $C_0 \subseteq C$, or again $\overline{\text{cone}}(C \setminus C_0) = \overline{\text{cone}}(C)$. The second fact translates then that the domain of convergence $\Omega$ of $h$ is contained in $C^\vee = \overline{D^\vee_\infty}$. Notice that by Lemma 2.5, we have that $\Omega + C^\vee = C^\vee$.

**2.4. Meromorphy of the Euler product.** — Before stating the main theorem of this section, we establish a result on an unusual kind of product of Dirichlet series.

*Lemma 2.8.* — Let $\sum_{n \in \mathbb{N}} b_n e^{-\lambda_n s}$ be a Dirichlet series with $0 < \lambda_0 < \lambda_1 < \dots$ and $b_0 \ne 0$. Let $\theta \ge 0$ be a real value at which this series converges absolutely.

<footnote>(***) This is verified if $Q(0, Y) = 1 - Y$. More generally, if $Q(0, 1) = 0$ and $\frac{\partial Q}{\partial Y}(0, 1) \ne 0$, we can always construct a solution $y(\mathbf{X}) \in \mathbb{Q}[[\mathbf{X}]]$ with $y(0) = 1$ and convergent around $0$. By the multivariable Eisenstein's Theorem, a dilation by a positive integer $a$ is sufficient to get $y(a\mathbf{X}) \in \mathbb{Z}[[\mathbf{X}]]$.</footnote>
Let $\sum_{\alpha \in \mathbb{N}^k} c(\alpha) e^{-\langle \alpha, s \rangle}$ be a Dirichlet series and let $C$ be the support of its co-efficient function. Let $s_0$ be a point of $C^\vee$ where this series converges. Set $m_0 = \min\{\langle \alpha, 1 \rangle | \alpha \in C\}$. Assume that $m_0 \ge 1$, that is $c(0) = 0$.

The compound Dirichlet series

$$ \sum_{n \in \mathbb{N}} \sum_{\alpha \in C} b_n c(\alpha) e^{-\lambda_n \langle \alpha, s \rangle} $$

converges normally on $\frac{1}{\lambda_0}s_0 + \frac{1}{m_0}\theta 1 + C^\vee$.

*Proof.* — By Lemma 2.5, the series $\sum_{\alpha \in \mathbb{N}^k} c(\alpha) e^{-\langle \alpha, s \rangle}$ converges normally on $s_0 + C^\vee$, and for any $n \in \mathbb{N}$ we have that $\sum_{\alpha \in \mathbb{N}^k} c(\alpha) e^{-\lambda_n \langle \alpha, s \rangle}$ converges on $\frac{1}{\lambda_n}s_0 + C^\vee$, therefore on $\frac{1}{\lambda_0}s_0 + C^\vee$.

Now we set $s = \frac{1}{\lambda_0}s_0 + \frac{1}{m_0}\theta 1$. For any $n \in \mathbb{N}$ and any $\alpha \in C$, we have

$$ \lambda_n \Re\langle \alpha, s \rangle = \frac{\lambda_n}{\lambda_0} \Re\langle \alpha, s_0 \rangle + \theta \lambda_n \frac{1}{m_0} \Re\langle \alpha, 1 \rangle \ge \Re\langle \alpha, s_0 \rangle + \theta \lambda_n, $$

since $s_0 \in C^\vee$ and $\theta \ge 0$. This implies that

$$ \sum_{n \in \mathbb{N}} \sum_{\alpha \in C} |b_n c(\alpha) e^{-\lambda_n \langle \alpha, s \rangle}| \le \left( \sum_{n \in \mathbb{N}} |b_n e^{-\lambda_n \theta}| \right) \left( \sum_{\alpha \in C} |c(\alpha) e^{-\langle \alpha, s_0 \rangle}| \right). $$

This proves that the double series converges in $s$ and converges normally on $s+C^\vee$. $\square$

We may now state and prove our convergence theorem.

**Theorem 2.9.** — Let $h(z)$ be defined by the infinite product $\prod_{\alpha \in \mathbb{N}^k} (1 - z^\alpha)^{c(\alpha)}$ with $c(\alpha) \in \mathbb{Z}$ and $c(0) = 0$, for $z$ in the neighborhood of $0$. Let $D_\infty$ be the cone of uniform convergence of the Dirichlet series $\sum_{\alpha \in \mathbb{N}^k} c(\alpha) e^{-\langle \alpha, s \rangle}$, which is defined as

$$ D_{\infty} = \{s \in \mathbb{C}^k \mid \Re\langle \alpha, s \rangle \ge 0 \text{ for all but finitely many } \alpha \in \mathbb{N}^k \text{ such that } c(\alpha) \ne 0\}, $$

and assume that the function $s \mapsto h(e^{-s_1}, \dots, e^{-s_k})$ is meromorphic on the open convex cone $\Gamma = \mathring{D}_{\infty}$.

Then, the Euler product

$$ \prod_p h(p^{-s_1}, \dots, p^{-s_k}) $$

defines a function $f$ that admits a meromorphic continuation on $\Gamma$, and the divisor of $f$ on $\Gamma$ is

$$ (9) \quad (f) = \sum_p \left[ (h_p) - \sum_{\alpha \in \mathbb{N}^k} c(\alpha) \sum_{\kappa \in \mathbb{Z}} H_\alpha\left(\frac{2\kappa\pi i}{\log p}\right) \right] - \sum_{\alpha \in \mathbb{N}^k} \sum_{\gamma \in G} c(\alpha) m_\gamma H_\alpha(\gamma), $$

where $h_p$ is the meromorphic function $s \mapsto h(p^{-s_1}, \dots, p^{-s_k})$ defined on $\Gamma$, $G$ is the set of zeros and poles of the Riemann zeta function, $m_\gamma$ is the multiplicity of $\gamma$ as a zero or its (negative) order as a pole ($m_1 = -1$) for $\zeta$, and

$$ H_{\lambda}(\gamma) = \{s \in \Gamma \mid \langle \lambda, s \rangle = \gamma\}. $$
This result requires some conditions on $h$.

The function $h$ is defined by an infinite product in the neighborhood of 0: by Lemma 2.3, this assumption is equivalent to the three assumptions presented at the beginning of this section.

The second assumption is that the function $s \mapsto h(e^{-s_1}, \dots, e^{-s_k})$ is meromorphic on $\tilde{D}_\infty$. Actually, this is not really a requirement, but the theorem would be even heavier to state without it. The proof shows that if $f$ is not meromorphic at some point of $\Gamma$, then one of the local factors $h_p$ is not meromorphic at that point. The converse may not be true since by default of meromorphy of different local factors at the same point may cancel altogether.

We give an alternate statement. By Corollary 2.7, we know that the function $s \mapsto h(e^{-s_1}, \dots, e^{-s_k})$ is meromorphic on $s_0 + \tilde{D}_\infty$, for some $s_0 \in \bar{D}_\infty$. In this case, the function $f$ is meromorphic on $\frac{1}{\log 2}s_0 + \tilde{D}_\infty$.

Every term between brackets in (9) is the scaled version of the same divisor, which is the divisor of the function $s \mapsto h(e^{-s_1}, \dots, e^{-s_k})$ subtracted by the part of this divisor identified in Corollary 2.7. Just like in this Corollary, only the terms $\alpha \in C \setminus D_\infty^\vee$ are relevant since the other ones do not intersect $\tilde{D}_\infty$. Note that if there exists some $\alpha_0 \in C \setminus D_\infty^\vee$, then it is not possible to remove the brackets, since $H_{\alpha_0}(0)$ would have infinite multiplicity in the resulting sum.

On the contrary, if $C \subseteq D_\infty^\vee$, we may replace the term between brackets simply by $(h_p)$. As noticed previously, this is the case when $h$ is a rational power series without cyclotomic factors. In this case, we can also simplify the expression of the domain $\Gamma$; this is the aim of a large part of the next section, which is finally attained in Proposition 3.17. In that case, we may write Theorem 2.9 in simpler form.

**Theorem 2.10.** — Let $h \in \mathbb{Z}[\mathbf{X}]$ be a rational power series, with constant term equal to 1, and with product expansion $h = \prod_{\alpha \in \mathbb{N}^k} (1 - z^\alpha)^{c(\alpha)}$. Let $q^+$ and $q^- \in \mathbb{Z}[\mathbf{X}]$ be its numerator and its denominator, respectively. Then, the Euler product

$$ \prod_p h(p^{-s_1}, \dots, p^{-s_k}) $$

defines a function $f$ that admits a meromorphic continuation on the interior $\Gamma$ of $(\text{supp } q^+ \cup \text{supp } q^-)^\vee$, and the divisor of $f$ on $\Gamma$ is

$$ (f) = \sum_p (h_p) - \sum_{\alpha \in \mathbb{N}^k} \sum_{\gamma \in G} c(\alpha) m_\gamma H_\alpha(\gamma), $$

with the same notations as in Theorem 2.9.

**Proof of Theorem 2.9.** — Like its associated product $\prod_{\alpha \in \mathbb{N}^k} (1 - z^\alpha)^{c(\alpha)}$, the power series $\sum_{\alpha \in \mathbb{N}^k} c(\alpha)z^\alpha$ is convergent on a neighborhood of 0, and there exists some real number $r \in ]0, 1[$ such that both expressions converge on the open polydisc of polyradius $(r, \dots, r)$. In particular, for $\rho > -\log r > 0$, the product $\prod_{\alpha \in \mathbb{N}^k} (1 - e^{-\langle \alpha, s \rangle})^{c(\alpha)}$ converges at $s_0 = (\rho, \dots, \rho)$.

At least formally, the function $f$ may be represented by the double product

$$ (10) \qquad \prod_p \prod_{\alpha} (1 - e^{-\log p \langle \alpha, s \rangle})^{c(\alpha)}. $$
This product converges wherever the double Dirichlet series

$$ (11) \qquad \sum_p \sum_{\alpha \in \mathbb{N}^k} c(\alpha) e^{-\log p \langle \alpha, s \rangle} $$

does. We apply Lemma 2.8 to the Dirichlet series $\sum_p p^{-s}$, which converges at $\theta$ if $\theta > 1$ and for which $\lambda_0 = \log 2$ since 2 is the smallest prime number, and to the Dirichlet series $\sum_{\alpha \in \mathbb{N}^k} c(\alpha) e^{-\langle \alpha, s \rangle}$, which converges at $\rho \mathbf{1}$ if $\rho > -\log r$ and for which $m_0 \ge 1$ since $c(\mathbf{0}) = 0$: we obtain that the double Dirichlet series (11) converges on $\frac{1}{\log 2} \rho \mathbf{1} + \theta \mathbf{1} + C^\vee$.

We deduce that the double product (10) converges uniformly on this domain. In particular, $f$ is meromorphic on this domain and can be expressed in two different ways as a convergent infinite product

$$ (12) \qquad f(s) = \prod_{p \text{ prime}} h(p^{-s_1}, \dots, p^{-s_k}) = \prod_{\alpha \in C} \zeta(\langle \alpha, s \rangle)^{-c(\alpha)}. $$

Notice that on this domain, we have $\Re\langle\alpha, s\rangle > (\theta + \frac{1}{\log 2}\rho)\langle\alpha, \mathbf{1}\rangle > 1$ for any $\alpha \in C$, therefore using the rightmost product of (12), we deduce that the divisor of $f$ is null on this domain.

If $C$ is finite, $h$ is a quotient of polynomials, and the divisor of $s \to h(e^{-s_1}, \dots, e^{-s_k})$ on $\mathbb{C}^k$ is exactly

$$ \sum_{\alpha \in C} c(\alpha) \sum_{\kappa \in \mathbb{Z}} H_{\alpha}(2\kappa\pi i). $$

Therefore, the first sum of (9) is null in this case. Since $C$ is finite, we have $D_\infty = \mathbb{C}^k$ and the product of zeta functions in (12) gives the meromorphic extension of $f$ to the whole space $\mathbb{C}^k$. From this product, the divisor of $f$ over $\mathbb{C}^k$ is

$$ - \sum_{\alpha \in \mathbb{N}^k} c(\alpha) \sum_{\gamma \in G} m_{\gamma} H_{\alpha}(\gamma), $$

which is the second part of (9).

If $C$ is not finite, it is countable and we fix a bijection $n \mapsto \alpha_n$ from $\mathbb{N}$ to $C$. We set $c_n = c(\alpha_n)$ for all $n \in \mathbb{N}$. For any $N \in \mathbb{N}$, we set $C_N$ as $\{\alpha_n \mid n \ge N\} \subseteq C$, $D_N = C_N^\vee$ and $m_N = \min\{\langle\alpha, \mathbf{1}\rangle \mid \alpha \in C_N\}$. It is clear that the sequence $(m_N)_{N \in \mathbb{N}}$ is nondecreasing and tends to infinity.

We fix two integral parameters $N \ge 0$ and $P \ge 2$, and set $\mu_{N,P} = \frac{1}{\log P} \rho + \frac{1}{m_N} \theta$. By absolute convergence of the double product (10) defining $f$, we may reorganize the terms to produce three subproducts

$$ \prod_{n<N} \left( \prod_p (1 - p^{-\langle \alpha_n, s \rangle})^{c_n} \right) \cdot \prod_{p<P} \left( \prod_{n \ge N} (1 - p^{-\langle \alpha_n, s \rangle})^{c_n} \right) \cdot \prod_{p \ge P} \prod_{n \ge N} (1 - p^{-\langle \alpha_n, s \rangle})^{c_n} $$

for all $s$ in $\mu_{0,2}\mathbf{1} + C^\vee$.

The leftmost term converges to

$$ \prod_{n<N} \zeta(\langle \alpha_n, s \rangle)^{-c_n} $$
which admits a meromorphic extension on $\mathbb{C}^k$, and it has for divisor

$$-\sum_{n<N} c_n \sum_{\gamma \in G} m_{\gamma} H_{\alpha_n}(\gamma).$$

In the middle term, each term indexed by $p < P$ converges to

$$h_p(s) \prod_{n<N} (1 - p^{-\langle \alpha_n, s \rangle})^{-c_n},$$

which admits a meromorphic extension on $\mathring{D}_{\infty}$ at least, and has for divisor on $\mathring{D}_{\infty}$

$$(h_p) - \sum_{n<N} \sum_{\kappa \in \mathbb{Z}} c_n H_{\alpha_n} \left( \frac{2\kappa\pi i}{\log p} \right).$$

Finally, by Lemma 2.8, the rightmost product converges uniformly on $\frac{1}{\log p} \rho \mathbf{1} + \frac{1}{m_N} \theta \mathbf{1} + C_N^\vee = \mu_{N,P} \mathbf{1} + D_N$, and by Corollary 2.7, the product is meromorphic on the interior of this cone with divisor

$$\sum_{p \ge P} \sum_{n \ge N} \sum_{\kappa \in \mathbb{Z}} c_n H_{\alpha_n} \left( \frac{2\kappa\pi i}{\log p} \right).$$

Altogether, we obtain a meromorphic extension of $f$ on $\mu_{N,P} \mathbf{1} + \mathring{D}_N$ with divisor

$$(13) - \sum_{n<N} \sum_{\gamma \in G} c_n m_{\gamma} H_{\alpha_n}(\gamma) + \sum_{p<P} \left[ (h_p) - \sum_{n<N} \sum_{\kappa \in \mathbb{Z}} c_n H_{\alpha_n} \left( \frac{2\kappa\pi i}{\log p} \right) \right] + \sum_{p \ge P} \sum_{n \ge N} \sum_{\kappa \in \mathbb{Z}} c_n H_{\alpha_n} \left( \frac{2\kappa\pi i}{\log p} \right).$$

For any $s$ in this domain and for any $n \ge N$, we have

$$\Re\langle \alpha_n, s \rangle \ge \mu_{N,P} \langle \alpha_n, 1 \rangle \ge m_n \mu_{N,P} > \theta > 1.$$

Therefore, any hyperplane of equation $\langle \alpha_n, s \rangle = \lambda$ with $n \ge N$ and $\Re(\lambda) \le 1$ has an empty intersection with the domain $\mu_{N,P} \mathbf{1} + \mathring{D}_N$. For this reason, for any $n \ge N$, we may add or remove the terms $H_{\alpha_n}(\gamma)$ or $H_{\alpha_n}(\frac{2\kappa\pi i}{\log p})$ from the divisor (13). We obtain an expression of $f$ independent of $N$

$$(f) = - \sum_{n \in \mathbb{N}} c_n \sum_{\gamma \in G} m_{\gamma} H_{\alpha_n}(\gamma) + \sum_{p < P} \left[ (h_p) - \sum_{n \in \mathbb{N}} \sum_{\kappa \in \mathbb{Z}} c_n H_{\alpha_n} \left( \frac{2\kappa\pi i}{\log p} \right) \right].$$

Finally, applying Corollary 2.7 to the Dirichlet series $\sum_n c_n e^{-\langle \alpha_n, s \rangle}$ which converges at $\rho \mathbf{1}$, we deduce that for any prime $p$, the divisor of $(h_p)$ on $\frac{1}{\log p} \rho \mathbf{1} + \mathring{D}_{\infty}$ is equal to

$$(h_p) = \sum_{n \in \mathbb{N}} \sum_{\kappa \in \mathbb{Z}} c_n H_{\alpha_n} \left( \frac{2\kappa\pi i}{\log p} \right).$$

If $p \ge P$, then $\mu_{N,P} > \frac{1}{\log P} \rho > \frac{1}{\log p} \rho$, thus $\mu_{N,P} \mathbf{1} \in \frac{1}{\log p} \rho \mathbf{1} + \mathring{D}_{\infty}$, and since $D_N \subseteq D_{\infty}$, we have $\mu_{N,P} \mathbf{1} + D_N \subseteq \frac{1}{\log p} \rho \mathbf{1} + \mathring{D}_{\infty}$. We may add or remove the term
$(h_p) - \sum_{n \in \mathbb{N}} \sum_{\kappa \in \mathbb{Z}} c_n H_{\alpha_n} (\frac{2\kappa\pi i}{\log p})$, which is null, from the divisor of $f$ on $\mu_{N,P} \mathbf{1} + \mathcal{D}_N^\circ$.
We obtain the following expression of $(f)$, which is independent of $P$:

$$ (f) = - \sum_{n \in \mathbb{N}} c_n \sum_{\gamma \in G} m_\gamma H_{\alpha_n}(\gamma) + \sum_p \left[ (h_p) - \sum_{n \in \mathbb{N}} \sum_{\kappa \in \mathbb{Z}} c_n H_{\alpha_n} \left( \frac{2\kappa\pi i}{\log p} \right) \right]. $$

We deduce that $f$ has a meromorphic extension on the union

$$ \Gamma = \bigcup_{N,P} (\mu_{N,P} \mathbf{1} + \mathcal{D}_N^\circ), $$

and its divisor on this domain is given by the expression above. It remains to prove that $\Gamma = \mathcal{D}_\infty^\circ$.

For any $N > N'$ and $P$, we have $\mu_{N',P} \ge \mu_{N,P} > 0$. In particular, $\mu_{N',P} \mathbf{1} + D_N \subseteq \mu_{N,P} \mathbf{1} + D_N \subseteq D_N$, and taking the union over $N$, we have

$$ \mu_{N',P} \mathbf{1} + D_{\infty} \subseteq \bigcup_{N} (\mu_{N,P} \mathbf{1} + D_{N}) \subseteq D_{\infty}. $$

Since $\mu_{N',P}$ tends to 0 as $N'$ and $P$ both grow to infinity, we have that

$$ \bigcup_{N',P} (\mu_{N',P} \mathbf{1} + D_{\infty}) = \mathcal{D}_{\infty}^{\circ}. $$

Using the two previous formulas, we derive that

$$ \mathcal{D}_{\infty}^{\circ} = \bigcup_{N,P} (\mu_{N,P} \mathbf{1} + D_N), $$

which leads directly to $\Gamma = \mathcal{D}_{\infty}^{\circ}$. $\square$

<figure><img src="image_4.png" /></figure>

Even if it may not seem like it, the roles of $h$ and $\zeta$ are relatively symmetric in this theorem. Actually, one may replace $\zeta$ by any function $\xi$, which is meromorphic over $\mathbb{C}$, and which can be expressed as $\prod_{i \in I} (1 - p_i^{-s})^{-m_i}$ for $\Re e(s) > \sigma$, where the exponents $m_i$ are all integers, the real numbers $p_i$ are all larger than 1 and $\sigma \in \mathbb{R}$. We denote by $\sum_{\gamma \in G_\xi} n_\gamma[\gamma]$ the divisor of $\xi$ over $\mathbb{C}$. In that case, and with the same assumptions on $h$ and the same definition of $\Gamma$, the same proof shows that the product $\prod_{i \in I} h(p_i^{-s_1}, \dots, p_i^{-s_k})^{m_i}$ defines a meromorphic function over $\Gamma$ with a divisor

$$ \sum_{i \in I} m_i \left[ (h_{p_i}) - \sum_{\alpha \in \mathbb{N}^k} c(\alpha) \sum_{\kappa \in \mathbb{Z}} H_{\alpha} \left( \frac{2\kappa\pi i}{\log p_i} \right) \right] - \sum_{\alpha \in \mathbb{N}^k} c(\alpha) \sum_{\gamma \in G_\xi} n_\gamma H_{\alpha}(\gamma). $$

This idea to replace $\zeta$ by other meromorphic functions also appears in [Dah52], or to a larger extent in [Kur86].

To conclude this part, we give a statement which is directly derived from the proof of the previous theorem, and in which the role played by the two functions has been made perfectly symmetric. To make the statement shorter, we fix all exponents $c(\alpha)$ to $-1$: we have remarked that there is no loss of generality by doing so.
**Corollary 2.11.** — Let two functions $h$ and $h'$ such that there exist two abscissae $\sigma_a \ge \sigma_m \ge 0$ and a real number $\lambda > 0$ such that $h(s)$ is defined by the converging product

$$ \prod_{j \in J} (1 - e^{-\omega_j s})^{-1} $$

if $\Re(e(s)) > \sigma_a$, where the real parameters $\omega_j$ verify $\omega_j \ge \lambda$, and such that $h$ admits a meromorphic extension on the half plane $\Re(e(s)) > \sigma_m$ with divisor $(h)$ (and respectively with $\sigma'_a, \sigma'_m, \lambda', J'$ and $\omega'_j$ for the function $h'$). The double product

$$ \prod_{j \in J} \prod_{j' \in J'} (1 - e^{-\omega_j \omega_{j'} s})^{-1} $$

is convergent for $\Re(e(s)) > \frac{\sigma'_a}{\lambda} + \frac{\sigma'_a}{\lambda'}$ to a function $f$ which admits a meromorphic extension on the half plane $\Re(e(s)) > \max(\frac{\sigma'_m}{\lambda}, \frac{\sigma'_m}{\lambda'})$ whose divisor is

$$ (f) = \sum_{j \in J} S_{\frac{1}{\omega_j}}(h') + \sum_{j' \in J'} S_{\frac{1}{\omega_{j'}}}(h), $$

where $S_\lambda$ is the scaling operator $s \mapsto \lambda s$ acting $\mathbb{Z}$-linearly on the components on the divisor.

Notice that if $\Re(e(s)) > \frac{\sigma'_a}{\lambda} + \frac{\sigma'_a}{\lambda'}$, $f(s)$ may also be defined by the two converging products

$$ f(s) = \prod_{j \in J} h'(\omega_j s) = \prod_{j' \in J'} h(\omega_{j'} s). $$

In the context of additive arithmetical semigroups (see [Kno75]), the function $h$ in the corollary is the zeta function associated to the free abelian semigroup $G = \mathbb{N}^{(J)}$ generated by $J$ associated to the morphism $\mathbb{N}^{(J)} \to \mathbb{R}_+$ defined by $j \mapsto \omega_j$, and similarly for the function $h'$ and the arithmetical semigroup $G' = \mathbb{N}^{(J')}$. The function $f$ is the zeta function associated to the tensor product $G \otimes G'$, isomorphic to $\mathbb{N}^{(J \times J')}$ associated to the morphism $(j, j') \mapsto \omega_j \omega_{j'}$.

### 3. Geometry of C

Theorem 2.9 gives a domain $\Gamma$ on which the Euler product $\prod_p h_p$ is meromorphic. It gives also a description of the divisor of this Euler product over $\Gamma$. However, both objects are mainly described in terms of the exponential support $C$ of $h$. The exponential support is generally infinite (except in the trivial case when $h$ is a quotient of products of cyclotomic polynomials), and usually, it is not part of the usual description of $h$ and it has to be computed, which can be rather ineffective. Moreover, in order to determine the domain $\Gamma$, or to exhibit accumulation in the divisor and detect singular points, the computation of infinitely many elements is required.

The goal of this section is twofold. First, we will give a simple description of $\Gamma$ as a rational convex cone when $h$ is a rational function, only in terms of the (finite)
supports of its numerator and denominator. More precisely, if $h$ has no cyclotomic factors, we will show that the dual $\Gamma^\vee$ of $\Gamma$, defined by

$$ \Gamma^\vee = \{\boldsymbol{\mu} \in \mathbb{R}^k \mid \forall \boldsymbol{s} \in \Gamma, \Re\langle\boldsymbol{\mu}, \boldsymbol{s}\rangle \ge 0\} $$

is the convex cone generated by the supports of the numerator and denominator of $h$. Second, we will show that the exponential support $C$ of a rational function $h$ contains infinite subsets of elements arranged in straight lines. More precisely, for any $\boldsymbol{\nu} \in \mathbb{N}^k$ generating an extreme ray $\boldsymbol{\nu}\mathbb{R}_+$ of the closed convex cone $\overline{\mathrm{cone}}(C)$, we will show that there exists a $\boldsymbol{\mu} \in \mathbb{Z}^k$ such that the ray $\boldsymbol{\mu} + \boldsymbol{\nu}\mathbb{R}_+$ goes through infinitely many points of $C$.

Those infinite subsets on straight lines are important in order to prove our maximality result: intuitively, they correspond to infinite families of hyperplanes in the divisor of $f$, which will generate cluster points, therefore singular points, on the boundary of $\Gamma$.

This kind of alignments naturally appear in the support of $h$ if $h$ is rational, and they play a key role in achieving our first goal.

Our strategy to complete both goals is as follows. We notice that, by definition of $\Gamma$ as limit inferior and [Zăl02, Theorem 1.1.9], we have

$$ \Gamma^{\vee} = \bigcap_{\substack{E \subseteq C \\ E \text{ finite}}} \overline{\mathrm{cone}}(C \setminus E), $$

and therefore $\Gamma^{\vee} \subseteq \overline{\mathrm{cone}}(C)$. For the reciprocal inclusion, it suffices to prove that any extreme ray $\boldsymbol{\nu}\mathbb{R}_+$ of $\overline{\mathrm{cone}}(C)$ is included in $\Gamma^{\vee}$.

Except in favorable cases when $C$ contains infinitely many elements on the extreme ray $\boldsymbol{\nu}\mathbb{R}_+$, it comes down to prove that $\boldsymbol{\nu} \in \overline{\mathrm{cone}}(C \setminus \boldsymbol{\nu}\mathbb{R}_+)$. It turns out that it is easy to define a function $h^\circ$ with $C \setminus \boldsymbol{\nu}\mathbb{R}_+$ as exponential support. But in most of the cases, the exponential support is not explicit enough to reach this conclusion, and we turn to the support $B^\circ$ of $h^\circ$ instead. It happens that for any function, the support and the exponential support generate the same convex cone: thus, $\overline{\mathrm{cone}}(C \setminus \boldsymbol{\nu}\mathbb{R}_+) = \overline{\mathrm{cone}}(B^\circ)$ and it remains to prove that $\boldsymbol{\nu} \in \overline{\mathrm{cone}}(B^\circ)$.

At this point, we will make additional assumptions on $h$ and use polynomial arithmetic to find a line directed by $\boldsymbol{\nu}$ which go through infinitely many elements of $B^\circ$. If $h$ is polynomial, arguments are fairly straightforward, but even when $h$ is rational, our arguments are already substantially more involved. The seemingly simple fact of connecting the support of $h$ with the supports of its numerator and denominator is already a bit of a challenge.

To get over this difficulty, we use a special property of $\mathbb{Z}$, and of $\mathbb{Z}[X]$ actually, known as the *Fatou property*, and we generalize this property to multivariate formal series. From here, we derive that the supports of the denominator and numerator are contained in $\overline{\mathrm{cone}}(B^\circ)$, and that $B^\circ$ has infinitely many elements on a line directed by $\boldsymbol{\nu}$. This is sufficient to conclude that

$$ \Gamma^{\vee} = \overline{\mathrm{cone}}(C) = \overline{\mathrm{cone}}(B) = \mathrm{cone}(B^{+}) + \mathrm{cone}(B^{-}), $$

where $B^+$ and $B^-$ are the supports of the numerator and denominator of $h$, when $h$ has no cyclotomic factors.
<figure><img src="image_5.png" /><figcaption>FIGURE 2. Support, exponential support and corresponding coefficients for the polynomial $h(X_1, X_2) = 1 - X_2 + X_2^2 + 2X_1X_2$</figcaption></figure>

Finally, the last step to achieve our second goal is to study the rationality of the subseries of $h$ of the form $\sum_{n \in \mathbb{N}} b(\mu + n\nu)T^n$, where $\mu$ and $\nu$ are in $\mathbb{N}^k$, and to carry the results to similar series involving the exponent function.

**3.1. First steps.** — In this subsection, we gradually reformulate the assertion that a non-zero vector $\nu \in \mathbb{R}^k$ directing an extreme ray of $\text{cone}(C)$ belongs to $\Gamma^\vee$, until we are able to give a simple, arithmetic criterion in two particular cases: when $h$ is a polynomial and when $h$ is a rational power series in two variables. The general case will be treated in the next subsection.

**3.1.1. Step 1: The disjunction.** — We first state a geometric criterion to decide whether an extreme ray of $\text{cone}(C)$ is in $\Gamma^\vee$ or not.

**Lemma 3.1.** — Let $\nu \in \mathbb{R}^k$ be a non-zero vector such that $\nu\mathbb{R}_+$ is an extreme ray of the closed convex cone $\text{cone}(C)$. We have

$$ \nu \in \bigcap_{\substack{E \subseteq C \\ E \text{ finite}}} \text{cone}(C \setminus E) $$

if and only if the set $C \cap \nu\mathbb{R}_+$ is infinite or $\nu \in \text{cone}(C \setminus \nu\mathbb{R}_+)$.

*Proof.* — The necessity is immediate: if the set $C \cap \nu\mathbb{R}_+$ is a finite subset of $C$, and $\nu$ is a member of $\text{cone}(C \setminus E)$ for any finite subset $E$ of $C$, then

$$ \nu \in \text{cone}(C \setminus (C \cap \nu\mathbb{R}_+)) = \text{cone}(C \setminus \nu\mathbb{R}_+) $$
The first term of the disjunction is also immediately sufficient: if the set $C \cap \nu\mathbb{R}_+$ is infinite, so is the set $(C \setminus E) \cap \nu\mathbb{R}_+$ for any finite subset $E$ of $C$, and $\nu \in \text{cone}(C \setminus E)$ for any such $E$.

It remains to prove that the second term of the disjunction is also sufficient. Since $\nu\mathbb{R}_+$ is an extreme ray of $\overline{\text{cone}}(C)$, the set $\overline{\text{cone}}(C) \setminus \nu\mathbb{R}_+^*$ is also a convex cone, and it includes the set $C \setminus \nu\mathbb{R}_+$, which we denote $C^o$. Therefore, $\nu \notin \text{cone}(C^o)$. Suppose that $\nu \in \overline{\text{cone}}(C^o)$. For any finite subset $E \subseteq C^o$, we have $\overline{\text{cone}}(E) = \text{cone}(E) \subseteq \text{cone}(C^o)$, thus $\nu \notin \overline{\text{cone}}(E)$. We also have that $\nu \in \overline{\text{cone}}(C^o) = \overline{\text{cone}}(E) + \overline{\text{cone}}(C^o \setminus E)$. And since the ray $\nu\mathbb{R}_+$ is extremal in $\overline{\text{cone}}(C^o)$, it is inside at least one of the terms of the sum $\overline{\text{cone}}(E) + \overline{\text{cone}}(C^o \setminus E)$. But since $\nu \notin \overline{\text{cone}}(E)$, we conclude that $\nu \in \overline{\text{cone}}(C^o \setminus E)$, and this for any finite subset $E$ of $C^o$. This implies that

$$ \nu \in \bigcap_{\substack{E \subseteq C^o \\ E \text{ finite}}} \overline{\text{cone}}(C^o \setminus E) \subseteq \Gamma^\vee. \quad \square $$

Notice that when $\nu$ is not rational, this criterion states that $\nu \in \Gamma^\vee$ if and only if $\nu \in \overline{\text{cone}}(C)$, which means that $\overline{\text{cone}}(C)$ and $\Gamma^\vee$ have the same non-rational extreme rays. However, we will see that $\overline{\text{cone}}(C)$ is a rational cone if $h$ is rational, so the non-rational extreme rays are not particularly of interest.

For these reasons, we will only consider rational extreme rays $\nu\mathbb{R}_+$ from now on, and the direction vectors $\nu$ will be chosen primitive, that is with integral coefficients that are (setwise) coprime.

**3.1.2. Step 2: Rewriting in terms of $h$ and of its support.** — Let $\nu$ be a primitive integer vector such that $\nu\mathbb{R}_+$ is an extreme ray of $\overline{\text{cone}}(C)$. By Lemma 3.1, whether $\nu$ is in $\Gamma^\vee$ or not can be decided by considering the sets $C \cap \nu\mathbb{R}_+$ and $C \setminus \nu\mathbb{R}_+$ separately.

Now, we recall that $C$ is the exponential support of the power series $h$, defined as the support $\{\alpha \in \mathbb{N}^k | c(\alpha) \neq 0\}$ of the exponent function $c: \mathbb{N}^k \to \mathbb{Z}$ present in the representation of $h$ as a formal product

$$ h = \prod_{\alpha \in \mathbb{N}^k} (1 - \mathbf{X}^\alpha)^{c(\alpha)}. $$

The partition

$$ (14) \qquad C = (C \cap \nu\mathbb{R}_+) \cup (C \setminus \nu\mathbb{R}_+) $$

from Lemma 3.1 motivates us to write $h$ as a product

$$ (14') \qquad h = \left( \prod_{\alpha \in C \cap \nu\mathbb{R}_+} (1 - \mathbf{X}^\alpha)^{c(\alpha)} \right) \times \left( \prod_{\alpha \in C \setminus \nu\mathbb{R}_+} (1 - \mathbf{X}^\alpha)^{c(\alpha)} \right). $$

As we expect $h$ to be more easily represented as a formal power series, we want to rephrase this decomposition in terms of the coefficient function $b$ of $h$, defined by

$$ h = \sum_{\beta \in \mathbb{N}^k} b(\beta) \mathbf{X}^\beta. $$
Lemmata 2.1 and 2.2 provide formulas to derive the coefficient function b from the exponent function c and vice versa. However, instead of computing their values using those formulas, we are tackling the simpler problem of studying their supports, independently of their magnitude.

**Lemma 3.2.** — Let $h \in \mathbb{Z}[[\mathbf{X}]]$ with $h(0) = 1$, with support $B$ and with exponential support $C$. Then, the monoid generated by $B$ is equal to the monoid generated by $C$. *Proof.* — The fact that $B$ is contained in the monoid generated by $C$ is a simple consequence of the conditions given in the sum at formula (4). Conversely, $C$ is contained in the monoid generated by $B$ by formula (6). □

In particular, we have $\mathrm{cone}(C) = \mathrm{cone}(B)$, and $\nu\mathbb{R}_+$ is an extreme ray of $\mathrm{cone}(B)$. We now state a useful result about the behavior of coefficients along such an extreme ray in a product.

**Lemma 3.3.** — Let $K \subseteq (\mathbb{R}_+)^k$ be a convex cone, and $\nu$ be a primitive vector of $\mathbb{N}^k$ such that $\nu\mathbb{R}_+$ is an extreme ray of $K$. For any power series $h \in \mathbb{Z}[[\mathbf{X}]]$ with support in $K$, we define $h_\nu$ as

$$h_\nu = \sum_{n \ge 0} b(n\nu) \mathbf{X}^{n\nu} \in \mathbb{Z}[[\mathbf{X}]],$$

where b is the coefficient function of h. Then, for any two power series h and h' with support in K, we have $(hh')_\nu = h_\nu h'_\nu$.

*Proof.* — Every element in the support of $hh'$ has to be written as the sum of an element in $\mathrm{supp}(h) \subset K$ and of an element in $\mathrm{supp}(h') \subset K$. Since $K$ is closed under addition, $\mathrm{supp}(hh') \subset K$.

Let $n \in \mathbb{N}$. The coefficient of index $n\nu$ of $hh'$ is

$$\sum_{\alpha + \alpha' = n\nu} b(\alpha)b'(\alpha'),$$

where $b$ (resp. $b'$) is the coefficient function of $h$ (resp. $h'$). Since the supports of $h$ and $h'$ are included in $K$, we may restrict the sum to couples $(\alpha, \alpha')$ of elements of $K \cap \mathbb{N}^k$. If the sum of two elements of $K$ is in the extreme ray $\nu\mathbb{R}_+$, then each of these elements is in the extreme ray $\nu\mathbb{R}_+$. Thus, we may restrict the sum to $\alpha$ and $\alpha'$ of the form $m\nu$ and $m'\nu$ respectively, and we get the following expression for the coefficient of index $n\nu$ of $hh'$

$$\sum_{m+m'=n} b(m\nu)b(m'\nu),$$

which is the coefficient of index $n\nu$ of $h_\nu h'_\nu$. □

<figure><img src="image_6.png" /></figure>

We are now in position to identify the left-hand side term of the product (14').

**Lemma 3.4.** — Let $K \subseteq (\mathbb{R}_+)^k$ be a convex cone, and $\nu$ be a primitive vector of $\mathbb{N}^k$ such that $\nu\mathbb{R}_+$ is an extreme ray of $K$. For any power series $h \in \mathbb{Z}[[\mathbf{X}]]$ with $h(0) = 1$ and with support in $K$, we have

$$\prod_{n \ge 1} (1 - \mathbf{X}^{n\nu})^{c(n\nu)} = \sum_{n \ge 0} b(n\nu) \mathbf{X}^{n\nu} \in \mathbb{Z}[[\mathbf{X}]],$$

where $b$ (respectively $c$) is the coefficient (respectively exponent) function of $h$.
*Proof.* — Notice that the right-hand side of the formula is precisely the power series $h_ν$ of Lemma 3.3.

By Lemma 2.2, $h$ can be represented as an infinite product, and we denote by $c$ the associated exponent function. By Lemma 3.2, the exponential support $C$ of $h$ is inside the cone $K$. Consider the decomposition (14') of $h$ as a product of two terms. By Lemma 2.2, both terms can be expressed as power series, and by Lemma 3.2, their support is contained in $ν\mathbb{R}_+$ for the left-hand side, and in $K \setminus ν\mathbb{R}_+$ for the left-hand side. In particular, both supports are inside $K$, so we may apply Lemma 3.3 to this product to get

$$h_{\nu} = \left( \prod_{\alpha \in C \cap \nu \mathbb{R}_{+}} (1 - \mathbf{X}^{\alpha})^{c(\alpha)} \right)_{\nu} \times \left( \prod_{\alpha \in C \setminus \nu \mathbb{R}_{+}} (1 - \mathbf{X}^{\alpha})^{c(\alpha)} \right)_{\nu}.$$

Consider first the right-hand side: since its support is inside $K \setminus ν\mathbb{R}_+$, the last product is trivially equal to 1. On the contrary, the support of the left-hand side is inside $ν\mathbb{R}_+$, thus

$$\left( \prod_{\alpha \in C \cap \nu \mathbb{R}_+} (1 - \mathbf{X}^{\alpha})^{c(\alpha)} \right)_{\nu} = \prod_{\alpha \in C \cap \nu \mathbb{R}_+} (1 - \mathbf{X}^{\alpha})^{c(\alpha)}.$$

We conclude that

$$\sum_{n \ge 0} b(n\nu) \mathbf{X}^{n\nu} = h_{\nu} = \prod_{\alpha \in C \cap \nu \mathbb{R}_+} (1 - \mathbf{X}^{\alpha})^{c(\alpha)}. \quad \square$$

In other words, the subseries $h_ν$ of $h$ is also its subproduct. To emphasize that $h_ν$ is actually a formal series in one variable $\mathbf{X}^ν$, we often write $h_ν$ as an element of $\mathbb{Z}[[T]]$ where $T = \mathbf{X}^ν$.

We set $h^° = h/h_ν$. This is the right-hand side of the decomposition (14') of $h$. It is a subproduct of $h$, and its exponential support $C^°$ is exactly $C \setminus ν\mathbb{R}_+$. However, it is generally not a subseries of $h$. Its support $B^°$ still verifies $\text{cone}(B^°) = \text{cone}(C^°)$ by Lemma 3.2.

We are now able to rewrite Lemma 3.1 without any reference to the exponential support of the function $h$.

**Lemma 3.1'.** — Let $h \in \mathbb{Z}[[\mathbf{X}]]$ with $h(0) = 1$, and set $\Gamma$ as the domain described in Theorem 2.9. Let $ν \in \mathbb{N}^k$ be a primitive vector such that $ν\mathbb{R}_+$ is an extreme ray of $\text{cone}(B)$, where $B$ is the support of $h$. Set

$$h_{\nu} = \sum_{n \ge 0} b(n\nu) \mathbf{X}^{n\nu} \quad \text{and} \quad h^{\circ} = h/h_{\nu} \in \mathbb{Z}[[\mathbf{X}]].$$

We have $ν \in \Gamma^\vee$ if and only if $h_ν$ is not a quotient of products of cyclotomic polynomials, or $ν$ is an element of $\text{cone}(B^°)$, where $B^°$ is the support of $h^°$.

*Proof.* — In Lemma 3.1, $ν\mathbb{R}_+$ is chosen as an extreme ray of $\text{cone}(C)$, where $C$ is the exponential support $h$. By Lemma 3.2, this assumption is equivalent to being an extreme ray of $\text{cone}(B)$.

In Lemma 3.1, the necessary and sufficient condition for $ν \in \Gamma^\vee$ is that the set $C \cap ν\mathbb{R}_+$ is infinite or $ν \in \text{cone}(C \setminus ν\mathbb{R}_+)$.
Any finite product of the form $\prod_n (1 - T^n)^{a_n}$ may be written as a finite product of the form $\prod_n \Phi_n(T)^{a'_n}$, and conversely. By contraposition, the first part of the disjunction is equivalent to the fact that $h_\nu$ is not a quotient of products of cyclotomic polynomials.

For the latter part, we only observe that

$$ \overline{\text{cone}}(C \setminus \nu \mathbb{R}_+) = \overline{\text{cone}}(C^\circ) = \overline{\text{cone}}(B^\circ), $$

where $C^\circ$ is the exponential support of $h^\circ$ and $B^\circ$ is its support. $\square$

**3.1.3. Step 3: Finding alignments.** — Lemma 3.1' takes advantage of the decomposition of $h$ as $h_\nu h^\circ$, which corresponds to the partition (14) of the exponential support $C$ of $h$.

Exponential support behaves particularly well with respect to products and quotients: the product (respectively quotient) of two functions corresponds to the sum (respectively difference) of their exponent functions. Therefore, for two functions $h$ and $h'$ with exponential support $C$ and $C'$ respectively, we have

$$ C \Delta C' \subseteq \text{xsupp}(hh') \subseteq C \cup C', $$

(resp. $C \Delta C' \subseteq \text{xsupp}(h/h') \subseteq C \cup C'$) where $\Delta$ denotes the symmetric difference. As an example of that, the exponential supports $C$ of $h$ and $C^\circ$ of $h^\circ$ are quite similar.

On the contrary, the support of a formal series does not behave as well as the exponential support with respect to products and quotients, with the exception of results as Lemma 3.3, which can be generalized for any face of the cone $K$. Actually, the supports $B$ of $h$ and $B^\circ$ of $h^\circ$ may differ considerably (see Figures 2 and 4).

In general, the previous relations obtained for the exponential support of a product have weaker counterparts in terms of supports: if $h$ and $h'$ are formal series of respective supports $B$ and $B'$, we have

$$ (15) \qquad B \dot{=} B' \subseteq \text{supp}(hh') \subseteq B + B', $$

where $A \dot{=} A'$ denotes the set of elements with an unique representation as a sum of an element $a \in A$ and an element $a' \in A'$.

This relation verified by $\text{supp}(hh')$ is generally not sufficient to determine its geometry, unless we are dealing with finite supports.

A basic result in this setting is a theorem of Ostrowski regarding Newton polytopes. For a polynomial $h$, we define its *Newton polytope* $\Delta(h)$ as the convex hull of its support. Ostrowski's Theorem (cf. [Ost75]) states that for two polynomials $h$ and $h'$, we have

$$ (16) \qquad \Delta(hh') = \Delta(h) + \Delta(h'). $$

Unfortunately, for the product $h = h_\nu h^\circ$, this result is helpful only in trivial cases.

However, we state a corollary of Ostrowski's Theorem which will be extremely useful.

**Corollary 3.6.** — Let $K \subseteq \mathbb{R}^k$ a convex cone, $q$ and $q'$ two polynomials of $\mathbb{Z}[\mathbf{X}]$. If $\text{supp}(qq') \subset K$ and $q'(\mathbf{0}) \neq 0$, then $\text{supp}(q) \subset K$.
*Proof.* — We have that $0 \in \Delta(q')$. Using Ostrowski's Theorem (16), we get

$$ \mathrm{supp}(q) \subseteq \Delta(q) \subseteq \Delta(q) + \{\mathbf{0}\} \subseteq \Delta(q) + \Delta(q') = \Delta(qq') \subset K. \quad \square $$

When $h$ is polynomial. — Before handling the general case with $h$ rational, we deal with two easier cases, which are particularly revealing: when $h$ is a polynomial, and when $h$ is a rational function in two variables.

**Lemma 3.7.** — Let $h$ be a polynomial in $\mathbb{Z}[\mathbf{X}]$, $\nu \in \mathbb{N}^k$ a nonzero vector, and $q$ a polynomial in the single variable $\mathbf{X}^\nu$ and with constant term 1. The rational function $h' = \frac{h}{q}$ has a representation as a formal series, and we denote $B'$ its support.

Then, we have the following exclusive disjunction:

- either $q$ divides $h$ in $\mathbb{Z}[\mathbf{X}]$,

- or there exists an integer vector $\mu$ such that $B' \cap (\mu + \nu\mathbb{N})$ is infinite.

*Proof.* — First notice that if $q$ divides $h$, then $h'$ is a polynomial and its support $B'$ is finite. We now show the reciprocal.

Since $q$ has constant term 1, $1/q$ can be represented as a power series in $\mathbf{X}^\nu$, and so can the rational function $h/q$.

Let $\mu \in B'$. Assume that $B' \cap (\mu + \nu\mathbb{N})$ is finite. In particular there is a maximal integer $n$ such that $\mu + n\nu \in B'$: we denote this maximum by $n^+$ and we set $\mu^+ = \mu + n^+\nu$ as the corresponding element in $B'$.

Let $d$ be the degree of the polynomial $q$, as a polynomial in $\mathbf{X}^\nu$. Any element of the support of $q$ is written as $(d-m)\nu$ for some integer $m \in [0, d]$. The vector $\alpha = \mu^+ + d\nu$ belongs to $B' + \mathrm{supp}(h)$ and any representation of $\alpha$ as the sum of an element of $B'$ and an element of $\mathrm{supp}(h)$ has the form $(\mu^+ + m\nu) + ((d-m)\nu)$ for some $m \in [0, d]$. By maximality of $n^+$, only the choice $m=0$ provides such a representation. By (15), this unicity implies that $\alpha$ belongs to the support of $h'q = h$, that is $\alpha \in B$.

We deduce that there exists an element $\alpha \in B$ such that $\mu \in \mathbb{N}^k \cap (\alpha - \nu\mathbb{N})$ since $\mu = \alpha - (d+n^+)\nu$. Since $\nu$ is a nonzero vector with nonnegative coefficients, the intersection $\mathbb{N}^k \cap (\alpha - \nu\mathbb{N})$ is a finite set.

We have shown that any $\mu \in B'$ such that the set $B' \cap (\mu + \nu\mathbb{N})$ is finite belongs the set

$$ \bigcup_{\alpha \in B} \mathbb{N}^k \cap (\alpha - \nu \mathbb{N}), $$

which is a finite union of finite sets. Therefore, unless $B'$ contains infinitely many elements on a line $\mu + \nu\mathbb{N}$, $B'$ is a subset of the above set, which is finite, thus $h'$ is a polynomial, which verifies $h = h'q$, and $q$ is a factor of $h$. $\square$

Lemma 3.7 is sufficient to retrieve the description of the domain $\Gamma$ on which the Euler product $f$ has a meromorphic extension, when $h$ is a non-trivial polynomial without cyclotomic factor.

**Proposition 3.8.** — Let $h \in \mathbb{Z}[\mathbf{X}]$ with $h(0) = 1$ without cyclotomic factor and with support $B$. Then, $\mathrm{cone}(B) = \Gamma^\vee$.

*Proof.* — Let $\nu \in \mathbb{N}^k$ be the primitive vector directing an extreme ray of $\mathrm{cone}(B) = \mathrm{cone}(B)$, and consider $h^\circ$ and $h_\nu$ defined as above; the latter can be considered as a
non-constant polynomial in the variable $X^\nu$. If $h_\nu$ divides $h$, it cannot be a product of cyclotomic polynomials, and we conclude that $\nu \in \Gamma^\vee$, by the first part of the disjunction in Lemma 3.1'.

If $h_\nu$ does not divide $h$, then, by Lemma 3.7, there is a ray $\mu + \nu \mathbb{R}_+$ containing infinitely many elements of the support $B^\circ$ of $h^\circ$, therefore $\nu \in \text{cone}(B^\circ)$, and we also conclude that $\nu \in \Gamma^\vee$, by the second part of the disjunction in Lemma 3.1'.

Since this is true for any extreme ray of cone($B$), we deduce that $\text{cone}(B) = \Gamma^\vee$. $\square$

As obtained by Delabarre [Del13], when $h$ is a polynomial without cyclotomic factor, the associated Euler product $f$ has a meromorphic extension on the dual cone of the support of $h$.

*When $h$ is rational in two variables.* — We will now turn our attention to the case of a rational function $h \in \mathbb{Q}(X_1, X_2)$ in two variables, which can be represented as a power series with integer coefficients, and such that $h(0, 0) = 1$.

Moreover we assume that $h$ can be written as the quotient of two coprime polynomials $q^+$ and $q^-$ in $\mathbb{Z}[X_1, X_2]$ such that $q^+(0, 0) = q^-(0, 0) = 1$. The existence of such a representation will be obtained later in a broader setting by generalizing the property of Fatou in Proposition 3.12.

Each polynomial $q^+$ and $q^-$ has a support denoted $B^+$ and $B^-$ respectively, each one generating a cone in $\mathbb{R}^2$. Convex cones in $\mathbb{R}^2$ are particularly easy to describe: they have at most two extreme rays. Since $q^-(0, 0) = 1$, the inverse $1/q^-$ is represented as a power series in $\mathbb{Z}[[X_1, X_2]]$, and the convex hull of the support of $1/q^-$ is exactly the cone generated by $B^-$. Therefore, the support $B$ of $h$ has to be inside $\text{conv}(B^+) + \text{cone}(B^-)$, thus

$$ (17) \qquad \text{cone}(B) \subseteq \text{cone}(B^+) + \text{cone}(B^-) = \text{cone}(B^+ \cup B^-). $$

However, as shown by the example $\frac{1+X_1+X_2-X_1X_2}{1+X_1+X_2}$ developed in the introduction, the previous inclusion is not always an equality, and the extreme rays of $\text{cone}(B)$ are not at our disposal a priori. We will use the extreme rays of $\text{cone}(B^+ \cup B^-)$ instead.

In this case, we are able to prove a result similar to Lemma 3.7.

**Lemma 3.9.** — Let $q^+$ and $q^-$ be two coprime polynomials in $\mathbb{Z}[X_1, X_2]$ with constant terms equal to 1, and $B^+$ and $B^-$ their respective supports. Their quotient $h = \frac{q^+}{q^-}$ has a representation as a power series in $\mathbb{Z}[[X_1, X_2]]$.

Let $\nu \in \mathbb{N}^2$ be a primitive vector such that $\nu \mathbb{R}_+$ is an extreme ray of $\text{cone}(B^+ \cup B^-)$. Define $h_\nu$, $q_\nu^+$ and $q_\nu^-$ as in Lemma 3.4. The rational function $h^\circ = \frac{h}{h_\nu}$ has a representation as a formal series, with $B^\circ$ as its support.

We have the following exclusive disjunction:

- either $q_\nu^+$ divides $q^+$ and $q_\nu^-$ divides $q^-$,

- or there exists an integer vector $\mu \in \mathbb{N}^2$ such that $B^\circ \cap (\mu + \nu \mathbb{N})$ is infinite.

The outline of the proof contains two main steps: apply a suitable change of indeterminates and use Fatou's lemma. This will serve as template for the general case.

Fatou proved (cf. [Fat04, pp. 368–371]) that if $P$ and $Q$ are two coprime polynomials in $\mathbb{Q}[X]$ with $Q(0) = 1$, such that the power series expansion of $P/Q$ has
coefficients in $\mathbb{Z}$, then the polynomials $P$ and $Q$ are in $\mathbb{Z}[X]$. This property has been generalized to other integral domains than $\mathbb{Z}$, in particular to any unique factorization domain such as $\mathbb{Z}[Y]$: if we set $R$ as the integral domain $\mathbb{Z}[Y]$, and $F$ as its fraction field $\mathbb{Q}(Y)$, any pair of coprime polynomials $P$ and $Q$ in $F[X]$ with $Q(0) = 1$ and such that the power series expansion of $P/Q$ has coefficients in $R$, are indeed polynomials in $R[X]$.

*Proof.* — If $q_\nu^+$ divides $q^+$ and $q_\nu^-$ divides $q^-$, we set $q^{\oplus} = \frac{q_\nu^+}{q_\nu^-}$ and $q^{\ominus} = \frac{q_\nu^-}{q_\nu^+}$, both polynomials in $\mathbb{Z}[X_1, X_2]$; we denote their supports by $B^{\oplus}$ and $B^{\ominus}$ respectively. The supports of $q^+$, $q^-$, $h$, $q^{\oplus}$ and $q^{\ominus}$ are all inside the cone $K = \text{cone}(B^+ \cup B^-)$ of extreme ray $\nu\mathbb{R}_+$. By Lemma 3.3, we have

$$h_\nu = \frac{q_\nu^+}{q_\nu^-}, \quad \text{thus} \quad h^\circ = \frac{q^{\oplus}}{q^{\ominus}}.$$

Also, $B^{\oplus}$ and $B^{\ominus}$ do not contain any non-trivial multiple of $\nu$. Thus, they are inside $K \setminus \nu\mathbb{R}_+^*$, which is a convex cone by the extremality of the ray $\nu\mathbb{R}_+$ in $K$. Consequently, $\text{cone}(B^{\oplus} \cup B^{\ominus}) \subseteq K \setminus \nu\mathbb{R}_+^*$.

Applying (17) to $h^\circ$, we get $\text{cone}(B^\circ) \subseteq \text{cone}(B^{\oplus}) + \text{cone}(B^{\ominus})$ (thus $B^\circ \subseteq K$). Since both cones in the right-hand term are finitely generated and therefore closed, we also have $\overline{\text{cone}}(B^\circ) \subseteq \text{cone}(B^{\oplus}) + \text{cone}(B^{\ominus})$. Therefore $\overline{\text{cone}}(B^\circ) \subseteq K \setminus \nu\mathbb{R}_+^*$, which means that $\nu$ does not belong to $\overline{\text{cone}}(B^\circ)$. In consequence, the set $B^\circ \cap (\mu + \nu\mathbb{N})$ can not be infinite for any $\mu \in \mathbb{N}^2$.

Now, let us assume that for any $\mu \in \mathbb{N}^2$, the set $B^\circ \cap (\mu + \nu\mathbb{N})$ is finite. We first change our indeterminates.

Since $\nu\mathbb{R}_+$ is an extreme ray of the cone $K = \text{cone}(\text{supp } q^+ \cup \text{supp } q^-)$, which includes $B^\circ$, we have that for any $\beta = (\beta_1, \beta_2) \in B^\circ$ the sign of $\nu_1\beta_2 - \nu_2\beta_1$ does not change (it may be zero though): we set $\epsilon = 1$ or $\epsilon = -1$ according to this sign.

Since the coordinates of $\nu$ are coprime, there exists a vector $\mu \in \mathbb{Z}^2$ such that $\det(\nu, \mu) = \epsilon$. Consequently, the family $\{\nu, \mu\}$ forms a basis of the lattice $\mathbb{Z}^2$, in which every element of $K \cap \mathbb{Z}^2$ has a nonnegative coordinate with respect to the basis vector $\mu$. Since $K \subseteq (\mathbb{R}_+)^2$, there exists a negative integer $m \in \mathbb{Z}$ such that $\mu + m\nu$ is not in $K$: set $\nu' = \mu + m\nu$, so that $\{\nu, \nu'\}$ forms a basis of the lattice $\mathbb{Z}^2$, and every element of $K \cap \mathbb{Z}^2$ has nonnegative integer coordinates in this basis. We set $Y_1 = X^\nu$ and $Y_2 = X^{\nu'}$, so that any power series with support inside $K$ may be regarded as a power series in the indeterminates $Y_1$ and $Y_2$: this change of variables is indeed a $\mathbb{Z}$-algebra homomorphism.

In this setting, we have $h_\nu(Y_1, Y_2) = h(Y_1, 0)$, therefore $h^\circ(Y_1, 0) = 1$, and by the fact that $B^\circ \cap (m\nu' + \nu\mathbb{N})$ is finite for any $m \in \mathbb{N}$, the coefficient in $Y_2^m$ of $h^\circ(Y_1, Y_2)$ when regarded as a power series in $(\mathbb{Z}[[Y_1]])[[Y_2]]$ is a polynomial in $Y_1$. As a result, we can consider $h^\circ(Y_1, Y_2)$ as a power series in $Y_2$ with coefficients in $\mathbb{Z}[Y_1]$ and with constant term equal to 1.

We set $R = \mathbb{Z}[Y_1]$ and $F = \mathbb{Q}(Y_1)$ its quotient field. We now consider every power series of $\mathbb{Z}[[X_1, X_2]]$ with support inside $K$ as a power series in $R[[Y_2]]$, and make explicit the variable. For example, the polynomials $q^+$ and $q_\nu^+ \in \mathbb{Z}[X_1, X_2]$ are now considered as $q^+(Y_2) \in R[Y_2]$ and as its constant term $q^+(0) \in R$ respectively. The
power series $h^\circ$ is considered as $h^\circ(Y_2)$ which can be seen as the element $\frac{h(Y_2)}{h(0)}$ of $F(Y_2)$ with constant term equal to 1, and as an element of $R[[Y_2]]$. Since the integral domain $R = \mathbb{Z}[Y_1]$ verifies Fatou's property, there are two polynomials $p^+(Y_2)$ and $p^-(Y_2)$ in $R[Y_2]$, coprime in $F[Y_2]$ with $p^+(0) = 1$ and $p^-(0) = 1$, such that $h^\circ(Y_2) = \frac{p^+(Y_2)}{p^-(Y_2)}$. On the other hand, we have

$$h^\circ(Y_2) = \frac{h(Y_2)}{h(0)} = \frac{q^+(Y_2)/q^+(0)}{q^-(Y_2)/q^-(0)}.$$

Since $F[Y_2]$ has unique factorization, there exists a polynomial $r(Y_2) \in F[Y_2]$ such that

$$q^+(Y_2) = q^+(0)r(Y_2)p^+(Y_2) \quad \text{and} \quad q^-(Y_2) = q^-(0)r(Y_2)p^-(Y_2).$$

The polynomial $r(Y_2)$ divides $q^+$ and $q^-$ in $F[Y_2]$. Note that $F$ is the fraction field of the UFD $\mathbb{Z}[Y_1]$. By Gauss's Lemma [DF04, §9.3], the primitive part $\tilde{r}(Y_1, Y_2)$ of $r(Y_2)$ divides $q^+(Y_1, Y_2)$ and $q^-(Y_1, Y_2)$ in $\mathbb{Z}[Y_1, Y_2]$. By Corollary 3.6, the support of $\tilde{r}(Y_1, Y_2)$ is in the cone generated by the supports of $q^+(Y_1, Y_2)$ and $q^-(Y_1, Y_2)$. In particular, every monomial of $\tilde{r}(X^\nu, X^{\nu'})$ is written as $X^\alpha$ with $\alpha \in K \cap \mathbb{Z}^2$. We conclude that $\tilde{r}(X^\nu, X^{\nu'})$ in a polynomial of $\mathbb{Z}[X_1, X_2]$ which divides $q^+$ and $q^-$, which are coprime in $\mathbb{Z}[Y_1, Y_2]$. We deduce that the primitive part of $r(Y_2)$ is equal to one, which means that $r(Y_2)$ is a polynomial of degree 0 as in $F[Y_2]$. Since we have $r(0) = 1$, we deduce that $r = 1$ and

$$q^+(Y_2) = q^+(0)p^+(Y_2) \quad \text{and} \quad q^-(Y_2) = q^-(0)p^-(Y_2).$$

The terms of these identities are in $\mathbb{Z}[Y_1, Y_2]$, which gives that $q_\nu^+$ divides $q^+$, and $q_\nu^-$ divides $q^-$, by substituting the original indeterminates back. $\square$

Using this lemma, along with Lemma 3.1', we may reason as for Proposition 3.8 and prove that the dual cone of the domain $\Gamma$ defined in Theorem 2.9 is the cone generated by the support of $q^+$ and the support of $q^-$, when $h$ is the quotient of two coprime polynomials $q^+$ and $q^-$ of $\mathbb{Z}[X_1, X_2]$ with constant terms equal to 1 and without cyclotomic factor.

**3.2. The general case.** — In this section, we generalize the two main ingredients in the proof of Lemma 3.9 to higher dimensions: the change of coordinates and the property of Fatou. We conclude by giving a proof of the generalization of Lemmata 3.7 and 3.9.

**3.2.1. Change of coordinates.** — To correctly convey this idea, we need to generalize the notions of polynomials and power series, mainly by generalizing the notion of monomials.

For $K$ a convex cone of $\mathbb{R}^k$, we define the algebra of generalized polynomials $\mathbb{Z}[X; K]$ in $k$ variables with support in $K$, namely those whose elements are finite sums

$$(18) \qquad \sum_{\alpha \in \mathbb{Z}^k \cap K} b_\alpha X^\alpha$$
with coefficients in $\mathbb{Z}$. This algebra is the standard **semigroup ring** associated to the monoid $\mathbb{Z}^k \cap K$ over the domain $\mathbb{Z}$, as defined in [Gil84, §7] for example. To avoid confusion between the different occurences of the ring of integers, we extend this definition to the algebra $R[X; K]$ of generalized polynomials with coefficients in an integral domain $R$.

The set of infinite sums of the form (18) can also be equipped of the natural algebra structure, provided that $K$ does not contain any affine line (see [AMK13] for an expository account): we name it the algebra of **generalized power series in $k$ variables**, and denote it $R[[X; K]]$.

If $K = (\mathbb{R}_+)^k$, we retrieve the usual ring of power series in $k$ variables $R[[X; K]] = R[[X]]$.

More generally, when the cone $K$ is generated by a basis $v_1, \dots, v_k$ of $\mathbb{Z}^k$, the associated power series ring is isomorphic to the usual ring of power series in $k$ variables. Let us consider the change-of-basis linear map associated to this basis

$$ \phi(x_1, \dots, x_k) = \sum_{i=1}^{k} x_i v_i, $$

where $\phi((\mathbb{R}_+)^k) = K$, and let us set $Y_i = X^{v_i}$ as the new indeterminates for our polynomial/power series ring. For any $\beta \in \mathbb{Z}^k$, we have $X^{\phi(\beta)} = Y^\beta$. Since $\phi$ is an isomorphism from $\mathbb{Z}^k$ to itself, every monomial may be rewritten as $X^\alpha = Y^{\phi^{-1}(\alpha)}$. This defines the isomorphism

$$ (19) \qquad \phi_*: \begin{array}{ccc} R[[X; K]] & \to & R[[Y; \phi^{-1}(K)]] \\ \sum_{\alpha \in K} b_\alpha X^\alpha & \mapsto & \sum_{\alpha \in K} b_\alpha Y^{\phi^{-1}(\alpha)} \end{array} $$

where $\phi^{-1}(K)$ is actually $(\mathbb{R}_+)^k$. Hence, we have an isomorphism between $R[[X; K]]$ and $R[[Y]]$.

Changing the name of the indeterminates from $X$ to $Y$ is purely cosmetic here, but it is useful for keeping track of which algebra we are operating in.

If a cone $K$ lies inside another cone $K'$, we have a trivial injective algebra morphism

$$ R[[X; K]] \to R[[X; K]]. $$

It is well-known(*) that a cone is line-free if and only if it is contained in a simplicial cone $K'$. By tweaking slightly this statement, we can even ask for this simplicial cone to be generated by a basis of $\mathbb{Z}^k$. Using the corresponding isomorphism $\phi_*$, the algebra $R[[X; K']|$ is isomorphic to $R[[Y]]$. This means that for any line-free cone $K$, the algebra $R[[X; K]]$ can be interpreted as a sub-algebra of the usual power series algebra $R[[Y]]$.

In our setting, the cone $K$ is generated by the support of two polynomials in $R[X]$, it is therefore rational and line-free. When a cone $K$ is rational (or, more generally, when it is closed), $K$ is line-free if and only if it does not contain any nontrivial vector subspace, which is equivalent to $K \cap (-K) = \{0\}$, or to the fact that $\{0\}$ is a face of

<footnote>(*)see [Grü03, §2.5, exercise 9]. Basically, a cone $K$ is line-free if and only if its closure $\bar{K}$ does not contain any non trivial vector subspace, which is equivalent to the fact that its dual cone $\bar{K}^>$ contains a basis of the ambient vector space.</footnote>
K.

We also want the chosen extreme ray $\nu\mathbb{R}_+$ of the cone $K$ to be an extreme ray of the containing cone $K'$. The point of this assumption is the following: given a power series $g \in \mathbb{Z}[[\mathbf{X}; K]]$, and assuming that $K'$ is generated by the basis $v_1, \dots, v_k$ of $\mathbb{Z}^k$ with $v_1 = \nu$, the image of $g$ and $g_\nu$ by $\phi_*$ verifies $\phi_*g_\nu(\mathbf{Y}) = \phi_*g(Y_1, 0, \dots, 0)$.

We will actually prove a slightly more precise result.

**Lemma 3.10.** — Let $K$ be a full-dimensional line-free rational convex cone in $\mathbb{R}^k$. Let

$$F_0 = \{0\} \subset F_1 \subset \dots \subset F_k = K$$

a maximal chain of faces of $K$. Then, there exists a basis $v_1, \dots, v_k$ of $\mathbb{Z}^k$ such that $F_i \subseteq \text{cone}(v_1, \dots, v_i)$ for every $i$ between 0 and $k$.

*Proof.* — We will prove this result by induction on $k$. The statement is trivial for $k=1$. Assume that $k \ge 2$. We want to find a linear projection $\pi$ onto the hyperspace $H$ spanned by $F_{k-1}$ such that the projected cone $K' = \pi(K)$ verifies the assumptions of the Lemma. In particular, we want $F_i$ to be a face of $K'$ for any $i \le k-2$ in order to apply the induction hypothesis, and $\pi(\mathbb{Z}^k) = H \cap \mathbb{Z}^k$ to exploit its conclusion.

As a face of codimension 2, $F_{k-2}$ is the intersection of two facets of $K$ (see [Ful93, Section 1.2]), one of them being $F_{k-1}$. Let $H'$ be the hyperplane spanned by the other facet. Both $H$ and $H'$ are rational hyperplanes. In particular, $H \cap \mathbb{Z}^k$ is a free abelian group of rank $k-1$.

Let $\phi: \mathbb{Q}^k \to \mathbb{Q}$ (resp. $\phi'$) be a linear form of kernel $H$ (resp. $H'$) and which is nonnegative on $K$. The image of $\mathbb{Z}^k$ by $\phi$ is a subgroup of $\mathbb{Q}$ generated by at most $k$ rational numbers. In particular, $\phi(\mathbb{Z}^k)$ is a discrete subgroup of $\mathbb{Q}$, and it has a minimal positive element. Let $x \in \mathbb{Z}^k$ with $\phi(x) = \min \phi(\mathbb{Z}^k) \cap \mathbb{Q}^*_+$.

The free abelian group $H \cap \mathbb{Z}^k$ is of rank $k-1$, and is not contained in $H'$. Let $u \in H \cap \mathbb{Z}^k$ such that $\phi'(u) > 0$. There is $n \in \mathbb{N}$, such that $n\phi'(u) > \phi'(x)$. Set $v_k = x - nu$. We have $\phi(v_k) = \phi(x)$ so $\phi(\mathbb{Z}^k)$ is the subgroup of $\mathbb{Q}$ generated by $\phi(v_k)$, and $\phi'(v_k) = \phi'(x) - n\phi'(u) < 0$.

Let $H''$ be the hyperplane generated by $v_k$ and $H \cap H'$. For any $y \in H''$ there is a unique $t \in \mathbb{R}$ with $y - tv_k \in H \cap H'$. Assume also that $y \in K$. We deduce that $\phi(y) = t\phi(v_k) \ge 0$, so $t \ge 0$, and $\phi'(y) = t\phi'(v_k) \ge 0$, so $t \le 0$. This means that $K \cap H'' \subseteq K \cap H \cap H' = F_{k-2}$.

Let $K'$ be the image of $K$ by projection along $v_k \mathbb{R}$ onto $H$, $K'$ is also rational, since it is generated by the finite family of projected images of the generators of $K$. Since $v_k \in H''$, the projection of $H''$ is $H \cap H'$. In particular, $H \cap H'$ is a supporting hyperplane of $K'$ within $H$, and the intersection $H \cap H' \cap K'$ is the projection of $H'' \cap K$, which equals $F_{k-2}$. Finally, since $F_{k-1} = K \cap H$, we have $F_{k-1} \subseteq K'$. Therefore $K'$ is of dimension $k-1$ and $F_{k-2}$ is a facet of $K'$. Since every face of a face of a cone is a face of this cone, every $F_i$ with $i \le k-2$ is a face of $K'$. In particular, $F_0 = \{0\}$ is face of $K'$ which means that $K'$ is line-free.

Since $K'$ is a rational cone of dimension $k-1$ in $H$, containing no line, we can apply the lemma for the following chain of faces

$$F_0 = \{0\} \subset F_1 \subset \dots \subset F_{k-2} \subset F'_{k-1} = K'$$
<figure><img src="image_7.png" /><figcaption>FIGURE 3. Construction of the basis vector $v_k$ and of its corresponding projection. Everything has been projected along the affine hull of $F_{k-2}$ for readability.</figcaption></figure>

Therefore there exists a basis $v_1, \dots, v_{k-1}$ of $H \cap \mathbb{Z}^k$ such that $F_i \subseteq \text{cone}(v_1, \dots, v_i)$ for every $i$ between 0 and $k-2$, and $K' \subseteq \text{cone}(v_1, \dots, v_{k-1})$. Since $F_{k-1} \subseteq K'$, we also have $F_{k-1} \subseteq \text{cone}(v_1, \dots, v_{k-1})$

It remains to prove that the family $(v_1, \dots, v_k)$ is a basis of $\mathbb{Z}^k$, and that the cone $K$ is contained in $\text{cone}(v_1, \dots, v_k)$.

First, let $v \in \mathbb{Z}^k$. Recall that $\phi(\mathbb{Z}^k)$ is a discrete subgroup of $\mathbb{Q}$ generated by $\phi(v_k)$, thus $\phi(v)$ is a multiple of this generator, and there exists a unique $n_k \in \mathbb{Z}$ such that $\phi(v) = n_k \phi(v_k)$. We derive that $v - n_k v_k \in \ker \phi = H$. Since $v_1, \dots, v_{k-1}$ is a basis of $H \cap \mathbb{Z}^k$, there exist $k-1$ integers $n_1, \dots, n_{k-1}$ such that $v - n_k v_k = n_1 v_1 + \dots + n_{k-1} v_{k-1}$. We conclude that the family $(v_1, \dots, v_k)$ spans $\mathbb{Z}^k$, and since $\mathbb{Z}^k$ is of rank $k$, $(v_1, \dots, v_k)$ is a basis of $\mathbb{Z}^k$.

Now, let $x \in K$. Since $K'$ is the projected image of $K$ along $v_k$, $x$ can be represented as $x' + t_k v_k$ with $x' \in K'$ and $t_k \in \mathbb{R}$. Since $K' \subseteq \text{cone}(v_1, \dots, v_{k-1})$, there are $t_1, \dots, t_{k-1}$ nonnegative real numbers such that $x' = t_1 v_1 + \dots + t_{k-1} v_{k-1}$. The form $\phi$ has non negative value on $K$, positive at $v_k$ and zero on $K' \subseteq H$, thus $0 \le \phi(x) = t_k \phi(v_k)$ which implies that $t_k \ge 0$. Altogether, we have $x = t_1 v_1 + \dots + t_k v_k$ with nonnegative scalars $t_1, \dots, t_k$. This proves that $K$ is contained in $\text{cone}(v_1, \dots, v_k)$. $\square$

One may remark that the dual statement asserts that it also exists a basis $v'_1, \dots, v'_k$ of $\mathbb{Z}^k$ such that $v'_i \in F_i$ for any $i$ between 1 and $k$.

With the goal to apply this lemma to a change of indeterminates, we may rewrite the lemma 3.10 in a more suitable way.

**Corollary 3.11.** — Let $K$ be a full-dimensional line-less rational convex cone in $\mathbb{R}^k$. Let

$$F_0 = \{0\} \subset F_1 \subset \dots \subset F_k = K$$

a maximal chain of faces of $K$. There exists a change-of-basis isomorphism $\phi : \mathbb{Z}^k \to \mathbb{Z}^k$ such that $\phi_*$ injectively maps $\mathbb{Z}[[\mathbf{X}; K]]$ inside $\mathbb{Z}[[\mathbf{Y}]]$ and every sub-algebra $\mathbb{Z}[[\mathbf{X}; F_i]]$ inside $\mathbb{Z}[[Y_1, \dots, Y_i]]$.

**3.2.2.** *The property of Fatou.* — In this section we want to generalize the following characteristic property of Fatou rings to multivariable power series.

**Definition 3.1.** — A power series $g$ of $R[[X]]$ is said to be **rational** if there exists two polynomials $P, Q \in R[X]$ verifying $Qg = P$. The power series $g$ can be identified
with the quotient $\frac{P}{Q} \in F(X)$, where $F$ is the quotient field of $F$.

An integral domain $R$ (with quotient field $F$) is said to be a **Fatou ring** if any rational element of $R[[X]]$ can be identified as the quotient of two polynomials of $R[X]$ that are coprime in $F[X]$ and such that the constant term of the denominator is 1.

For more context on this notion, see [Eil74, Chapter XVI].

We are able to prove that, if $R$ is a Fatou ring, and $K$ a line-free cone of $\mathbb{R}^k$, then for any rational power series of $R[[X; K]]$ there exists two polynomials $P, Q \in R[X; K]$, which are coprime in $F[X; K']$ for any line-free convex cone $K'$ containing $K$, and with $Q(0) = 1$.

Even if this proof relies on some arguments already developed in this article, presenting it entirely would make this discourse even longer. We shall restrict ourselves to proving the case with $R$ being a UFD, and $K = K' = (\mathbb{R}_+)^k$:

**Proposition 3.12.** — Let $R$ be a UFD, and $F$ its fraction field. For any rational power series $g \in R[[X]]$, there is an unique pair of coprime polynomials $P, Q \in R[X]$ with $Q(0) = 1$, such that $Qg = P$.

Consider the special case of a field, that is with $R = F$. Proving that a field is a Fatou ring is straightforward, because with one indeterminate, a zero constant term implies a factorization by the indeterminate. However, the corresponding statement of Proposition 3.12 in this case is surprisingly difficult to derive. We refer to [KKT14] for the only proof we know of. It is equivalent to the following statement.

**Proposition 3.13 ([KKT14])**. — Let $R$ be an integral domain, and $F$ its fraction field. For any rational power series $g \in R[[X]]$, there is a pair of polynomials $P, Q \in R[X]$, coprime in $F[X]$, with $Q(0) \neq 0$, such that $Qg = P$.

Their proof relies on the fact that a unique factorization domain is a Fatou ring. A proof of this fact may also be found in [Dre68, Lemme 3].

*Proof of Proposition 3.12.* — Let $g$ a rational power series in $R[[X]]$. By Proposition 3.13, there are polynomials $P'', Q'' \in R[X]$, coprime in $F[X]$, with $Q''(0) \neq 0$, such that $Q''g = P''$.

Consider the change of indeterminates $Y_i = X_i X_k^{-1}$ for any $i \neq k$ and $Y_k = X_k$. It corresponds to the $R$-algebra isomorphism

$$ \psi_* : \begin{array}{ccc} R[[X]] & \to & R[[Y; K]] \\ \sum_{\alpha \in \mathbb{N}^k} b_\alpha X^\alpha & \mapsto & \sum_{\alpha \in \mathbb{N}^k} b_\alpha Y_1^{\alpha_1} \cdots Y_{k-1}^{\alpha_{k-1}} Y_k^{\alpha_k + \cdots + \alpha_k} \end{array} $$

where $K$ is the cone by the vectors $e_k$ and all the $e_i + e_k$ for $i$ between 1 and $k-1$, with $(e_i)_{1 \le i \le k}$ being the canonical basis of $\mathbb{R}^k$.

Each term of $\psi_*g$ can be written as $a_n(Y_1, \dots, Y_{k-1})(Y_k)^n$ where $a_n$ is a monomial in $R[Y_1, \dots, Y_{k-1}]$ of total degree less or equal to $n$. We may as well regard $\psi_*g$ as a power series of $R'[[Y_k]]$, with $R' = R[Y_1, \dots, Y_{k-1}]$.

On the other hand, since $\psi_*$ is an algebra homomorphism, we have

$$ \psi_* P'' = \psi_*(Q''g) = \psi_* Q'' \cdot \psi_* g, $$

where $\psi_* P''$ and $\psi_* Q''$ are elements in $R[[Y]]$ with finite supports, and can be seen as elements of $R'[Y_k]$. Therefore, $\psi_* g$ is a rational power series of $R'[[Y_k]]$.
Since $R'$ is a UFD, it is a Fatou ring, and there are two polynomials $P'$ and $Q'$ in $R'[Y_k]$, coprime in $F'[Y_k]$ where $F'$ is the quotient field of $R'$, with $Q'(0) = 1 \in R'$, such that $Q' \cdot \psi_*g = P'$.

We now make full use of the fact that $R'$ is a unique factorization domain. Since $Q' \in R'[Y_k]$ has constant term 1, its content is 1 and the contents of $P'$ and $Q'$ are coprime in $R'$. Since $P'$ and $Q'$ are coprime in $F'[Y_k]$, their primitive parts are coprime in $R'[Y_k]$. Finally, we deduce that $P'$ and $Q'$ are coprime in $R'[Y_k]$, that is in $R[Y]$. From now on, $P'$ and $Q'$ are considered as polynomials in $R[Y]$.

From the relations $\psi_*Q \cdot \psi_*g = \psi_*P$ and $Q' \cdot \psi_*g = P'$, we deduce that $P' \cdot \psi_*Q = Q' \cdot \psi_*P$. Since $R[Y]$ is a UFD, and since $P'$ and $Q'$ are coprime in this domain, there exists a polynomial $S \in R[Y]$ such that $\psi_*Q'' = SQ'$ and $\psi_*P'' = SP'$.

Since $Q'(0) = 1$, we have $S(0) = \psi_*Q''(0) = Q''(0) \neq 0$. By the Corollary 3.6 of Ostrowski's Theorem, we deduce that $\text{supp} Q' \subset K$ from the fact that $\text{supp}(\psi_*Q'') \subset K$. Moreover, we have

$$ \text{supp} P' = \text{supp}(Q'\psi_*g) \subseteq \text{supp}(Q') + \text{supp}(\psi_*g) \subseteq K + K = K. $$

We deduce that $P'$ and $Q'$ are elements of $R[Y; K]$, thus of $R[[Y; K]]$ also. Using the reciprocal $\psi_*^{-1}$ of the algebra isomorphism $\psi_*$, we may define $P = \psi_*^{-1}P'$ and $Q = \psi_*^{-1}Q'$ which are polynomials in $R[[X]]$, therefore in $R[X]$.

Finally, we show that $P$ and $Q$ are the polynomials we are looking for. Since $\psi_*^{-1}$ is an algebra homomorphism, we have

$$ P = \psi_*^{-1}P' = \psi_*^{-1}(Q' \cdot \psi_*g) = (\psi_*)^{-1}Q' \cdot g = Qg. $$

Every common divisor of $P$ and $Q$ is send by $\psi_*$ on a common divisor of $P'$ and $Q'$, which can only be a unit. Therefore $P$ and $Q$ are coprime in $R[X]$. The constant term of $Q$ is $Q'(0)$, namely 1.

Finally the uniqueness of these polynomials $P$ and $Q$ relies on the fact that $R$ is a UFD. Let $P_0$ and $Q_0$ two polynomials of $R[X]$, coprime in $F[X]$, with $Q_0(0) = 1$ and $Q_0g = P_0$. We have $QP_0 = QQ_0g = PQ_0$, and since $P$ and $Q$ are coprime in $R[X]$, there is a polynomial $S_0 \in R[X]$ such that $P_0 = S_0P$ and $Q_0 = S_0Q$. Since $P_0$ and $Q_0$ are coprime in $R[X]$, $S_0$ is a constant polynomial, and by specifying at 0 the relation $Q_0 = S_0Q$, we get that $S_0 = 1$. Therefore $P_0 = P$ and $Q_0 = Q$. $\square$

When $R = \mathbb{Z}$, this result resolves our problem of representing the rational power series $h \in \mathbb{Z}[[X]]$ verifying $h(0) = 1$ as a quotient of two coprime polynomials in $\mathbb{Z}[X]$ with constant term 1.

**3.2.3. The main result.** — Before stating our main result concerning the support $B^\circ$, we recall the different objects and properties we have considered so far.

Let $h \in \mathbb{Z}[[X]]$ a rational power series.

By Proposition 3.12, there exist two unique coprime polynomials $q^+$ and $q^- \in \mathbb{Z}[X]$ with $q^-(0) = 1$ and such that $q^-h = q^+$. From now on, we shall refer to these polynomials, respectively, as the numerator and the denominator of $h$.

We set $K = \overline{\text{cone}}(\text{supp} h)$, and let $\nu \in \mathbb{N}^k$ be the primitive vector directing an extreme ray of $K$. From Lemma 3.1, we are only interested in rational extreme rays
of this cone. But by Theorem 1.1 of [KKT14], we have that

$$ (20) \qquad \overline{\text{cone}}(\text{supp } h) = \text{cone}(\text{supp } q^+) + \text{cone}(\text{supp } q^-). $$

In particular, each extreme ray of K is rational; it is even generated by an element of the support of $q^+$ or of $q^-$.

Once $h$ and $\nu$ are fixed, we choose a convenient change of coordinates $\phi_*$.

Since $K$ is rational and line-less (it is contained in $(\mathbb{R}_+)^k$), and $\nu\mathbb{R}_+$ is an extreme ray of this cone (that is a face of dimension 1), we can apply Corollary 3.11 with a maximal chain of faces of $\overline{\text{cone}}(\text{supp } h)$ starting from $F_1 = \nu\mathbb{R}_+$. If $K$ is not full-dimensional, say of dimension $d < k$, we can build a larger cone $\tilde{K} \subset (\mathbb{R}_+)^k$ of dimension $k$ by appending $k-d$ vectors, such that $\tilde{K}$ admits $K$ as a face, and apply Corollary 3.11 to it. In any case, we have $h \in \mathbb{Z}[[\mathbf{X}; K]]$.

By Corollary 3.11, there exists an isomorphism $\phi: \mathbb{Z}^k \to \mathbb{Z}^k$, with $\phi(1, 0, \dots, 0) = \nu$, such that the application $\phi_*$ defined by (19) is an injective morphism of algebras from $\mathbb{Z}[[\mathbf{X}; K]]$ to $\mathbb{Z}[[\mathbf{Y}]]$, which maps $\mathbb{Z}[[\mathbf{X}; F_1]]$ to $\mathbb{Z}[[Y_1]]$.

Since the supports of $h$, $q^+$ and $q^-$ are inside $K$, their images $\phi_*h$, $\phi_*q^+$ and $\phi_*q^-$ are elements of $\mathbb{Z}[[\mathbf{Y}]]$. Since the supports of $q^+$ and $q^-$ are finite, so are the supports of $\phi_*q^+$ and $\phi_*q^-$, which means that $\phi_*q^+$ and $\phi_*q^-$ are two polynomials of $\mathbb{Z}[\mathbf{Y}]$. The relation $q^-h = q^+$ is mapped by the morphism of algebras to $\phi_*q^- \cdot \phi_*h = \phi_*q^+$. Therefore, $\phi_*h$ is a rational power series in $\mathbb{Z}[[\mathbf{Y}]]$.

Finally, we identify the $\mathbb{Z}$-algebra $\mathbb{Z}[[\mathbf{Y}]]$ to the isomorphic algebra $\mathbb{Z}[[Y_1]][[Y_2, \dots, Y_k]]$, and we define the coefficient function $\beta' \in \mathbb{N}^{k-1} \mapsto a_{\beta'} \in \mathbb{Z}[[Y_1]]$ of $\phi_*h$ so that

$$ \phi_*h = \sum_{(\beta_2, \dots, \beta_k) \in \mathbb{N}^{k-1}} a_{(\beta_2, \dots, \beta_k)} Y_2^{\beta_2} \cdots Y_k^{\beta_k}. $$

By Formula (19), we have that $\phi_*h = \sum_{\beta \in \mathbb{N}^k} b_{\phi(\beta)}\mathbf{Y}^{\beta}$, where the coefficient function $\alpha \mapsto b_{\alpha}$ of $h$ is extended on all of $\mathbb{Z}^k$ by taking the zero value. By equating the coefficients, we have that for any $(\beta_2, \dots, \beta_k) \in \mathbb{N}^{k-1}$

$$ a_{(\beta_2, \dots, \beta_k)} = \sum_{\beta_1 \in \mathbb{N}} b_{\phi(\beta_1, \beta_2, \dots, \beta_k)} Y_1^{\beta_1} \in \mathbb{Z}[[Y_1]]. $$

By writing $\phi(0, \beta_2, \dots, \beta_k)$ as $\mu$, and since $\phi(1, 0, \dots, 0) = \nu$, we can rewrite this power series as $\sum_{n \in \mathbb{N}} b_{\mu+n\nu} Y_1^n$. In particular, the support of $a_{(\beta_2, \dots, \beta_k)}$ as element of $\mathbb{Z}[[Y_1]]$ is the set $\{n \in \mathbb{N} \mid \mu + n\nu \in \text{supp}(h)\}$.

For $(\beta_2, \dots, \beta_k) = (0, \dots, 0)$, we get the constant term of $\phi_*h \in \mathbb{Z}[[Y_1]][[Y_2, \dots, Y_K]]$

$$ \phi_*h(Y_1, 0, \dots, 0) = a_{(0, \dots, 0)} = \sum_{n \in \mathbb{N}} b_{n\nu} Y_1^n. $$

This is also the image by $\phi_*$ of $h_\nu$, defined as $\sum_{n \in \mathbb{N}} b_{n\nu} \mathbf{X}^{n\nu}$.

**Proposition 3.14.** — Let $h = \sum_{\alpha \in \mathbb{N}^k} b_\alpha \mathbf{X}^\alpha \in \mathbb{Z}[[\mathbf{X}]]$ be a rational power series with denominator $q^-$, and let $\nu \in \mathbb{N}^k$ be the primitive vector directing an extreme ray of the $\overline{\text{cone}}(\text{supp } h)$. Then, there exists a vector $\mu \in \mathbb{Z}^k$ such that there are infinitely many points of $\text{supp}(h)$ on the half-line $\mu + \nu\mathbb{R}_+$ if and only if $q_\nu^- \neq 1$.
*Proof.* — Let $K^-$ denote cone(supp $q^-$). This convex cone is finitely generated and therefore closed. Recall that the cone $K = \text{cone}(\text{supp } h)$ contains $K^-$. Since $q^-(0) = 1$, $q^-$ is invertible in $\mathbb{Z}[[X]]$, and $\frac{1}{q^+} = 1 + \sum_{n \ge 1} (1 - q^-)^n$. Since the convex cone $K^-$ contains the support of $1 - q^-$, it also contains the support of its powers, therefore it contains the support of $\frac{1}{q^+}$ too. Consequently

$$ \text{supp}(h) \subseteq \text{supp}(q^+) + \text{supp}(1/q^-) \subseteq \text{supp}(q^+) + K^- $$

Let us first assume that there exists a vector $\mu \in \mathbb{Z}^k$ such that the set $\{n \in \mathbb{N} \mid \mu + n\nu \in \text{supp}(h)\}$ is infinite. Since $\text{supp}(h) \subseteq \text{supp}(q^+) + K^-$, for each of these $n$, there exists an element $\kappa_n \in \text{supp}(q^+)$, such that $\mu + n\nu - \kappa_n \in K^-$. Therefore, we have $\nu + \frac{1}{n}(\mu - \kappa_n) \in K^-$ for infinitely many $n$. Since the sequence $\mu - \kappa_n$ is bounded, the sequence $\nu + \frac{1}{n}(\mu - \kappa_n)$ converges to $\nu$, and since $K^-$ is closed, we conclude that $\nu \in K^-$. Thus, we have the following inclusions of rational cones

$$ \nu \mathbb{R}_+ \subseteq K^- \subseteq K. $$

Since $\nu \mathbb{R}_+$ is a face of $K$, there is a hyperplane $H$ such that $\nu \mathbb{R}_+ = H \cap K$. Thus, we have that $\nu \mathbb{R}_+ = H \cap K^-$, and we deduce that $\nu \mathbb{R}_+$ is an extreme ray of $K^-$. In particular, $\text{supp}(q^-)$, like any other set generating the convex cone $K^-$, contains a non zero multiple of $\nu$, which means that $q^- \ne 1$.

Now we assume that for any $\mu \in \mathbb{Z}^k$, the set $\{n \in \mathbb{N} \mid \mu + n\nu \in \text{supp}(h)\}$ is finite. Using the change of indeterminates described above, this means that for any $(\beta_2, \dots, \beta_k) \in \mathbb{N}^{k-1}$, the power series $a_{(\beta_2, \dots, \beta_k)} = \sum_{n \in \mathbb{N}} b_{\mu+n\nu} Y_1^n$ has a finite support, thus it is a polynomial in $\mathbb{Z}[Y_1]$. Consequently, the power series

$$ \phi_* h = \sum_{(\beta_2, \dots, \beta_k) \in \mathbb{N}^{k-1}} a_{(\beta_2, \dots, \beta_k)} Y_2^{\beta_2} \cdots Y_k^{\beta_k} $$

is an element of $\mathbb{Z}[Y_1][[Y_2, \dots, Y_k]]$. It also verifies $\phi_* q^- \cdot \phi_* h = \phi_* q^+$, where $\phi_* q^+$ and $\phi_* q^-$ are elements of $\mathbb{Z}[Y_1][Y_2, \dots, Y_k]$, that is polynomials over the ring of coefficients $\mathbb{Z}[Y_1]$. Since $\mathbb{Z}[Y_1]$ is a UFD, we apply Proposition 3.12 and get an unique pair of coprime polynomials $(P, Q)$ of $\mathbb{Z}[Y_1][Y_2, \dots, Y_k]$, with $Q$ of constant term equal to 1 and such that $Q \cdot \phi_* h = P$. We deduce that $Q \cdot \phi_* q^+ = P \cdot \phi_* q^-$. By unique factorization and coprimality of $P$ and $Q$, there exists a polynomial $R \in \mathbb{Z}[Y]$ such that $PR = \phi_* q^+$ and $QR = \phi_* q^-$. Since $Q$ and $\phi_* q^-$ have their constant terms equal to 1, so has $R$.

Notice that $\text{supp}(\phi_* q^+) = \phi^{-1}(\text{supp} q^+) \subseteq \phi^{-1}(K)$. Using the Corollary 3.6 of Ostrowski's Theorem, we have that $\text{supp}(P) \subseteq \phi^{-1}(K)$, and similarly, the supports of $Q$ and $R$ are included in $\phi^{-1}(K)$. Therefore, the image of $P$, $Q$ and $R$ by the reciprocal isomorphism $\phi_*^{-1}$ (which stands for $(\phi_*)^{-1}$ as for $(\phi_*^{-1})_*$) are elements of $\mathbb{Z}[X; K] \subseteq \mathbb{Z}[X]$. By morphism of algebras, the relations $PR = \phi_* q^+$ and $QR = \phi_* q^-$ can be written as relations $\phi_*^{-1} P \cdot \phi_*^{-1} R = q^+$ and $\phi_*^{-1} Q \cdot \phi_*^{-1} R = q^-$ in $\mathbb{Z}[X]$. Since $q^+$ and $q^-$ are coprime, we have that $\phi_*^{-1} R = 1$, thus $R = 1$, hence $P = \phi_* q^+$ and $Q = \phi_* q^-$. But $Q$ has constant term 1 as element of $\mathbb{Z}[Y_1][Y_2, \dots, Y_k]$, that is
$Q(Y_1, 0, \dots, 0) = 1$. We conclude that

$$ \phi_* q_{\nu}^{-} = \phi_* q^{-}(Y_1, 0, \dots, 0) = Q(Y_1, 0, \dots, 0) = 1, $$

and therefore, $q_{\nu}^{-} = 1$.

<figure><img src="image_8.png" /></figure>

We deduce directly a substantial improvement of the relation (20), which extends the Ostrowski's Theorem (16) in some way.

**Corollary 3.15.** — Let $h \in \mathbb{Z}[\mathbf{X}]$ a rational power series with denominator $q^{-}$ and numerator $q^{+}$. Then, we have

$$ \overline{\text{conv}}(\text{supp } h) = \text{conv}(\text{supp } q^{+}) + \text{cone}(\text{supp } q^{-}). $$

*Proof.* — We first notice that $\text{supp}(h) \subseteq \text{supp}(q^{+}) + \text{supp}(1/q^{-})$ by application of (15), and that $\text{supp}(1/q^{-}) \subseteq \text{cone}(\text{supp } q^{-})$. We deduce that

$$ \overline{\text{conv}}(\text{supp } h) \subseteq \text{conv}(\text{supp } q^{+}) + \text{cone}(\text{supp } q^{-}) $$

where the right hand term is closed because the supports of $q^{+}$ and $q^{-}$ are finite. We denote $\overline{\text{conv}}(\text{supp } h)$ by $\Delta$, and $\text{cone}(\text{supp } q^{-})$ by $K^{-}$. It suffices to show that $\text{supp}(q^{+}) + K^{-} \subseteq \Delta$.

We show that $x + K^{-} \subset \Delta$ for any $x \in \Delta$, with extra assumptions at first. We give a proof assuming that $\text{supp}(q^{+}) \subseteq K^{-}$. In this case, the formula (20) yields

$$ K = \overline{\text{cone}}(\text{supp } h) = \text{cone}(\text{supp } q^{+}) + \text{cone}(\text{supp } q^{-}) = K^{-}. $$

Let $\nu$ be a primitive integer vector such that the ray $\nu\mathbb{R}_{+}$ is extreme in $K^{-}$. In particular, we have that $q_{\nu}^{-} \neq 1$ and that $\nu\mathbb{R}_{+}$ is also an extreme ray in $K$. By Proposition 3.14, there exist $\mu \in \mathbb{Z}^{k}$ and an infinite subset $S \subseteq \mathbb{N}$ such that $\mu + n\nu \in \text{supp}(h) \subseteq \Delta$ for any $n \in S$. Thus, for any $x \in \Delta$, any $\lambda \in \mathbb{R}_{+}$ and any $n \in S$ such that $n > \lambda$, we have that

$$ x + \lambda\nu + \frac{\lambda}{n}(\mu - x) = (1 - \frac{\lambda}{n})x + \frac{\lambda}{n}(\mu + n\nu) \in \Delta. $$

Since $\Delta$ is closed, the limit $x + \lambda\nu$ is also in $\Delta$. By convexity of $K^{-}$, we deduce that $x + K^{-} \subseteq \Delta$, for any $x \in \Delta$.

Now, we only assume that there exists a $\mu \in \mathbb{Z}^{k}$ such that $\text{supp}(q^{+}) \subseteq \mu + \text{span}(K^{-})$. Therefore, any $y \in \text{supp } q^{+}$ can be written as a linear combination $\mu + \sum_{x} \lambda_{yx}x$ with $\lambda_{yx} \in \mathbb{R}$ and where $x$ runs through $\text{supp}(q^{-})$. Choosing $n_{x} \in \mathbb{N}$ such that $n_{x} \ge |\lambda_{yx}|$ for all $y \in \text{supp } q^{+}$, the vector $\mu' = -\mu + \sum_{x} n_{x}x \in \mathbb{Z}^{k}$ verifies that $\mu' + \text{supp}(q^{+}) \subseteq K^{-}$. We deduce that $X^{\mu'}q^{+}$ has its support in $K^{-} \cap \mathbb{Z}^{k} \subseteq \mathbb{N}^{k}$ and therefore is a polynomial. It is the numerator of the rational power series $h' = X^{\mu'}h$. By the previous argument, for any $x' \in \overline{\text{conv}}(\text{supp } h')$, we have that $x' + K^{-} \subseteq \overline{\text{conv}}(\text{supp } h')$. Shifting back by $\mu'$, we obtain the desired conclusion.

Finally, we treat the general case. We decompose $q^{+}$ as $q_{1}^{+} + \dots + q_{r}^{+}$ such that each subpolynomial $q_{i}^{+}$ is made of the terms of $q^{+}$ whose exponents lie in an affine subspace of direction $\text{span}(K^{-})$. In particular, for each $i$, there exists a $\mu_{i} \in \mathbb{Z}^{k}$ such that $\text{supp}(q_{i}^{+}) = \text{supp}(q^{+}) \cap (\mu_{i} + \text{span } K^{-})$. For each $i$ we define $h_{i}$ as the power series expansion of $\frac{q_{i}^{+}}{q^{-}}$ such that $h = h_{1} + \dots + h_{r}$ and $\text{supp}(h_{i}) = \text{supp}(h) \cap (\mu_{i} + \text{span } K^{-})$ for all $i$. Notice that $\text{supp}(h)$ is the union of every $\text{supp}(h_{i})$, and that $\Delta = \text{conv}(\bigcup_{i} \Delta_{i})$ where $\Delta_{i} = \overline{\text{conv}}(\text{supp } h_{i})$. Let $y \in K^{-}$ and $x \in \Delta$. The element $x$ can be written as
$x = \lambda_1 x_1 + \dots + \lambda_r x_r$ with $x_i \in \Delta_i$ and $\lambda_i \in [0, 1]$ such that $\lambda_1 + \dots + \lambda_r = 1$. Since $\text{supp } q_i^+ \subseteq \mu_i + \text{span } K^-$ and $y \in K^-$, we have that $x_i + y \in \Delta_i$ for any $i$ and

$$x + y = \sum_{i} \lambda_{i}(x_{i} + y) \in \text{conv}(\bigcup_{i} \Delta_{i}) = \Delta.$$

This concludes our proof that $\Delta + K^- = \Delta$ for any rational power series $h$.

We now show $\text{supp}(q^+) \subseteq \Delta$. If $y$ does not belong to any $y' + K^-$ for any other $y' \in \text{supp}(q^+)$, the only decomposition of $y$ as the sum of an element $y'$ in $\text{supp}(q^+)$ and an element in $\text{supp}(1/q^-) \subseteq K^-$ is the sums $y + 0$. The unicity of this decomposition proves that $y \in \Delta$. Now, if $y \in y' + K^-$ for some other $y' \in \text{supp}(q^+)$, we set $y_0$ as the element with minimal sum of coordinates amongst the elements $y' \in \text{supp}(q^+)$ verifying $y \in y' + K^-$. If there is some other $y_1 \in \text{supp}(q^+)$ such that $y_0 \in y_1 + K^-$, we would have $y \in y_0 + K^- \subseteq y_1 + K^-$, and since $K^- \subseteq (\mathbb{R}^+)^k$, every coordinate of $y_1$ would be smaller than the corresponding coordinate of $y_0$. This would refute the minimality of $y_0$. Therefore $y_0$ does not belong to any $y' + K^-$ for any other $y' \in \text{supp}(q^+)$, and by the preceding argument, $y_0 \in \Delta$. This implies that $y \in y_0 + K^- \subseteq \Delta + K^- = \Delta$.

We deduce that $\text{supp}(q^+) \subseteq \Delta$ which implies that $\text{supp}(q^+) + K^- \subseteq \Delta + K^- = \Delta$ and concludes our argument. $\square$

We are now in position to prove the generalization of Lemma 3.9 to rational power series with more than two variables, by applying Proposition 3.14 above to $h^\circ$.

**Corollary 3.16.** — Let $h \in \mathbb{Z}[[X]]$ be a rational power series with constant term equal to 1. Let $\nu \in \mathbb{N}^k$ be the primitive vector directing an extreme ray of cone(supp $h$), and define $h_\nu$ as $\sum_{n \in \mathbb{N}} b(n\nu)X^{n\nu}$, where $b$ is the coefficient function of $h$, and $h^\circ \in \mathbb{Z}[[X]]$ as $h/h_\nu$.

We have the exclusive disjunction:

(i) either in $\mathbb{Z}[X]$, the polynomial $q_\nu^+$ divides $q^+$, and $q_\nu^-$ divides $q^-$.

(ii) or there exists $\mu \in \mathbb{N}^k$ such that the half-line $\mu + \nu \mathbb{R}_+$ contains infinitely many elements of $\text{supp}(h^\circ)$.

These two options correspond to the cases $\nu \notin \text{cone}(\text{supp } h^\circ)$ and $\nu \in \text{cone}(\text{supp } h^\circ)$ respectively.

Note that since $\nu \mathbb{R}_+$ is an extreme ray of the rational cone cone(supp $h$) generated by the supports of $q^+$ and $q^-$, we have $q_\nu^+ q_\nu^- \neq 1$ and the condition (i) is never trivial.

*Proof.* — By Lemma 3.3, we have $q_\nu^- h_\nu = q_\nu^+$ which implies that the power series $h_\nu$ is rational. Therefore $h^\circ$ is rational and verifies $q_\nu^+ q^- h^\circ = q_\nu^- q^+$.

Let $q^\oplus$ and $q^\ominus$ be the numerator and denominator of $h^\circ$, respectively. By unique factorization, there exists a polynomial $r \in \mathbb{Z}[X]$ such that $q_\nu^+ q^- = rq^\ominus$ and $q_\nu^- q^+ = rq^\oplus$.

By Proposition 3.14 applied to $h^\circ$, we have either $q_\nu^\ominus = 1$ or there exists $\mu \in \mathbb{N}^k$ such that the half-line $\mu + \nu \mathbb{R}_+$ contains infinitely many elements of $\text{supp}(h^\circ)$, which is exactly the option (ii).

Thus, it remains to prove that (i) is equivalent to $q_\nu^\ominus = 1$.

Notice that $q^-$ and $q^\ominus$ have constant term equal to 1 for being the denominators of rational power series. Since $h(0) = 1$ we also have that $q_\nu^+(0) = q^+(0) = 1$. Thus $r$
has constant term equal to 1.

The supports of $q_\nu^+q^-$ and of $q_\nu^-q^+$ are inside the cone $K$, then by Corollary 3.6 of Ostrowski's Theorem, the supports of $r$, of $q^\ominus$ and of $q^\oplus$ are all inside $K$. We then apply Lemma 3.3 and get that $q_\nu^+q_\nu^- = r_\nu q_\nu^\ominus = r_\nu q_\nu^\oplus$.

If $r$ has an irreducible factor which does not divide $q_\nu^+q_\nu^-$, then it has to divide $q^+$ and $q^-$, which are coprime polynomials. Therefore every factor of $r$ has support in $\nu\mathbb{N}$, and $r = r_\nu$.

If $q_\nu^+$ divides $q^+$ and $q_\nu^-$ divides $q^-$, then $q_\nu^+q_\nu^-$ divides $rq^\oplus$ and $rq^\ominus$. Since $q^\oplus$ and $q^\ominus$ are coprime, we deduce that $q_\nu^+q_\nu^-$ divides $r = r_\nu$. And from the relation $q_\nu^+q_\nu^- = r_\nu q_\nu^\ominus$ we deduce that $q_\nu^\ominus = 1$.

Conversely, if $q_\nu^\ominus = 1$, then $r = r_\nu = q_\nu^+q_\nu^-$. We deduce that $q^- = q_\nu^-q^\ominus$ and $q^+ = q_\nu^+q^\oplus$, which implies that $q_\nu^-$ divides $q^-$, and $q_\nu^+$ divides $q^+$. $\square$

As already noticed, this Corollary is actually sufficient to prove the characterization of $\Gamma$ we are aiming for.

**Proposition 3.17.** — Let $h \in \mathbb{Z}[[X]]$ be a rational power series, with constant term equal to 1. Let $q^+$ and $q^- \in \mathbb{Z}[X]$ be its numerator and its denominator, respectively. If $h$ has no cyclotomic factor (meaning that the product $q^+q^-$ has no cyclotomic factor), then the domain $\Gamma$ described in Theorem 2.9, on which the Euler product is meromorphic, is the interior of the dual of the rational cone spanned by the supports $B^+$ of $q^+$ and $B^-$ of $q^-$, that is

$$ \Gamma = \bigcap_{\substack{\alpha \in B^+ \cup B^- \\ \alpha \neq 0}} \{s \in \mathbb{C}^k \mid \Re\langle\alpha, s\rangle > 0\}. $$

*Proof.* — We have already noticed that $\Gamma^\vee \subseteq \overline{\text{cone}}(C)$ where $C$ is the exponential support of $h$. By Lemma 3.2 we have that $\overline{\text{cone}}(C) = \text{cone}(B)$ where $B$ is the support of $h$, and by (20) we have that $\overline{\text{cone}}(B) = \text{cone}(B^+ \cup B^-)$, which is a rational cone. It remains to prove that every extreme ray $\nu\mathbb{R}_+$ of cone$(B^+ \cup B^-)$ is in $\Gamma^\vee$.

Assume that $\nu \notin \Gamma^\vee$. By Lemma 3.1', $h_\nu$ is a quotient of products of cyclotomic polynomials, and $\nu$ is not in $\overline{\text{cone}}(\text{supp}(h^\circ))$. By Corollary 3.16, this latter condition implies that $q_\nu^+$ divides $q^+$ and $q_\nu^-$ divides $q^-$. Since $q^+$ and $q^-$ are coprime, so are $q_\nu^+$ and $q_\nu^-$, thus $q_\nu^+$ and $q_\nu^-$ are the numerator and denominator of $h_\nu$. We conclude that $q_\nu^+$ and $q_\nu^-$ are products of cyclotomic polynomials, not both reduced to 1, and that they divide $q^+$ and $q^-$ respectively. This contradicts the fact that $h$ has no cyclotomic factor. Therefore, we have that $\nu \in \Gamma^\vee$ for every extreme ray of cone$(B^+ \cup B^-)$, which proves that $\Gamma^\vee = \text{cone}(B^+ \cup B^-)$. $\square$

**3.3. Rationality in the support and the exponential support.** — In the previous section we have describing the domain $\Gamma$ in terms of $\text{supp}(q^+)$ and $\text{supp}(q^-)$, by proving that the intersection of $\text{supp}(h^\circ)$ with some half-line $\mu + \nu\mathbb{R}_+$ are infinite. In this section, we improve the statement about this intersection, by showing that it is a **syndetic** set, which means that the differences between pairs of consecutive elements are bounded. From there we prove the same property for the intersection of the exponential support $C$ of $h$ with some half-line $\mu + \nu\mathbb{R}_+$.
In the one variable setting, Proposition 3.14 reduces to the obvious statement that a rational power series of $Z[[X]]$ is a polynomial if and only if its denominator is equal to 1. However, in this case, we have two results describing the support of a non-polynomial rational power series $\sum_{n \in \mathbb{N}} a_n X^n$:

- Since the sequence $(a_n)_{n>\deg q^+}$ satisfies a linear recurrence relation of order $\deg q^-$, the sequence has a non-zero term amongst any $\deg q^-$ consecutive terms.

- By the celebrated Skolem-Mahler-Lech Theorem$^{(†)}$, we know that there exist two numbers $N \in \mathbb{N}$ and $d \ge 1$ such that the set $\{n \ge N \mid a_n = 0\}$ is a union of arithmetic progressions of common difference $d$. The same statement is also true for its complement $\{n \ge N \mid a_n \ne 0\}$.

We will show that both statements apply to the subsequence $(b_{\mu+n\nu})_{n \in \mathbb{N}}$ of the coefficients of $h$, for any $\mu \in \mathbb{Z}^k$, provided that its support is not finite. Actually, the first one, which is almost trivial, would be sufficient for our needs, but we will also refer to the second, much deeper one, only because it will slightly simplify some arguments.

The reason that these statements are valid for the sequences $(b_{\mu+n\nu})_{n \in \mathbb{N}}$ is the rationality of the corresponding power series.

**Proposition 3.18.** — Let $h = \sum_{\alpha \in \mathbb{N}^k} b_\alpha X^\alpha \in Z[[X]]$ be a rational power series, and let $\nu \in \mathbb{N}^k$ be the primitive vector directing an extreme ray of cone(supp $h$). Then, for any $\mu \in \mathbb{Z}^k$, the power series

$$ \sum_{n \in \mathbb{N}} b_{\mu+n\nu} T^n \in Z[[T]] $$

is rational.

*Proof.* — If the statement is verified for a particular $\mu \in \mathbb{Z}^k$, so that we get a rational power series in $Z[[T]]$ of numerator $p$ and denominator $q$, then it is also true for $\mu - \nu$ by considering $Tp(T)/q(T)$, and for $\mu + \nu$ by considering $p'/q$ with $p'(T) = \frac{p(T)-p(0)}{T} \in Z[T]$. Therefore this is true for any $\mu + m\nu$ with $m \in \mathbb{Z}$. The image by $\phi$, as described in the previous section, of the canonical basis of $\mathbb{Z}^k$ is a basis of $\mathbb{Z}^k$ whose first vector is $\nu$. We may restrict ourselves to prove the assertion for $\mu$ in the span of the other vectors of this basis, that is for $\mu$ written as $\phi(0, \beta_2, \dots, \beta_k)$ for $(\beta_2, \dots, \beta_k) \in \mathbb{Z}^{k-1}$. We may even restrict ourselves to $(\beta_2, \dots, \beta_k) \in \mathbb{N}^{k-1}$: otherwise, for any $\beta_1 \in \mathbb{Z}$, we will have $(\beta_1, \beta_2, \dots, \beta_k) \notin \mathbb{N}^k$, therefore $\phi(\beta_1, \beta_2, \dots, \beta_k) \notin \text{supp } h$ since $\text{supp } h \subseteq K \cap \mathbb{Z}^k \subseteq \phi(\mathbb{N}^k)$, and finally, the considered power series will be equal to zero, therefore rational.

We denote by $R$ the ring of rational power series in $Z[[Y_1]]$, or equivalently, the intersection $\mathbb{Q}(Y_1) \cap Z[[Y_1]]$ of subrings of $\mathbb{Q}((Y_1))$. Note that $Z[Y_1] \subset R$. The polynomials $\phi_*q^+$ and $\phi_*q^-$ may be regarded as elements of $Z[Y_1][Y_2, \dots, Y_k]$, therefore also as elements of $R[Y_2, \dots, Y_k]$. The constant term of $\phi_*q^-$ in this ring is $\phi_*q^-(Y_1, 0, \dots, 0)$. This constant term is a polynomial of $Z[Y_1]$, and since $q^-(0) = 1$, it has a constant term equal to one. In particular, this constant term is an invertible element of $R$,

<footnote>(†)For example, see [Eil74, Theorem 3.4].</footnote>
thus $\phi_*q^-$ is an inversible element of $R[[Y_2, \dots, Y_k]]$. Therefore, the quotient $\frac{\phi_*q^+}{\phi_*q^-}$, which is equal to $\phi_*h$, is an element of $R[[Y_2, \dots, Y_k]]$.

For any $(\beta_2, \dots, \beta_k) \in \mathbb{N}^{k-1}$, we define the coefficient function

$$a_{(\beta_2, \dots, \beta_k)} = \sum_{\beta_1 \in \mathbb{N}} b_{\phi(\beta_1, \beta_2, \dots, \beta_k)} Y_1^{\beta_1} \in \mathbb{Z}[[Y_1]],$$

so that

$$\phi_*h = \sum_{(\beta_2, \dots, \beta_k) \in \mathbb{N}^{k-1}} a_{(\beta_2, \dots, \beta_k)} Y_2^{\beta_2} \dots Y_k^{\beta_k}.$$

Since $\phi_*h$ belongs to $R[[Y_2, \dots, Y_k]]$, we can identify its coefficients as members of $\mathbb{Z}[[Y_1]]$, and we deduce that $a_{(\beta_2, \dots, \beta_k)}$ belongs to $R$ for any $(\beta_2, \dots, \beta_k) \in \mathbb{N}^{k-1}$. This means that for any $\mu$ of the form $\phi(0, \beta_2, \dots, \beta_k)$ with $(\beta_2, \dots, \beta_k) \in \mathbb{N}^{k-1}$, the power series in one variable

$$a_{(\beta_2, \dots, \beta_k)} = \sum_{\beta_1 \in \mathbb{N}} b_{\phi(\beta_1, \beta_2, \dots, \beta_k)} Y_1^{\beta_1} = \sum_{n \in \mathbb{N}} b_{\mu+n\nu} Y_1^n$$

is rational.

□

We can also provide another proof of the rationality of $a_{(\beta_2, \dots, \beta_k)}$ for $(\beta_2, \dots, \beta_k) \in \mathbb{N}^{k-1}$, by using the partial derivatives $\partial_i$ with respect to the variables $Y_i$. As a matter of fact, we have that

$$\frac{\partial_{\beta_2}^{\beta_2}}{\beta_2!} \cdots \frac{\partial_{\beta_k}^{\beta_k}}{\beta_k!} \phi_*h = \sum_{(\beta'_2, \dots, \beta'_k) \in \mathbb{N}^{k-1}} a_{(\beta_2+\beta'_2, \dots, \beta_k+\beta'_k)} Y_2^{\beta'_2} \cdots Y_k^{\beta'_k},$$

therefore $a_{(\beta_2, \dots, \beta_k)} = \frac{\partial_{\beta_2}^{\beta_2}}{\beta_2!} \cdots \frac{\partial_{\beta_k}^{\beta_k}}{\beta_k!} \phi_*h(Y_1, 0, \dots, 0)$. Since $\phi_*h = \frac{\phi_*q^+}{\phi_*q^-}$, we have that

$$(\phi_*q^-)^{1+\beta_2+\dots+\beta_k} \frac{\partial_{\beta_2}^{\beta_2}}{\beta_2!} \cdots \frac{\partial_{\beta_k}^{\beta_k}}{\beta_k!} \phi_*h \in \mathbb{Z}[\mathbf{Y}].$$

By specializing to $Y_2 = \dots = Y_k = 0$ and recalling that $\phi_*q^-(Y_1, 0, \dots, 0) = \phi_*q^-\bar{\nu}$, we get that $(\phi_*q^-\bar{\nu})^{1+\beta_2+\dots+\beta_k} a_{(\beta_2, \dots, \beta_k)}$ is a polynomial of $\mathbb{Z}[Y_1]$. This proves not only that $a_{(\beta_2, \dots, \beta_k)}$ is rational, but also that its denominator divides $(\phi_*q^-\bar{\nu})^{1+\beta_2+\dots+\beta_k}$. In particular, we set $\mu = \phi(0, \beta_2, \dots, \beta_k)$, and if the sequence $(b_{\mu+n\nu})_{n \in \mathbb{N}}$ has infinite support, it may admit only finitely many subsequences of $(1+\beta_2+\dots+\beta_k) \deg \phi_*q^-\bar{\nu}$ consecutive terms with zero values.

**3.3.1. Rationality in the exponential support.** — Let $h$ be a rational power series with constant term equal to 1, $B$ its support, and $K$ the closed convex cone generated by $B$. We have seen that $K$ is rational and line-free.

Let $\nu$ be a primitive integer vector such that $\nu\mathbb{R}_+$ is an extreme ray of $K$. Let $(v_1, \dots, v_k)$ be a basis of the lattice $\mathbb{Z}^k$ such that $v_1 = \nu$ and $K$ is contained in the cone spanned by $(v_1, \dots, v_k)$.

Let $M'$ the monoid generated by $(v_2, \dots, v_k)$. It is equipped with a canonical partial order $\leq$ defined as $\mu \leq \mu'$ if and only if $\mu' - \mu \in M'$, and $\prec$ its corresponding strict partial order (its irreflexive kernel). For any $(n_2, \dots, n_k)$ and $(n'_2, \dots, n'_k)$ in $\mathbb{N}^{k-1}$, we have that $n_2v_2 + \dots + n_kv_k \leq n'_2v_2 + \dots + n'_kv_k$ if and only if $n_i \leq n'_i$ for
<figure><img src="image_9.png" /><figcaption>FIGURE 4. Support, exponential support and corresponding coefficients for the rational function $h^\circ(X_1, X_2) = \frac{1-X_2+X_2^2+2X_1X_2}{1-X_2+X_2^2}$ associated to the polynomial $h(X_1, X_2) = 1 - X_2 + X_2^2 + 2X_1X_2$ of Figure 2 and to the direction $\nu = (0, 1)$</figcaption></figure>

every $i \in [2, k]$. In particular, for any $\boldsymbol{\mu} \in M'$, there are only finitely many elements $\boldsymbol{\mu}' \in M'$ such that $\boldsymbol{\mu}' \preceq \boldsymbol{\mu}$.

Every $\boldsymbol{\alpha} \in B$ is an element of $\mathbb{Z}^k \cap K$ and can be uniquely written as $n_1\boldsymbol{v}_1 + \cdots + n_k\boldsymbol{v}_k$ with $(n_1, \ldots, n_k) \in \mathbb{N}^k$, or again as $n\boldsymbol{\nu} + \boldsymbol{\mu}$, with $n \in \mathbb{N}$ and $\boldsymbol{\mu} \in M'$.

**Lemma 3.19.** — Let $c$ be the exponent function associated to $h$. Then, the support of $c$ is contained in $\mathbb{Z}^k \cap K$ and for all $\boldsymbol{\mu} \in M' \setminus \{\boldsymbol{0}\}$, we have

$$ \sum_{n \in \mathbb{N}} c_{\boldsymbol{\mu}+n\boldsymbol{\nu}} T^n = \sum_{d \ge 1} \sum_{r \ge 1} \frac{\mu(d)(-1)^r}{dr} \sum_{\substack{\boldsymbol{\mu}_1, \ldots, \boldsymbol{\mu}_r \in M' \setminus \{\boldsymbol{0}\} \\ d \sum \boldsymbol{\mu}_i = \boldsymbol{\mu}}} \prod_{i=1}^r \left( \sum_{m \in \mathbb{N}} b^{\boldsymbol{\mu}_i+m\boldsymbol{\nu}} (T^d)^m \right), $$

where $b^\circ$ is the coefficient function of $h^\circ = h/h_\nu$.

*Proof.* — As noted in Lemma 3.2, the exponential support of $h$ generates the same monoid than its support $B$, therefore any $\boldsymbol{\beta}$ in the exponential support of $h$ is in $\mathbb{Z}^k \cap K$, and it can be uniquely written as $\boldsymbol{\beta} = n\boldsymbol{\nu} + \boldsymbol{\mu}$, with $n \in \mathbb{N}$ and $\boldsymbol{\mu} \in M'$.

if $\boldsymbol{\mu} \neq \boldsymbol{0}$, and since $\boldsymbol{\nu}\mathbb{N} \cap M' = \{\boldsymbol{0}\}$, then $\boldsymbol{\beta} \notin \boldsymbol{\nu}\mathbb{R}_+$ and the exponent $c(\boldsymbol{\beta})$ is also equal to the exponent of $h^\circ$, which is defined by Equation (14'). We denote by $b^\circ$ the coefficient function of the power series $h^\circ$, whose support is also in $\mathbb{Z}^k \cap K$. By
Lemma 2.2, we have

$$c_{\beta} = \sum_{d \ge 1} \sum_{r \ge 1} \frac{\mu(d)(-1)^r}{dr} \sum_{\substack{\alpha_1, \dots, \alpha_r \in \mathbb{Z}^k \cap K \\ d \sum \alpha_i = \beta}} \prod_{i=1}^{r} b_{\alpha_i}^{\circ}.$$

Note that we may consider only $\alpha_i$ which are not collinear with $\nu$, since $b^{\circ}(n\nu) = 0$ if $n > 0$. We write $\beta$ as $n\nu + \mu$ and $\alpha_i = m_i\nu + \mu_i$, with $\mu_i \in M' \setminus \{0\}$. The condition $d \sum_{i=1}^r \alpha_i = \beta$ is then translated in two equations: $d \sum_{i=1}^r \mu_i = \mu$ and $d \sum_{i=1}^r m_i = n$, so that

$$c_{\mu+n\nu} T^n = \sum_{d \ge 1} \sum_{r \ge 1} \frac{\mu(d)(-1)^r}{dr} \sum_{\substack{\mu_1, \dots, \mu_r \in M' \setminus \{0\} \\ d \sum \mu_i = \mu}} \sum_{\substack{m_1, \dots, m_r \in \mathbb{N} \\ d \sum m_i = n}} \prod_{i=1}^{r} (b_{\mu_i+m_i\nu}^{\circ}(T^d))^{m_i}.$$

By summing over $n \in \mathbb{N}$, we notice that

$$\sum_{n \in \mathbb{N}} \sum_{m_1, \dots, m_r \in \mathbb{N}} \prod_{i=1}^{r} (b_{\mu_i+m_i\nu}^{\circ}(T^d))^{m_i} = \prod_{i=1}^{r} \left( \sum_{m \in \mathbb{N}} b_{\mu_i+m\nu}^{\circ}(T^d) \right)^{m_i},$$

which gives the targeted formula. $\square$

**Corollary 3.20.** — Let $c$ be the exponent function associated to $h$. For all $\mu \in \mathbb{Z}^k$ which is not collinear to $\nu$, the power series

$$\sum_{n \in \mathbb{N}} c_{\mu+n\nu} T^n \in \mathbb{Z}[[T]]$$

is rational.

*Proof.* — Let $(m_1, m_2, \dots, m_k) \in \mathbb{N}^k$ be the coordinates of $\mu$ in the basis $(v_1, \dots, v_k)$ of $\mathbb{Z}^k$. As for Proposition 3.18, the validity of the statement does not change by adding some multiple of $\nu$ to $\mu$, and we may as well suppose that $m_1 = 0$.

Furthermore, the statement is obviously verified if $\mu \notin M'$, so we may assume that $\mu \in M'$, that is that $m_i \ge 0$ for all $i \in [2, k]$.

Since $\mu \ne 0$, we can use the formula of Lemma 3.19. The relation $d \sum_{i=1}^r \mu_i = \mu$ with $\mu_i \ne 0$ implies that $d$ is bounded by the lcm of the $m_i$, $r$ is bounded by the sum $m_2 + \dots + m_k$ and each $\mu_i$ has to be chosen among at most $(m_2 + 1) \dots (m_k + 1)$ values.

By Proposition 3.18, for any $\mu' \in \mathbb{Z}^k$, the power series $\sum_{m \in \mathbb{N}} b_{\mu'+m\nu}^{\circ}(T^d)^m$ is rational. Therefore, the power series $\sum_{n \in \mathbb{N}} c_{\mu+n\nu} T^n$ is a finite sum of finite products of rational power series, hence also rational. $\square$

Notice that when $\mu = 0$, we don't expect the power series $\sum_{n \in \mathbb{N}} c_{n\nu} T^n$ to be rational.

**Corollary 3.21.** — Let $\mu \in M' \setminus \{0\}$. If for every $\mu' \in M' \setminus \{0\}$ with $\mu' \prec \mu$, the series $\sum_{n \in \mathbb{N}} b_{\mu'+n\nu}^{\circ} T^n$ is a polynomial, then the difference of series

$$\sum_{n \in \mathbb{N}} c_{\mu+n\nu} T^n - \sum_{n \in \mathbb{N}} b_{\mu+n\nu}^{\circ} T^n$$
is a polynomial. In particular, there exists $N \in \mathbb{N}$ such that $c_{\mu+n\nu} = b^{\circ}_{\mu+n\nu}$ for every $n \ge N$.

*Proof.* — In the formula of Lemma 3.19, the term with $r = d = 1$ is exactly $\sum_{n \in \mathbb{N}} b^{\circ}_{\mu+n\nu} T^n$. The condition $d \sum_{i=1}^r \mu_i = \mu$ implies that, if $d > 1$, then for any $i$ we have

$$ \mu_i \le \sum_{i=1}^{r} \mu_i < d \sum_{i=1}^{r} \mu_i = \mu, $$

and if $r > 1$, then for any $i$ we have

$$ \mu_i < \sum_{i=1}^{r} \mu_i \le d \sum_{i=1}^{r} \mu_i = \mu. $$

In any case, all the series occurring in the other term are polynomials. Using the same analysis than in the proof of the previous corollary, we conclude that the power series $\sum_{n \in \mathbb{N}} c_{\mu+n\nu} T^n - \sum_{n \in \mathbb{N}} b^{\circ}_{\mu+n\nu} T^n$ is a finite sum of finite products of polynomials, therefore it is a polynomial. $\square$

We can finally conclude on the existence of infinitely many elements in $C$ on a half-line directed by $\nu$, with bounded difference between consecutive terms.

**Proposition 3.22.** — Let $h \in \mathbb{Z}[[X]]$ be a rational power series with constant term equal to 1, let $q^+$ and $q^- \in \mathbb{Z}[X]$ be its numerator and denominator, respectively. Assume that $h$ has no cyclotomic factors. Let $C$ be the exponential support of $h$, and let $\nu \in \mathbb{N}^k$ be the primitive vector directing an extreme ray of $\text{cone}(C)$.

Then, exactly one of the two following statements is verified :

(i) either $\nu \mathbb{N}$ contains infinitely many elements of $C$,

(ii) or there exists $\mu_0 \in \mathbb{N}^k$ such that $\mu_0 + \nu \mathbb{N}$ contains infinitely many elements of $C$, with the following additional properties:

- There exist two positive integers $n_0$ and $d_0$ such that for every $n \ge n_0$, at least one of the $d_0$ values $\mu_0 + (n+1)\nu$, ..., $\mu_0 + (n+d_0)\nu$ belongs to $C$.

- There are only finitely many elements of $C$ of the form $t\mu_0 + x\nu$, with $t \in [0, 1[$ and $x \in \mathbb{R}_+$.

Both options correspond, respectively, to the case whether $h_\nu$ is a quotient of products of cyclotomic polynomials or not. The two technical additional properties in option (ii) will be of great use in the next section.

*Proof.* — Assertions (i) and (ii) cannot hold together since the second property of (ii) (with $t = 0$) contradicts (i). It remains to prove that (ii) holds if (i) does not. As remarked in the proof of Lemma 3.1', (i) is equivalent to the fact that $h_\nu$ is not a quotient of products of cyclotomic polynomials. Thus we assume that $C \cap \nu \mathbb{R}_+$ is finite, namely that $h_\nu$ is a quotient of products of cyclotomic polynomials.

The validity of (ii) doesn't change if we add or remove finitely many elements from $C$. Since $C = (C \cap \nu \mathbb{R}_+) \cup C^\circ$, where $C^\circ = \text{xsupp}(h^\circ)$, we will prove the statement (ii) for $C^\circ$ instead of $C$. We first show that it is impossible that $q_\nu^+$ divides $q^+$ and $q_\nu^-$ divides $q^-$ in order to use Corollary 3.16.

Assume that $q_\nu^+$ divides $q^+$ and $q_\nu^-$ divides $q^-$. Since $q^+$ and $q^-$ are coprime polynomials, $q_\nu^+$ and $q_\nu^-$ do not share a non-trivial common factor. Thus, they are the
numerator and the denominator of $h_ν$, which is a quotient of products of cyclotomic polynomials. This implies that $q_ν^+$ and $q_ν^-$ are products of cyclotomic polynomials. However, $q^+$ and $q^-$ do not have any cyclotomic factors, therefore $q^+ = q^- = 1$. On the other hand, by Lemma 3.2 and formula (20), we have that $\overline{\text{cone}}(C) = \overline{\text{cone}}(B) = \text{cone}(B^+ \cup B^-)$. Since $ν\mathbb{R}_+$ is an extreme ray of this cone, the generating set $B^+ \cup B^-$ contains a nonzero multiple of $ν$, which means that not both polynomials $q_ν^+$ or $q_ν^-$ are constant which contradicts that $q^+ = q^- = 1$.

This contraction proves that the assertion (i) in Corollary 3.16 is not verified, so the assertion (ii) holds: there exists $\mu \in \mathbb{N}^k$ such that the half-line $\mu + ν\mathbb{R}_+$ contains infinitely many elements of the support $B^°$ of $h^°$.

We may choose $\mu$ in $B^° \subseteq \overline{\text{cone}}(C)$, and by removing from $\mu$ a suitable multiple of $ν$, we may choose $\mu \in M'$. In that case, we have $\mu \neq 0$, because $ν\mathbb{N} \cap B^° = \{0\}$. The set of $\mu' \in M'$ with $\mu' \leq \mu$ is finite, so we can find $\mu_0 \in M' \setminus \{0\}$ which is minimal for the partial order relation $\leq$ among the set of $\mu \in M'$ for which $(\mu + ν\mathbb{R}_+) \cap B^°$ is not finite.

We verify the conditions of Corollary 3.21 to any $\mu \leq \mu_0$: if $\mu < \mu_0$, the symmetric difference between the sets $B^° \cap (\mu + ν\mathbb{N})$ and $C \cap (\mu + ν\mathbb{N})$ is finite, and since the former one is also finite, so is $C \cap (\mu + ν\mathbb{N})$. The symmetric difference between the sets $B^° \cap (\mu_0 + ν\mathbb{N})$ and $C \cap (\mu_0 + ν\mathbb{N})$ is also finite, but $B^° \cap (\mu_0 + ν\mathbb{N})$ is not finite, and $C \cap (\mu_0 + ν\mathbb{N})$ is neither.

The series $\sum_{n \in \mathbb{N}} c_{\mu+n\nu} T^n$ is not polynomial, and is rational by Corollary 3.20. The Skolem-Mahler-Lech Theorem applies, concluding that $C \cap (\mu_0 + ν\mathbb{N})$ contains an infinite arithmetic progression, which proves the first part of (ii).

Finally, there are finally many $t \in [0, 1[$ such that $t\mu_0$ is an element of $M'$, and for each of them, we have $t\mu_0 < \mu_0$, thus $C \cap (t\mu_0 + ν\mathbb{N})$ is finite. If $t\mu_0 \notin M'$, the previous intersection is empty. A finite union of finite sets is still finite, therefore there are finitely many elements of $C$ of the form $t\mu_0 + n\nu$, with $t \in [0, 1[$ and $n \in \mathbb{N}$. If $x \notin \mathbb{Z}$ then $t\mu_0 + x\nu \notin \mathbb{Z}^k$, and the last part of (ii) is verified. $\square$

## 4. Hyperplanes in the divisor of $f$

In this last section, we study hyperplanes in the divisor of the Euler product $f$ over $\Gamma$, in order to detect cluster points, therefore singularities of $f$ on $\partial\Gamma$.

This divisor is described in Theorem 2.9:

$$ (f) = \sum_{p} \left[ (h_p) - \sum_{\alpha \in \mathbb{N}^k} c(\alpha) \sum_{\kappa \in \mathbb{Z}} H_{\alpha} \left( \frac{2\kappa\pi i}{\log p} \right) \right] - \sum_{\alpha \in \mathbb{N}^k} \sum_{\gamma \in G} m_{\gamma} c(\alpha) H_{\alpha}(\gamma), $$

where the divisor of Riemann zeta function over $\mathbb{C}$ is $\sum_{\gamma \in G} m_{\gamma} \gamma$. Most of this expression is already mainly constituted of hyperplanes, with rational direction. Only the terms $(h_p)$ are not of this form. The following lemma 4.1 describes the hyperplanes that appear in those terms.

But since this point is crucial, we give some more details on divisors and hyper-surfaces. We limit the exposition to holomorphic functions, but it can be extended
to meromorphic functions by means of local quotients. We can refer the reader to [Huy05, §2.1] for the basics on complex manifolds and affine hypersurfaces.

Let $\Gamma$ be a domain of $\mathbb{C}^k$, and $f$ be an holomorphic function defined on $\Gamma$. The (analytic) order of vanishing of $f$ at $z$ is the minimal order $m = m_1 + \dots + m_k$ of a nonzero partial derivative of $f$ at $z$, that is such $\frac{\partial^m f}{\partial m_1 z_1 \dots \partial m_k z_k}(z) \neq 0$. The work of Weierstraß gives a geometric meaning of this order, through the divisor of $f$ as a formal sum of hypersurfaces: if $(f) = \sum_S m_S S$, then the order of $f$ at $z$ is $\sum_{S \ni z} m_S$, where the sum is restrained to hypersurfaces $S$ containing $z$.

These (analytic) hypersurfaces are defined locally as zeroes of holomorphic functions, that is: there exists a open cover $(U_i)_{i \in I}$ of $\Gamma$ and a family of holomorphic functions $g_i : U_i \to \mathbb{C}$ such that $S \cap U_i$ is the set of zeros of $g_i$ for any $i \in I$. A hypersurface is said to be *irreducible* if it cannot be written as a union of two proper hypersurfaces. It can easily be shown that an irreducible hypersurface is connex and it admits a representation by a family $g_i : U_i \to \mathbb{C}$ where the order of vanishing of $g_i$ is at most 1 for every $z \in U_i$.

If $f$ is a polynomial function, each irreducible hypersurface appearing in $(f)$ corresponds to an irreducible factor of $f$ that vanishes at some point on $\Gamma$. In that case, the order $m_S$ of this hypersurface is equal to the algebraic multiplicity of the corresponding factor in $f$.

Our final tool is the pullback by a nondegenerate holomorphic mapping (see [Huy05, §2.3]). A mapping $\Phi : \Omega \to \Gamma$ between two domains of $\mathbb{C}^k$ is *holomorphic* if every coordinate function $\Omega \to \mathbb{C}$ is holomorphic. We define the *pullback* of $f$ by $\Phi$ as the function $\Phi^*f = f \circ \Phi : \Omega \to \mathbb{C}$. We also define the pullback of a hypersurface $S$ defined by the local data $(U_i, g_i)_{i \in I}$ as the hypersurface $\Phi^*S$ defined by the local data $(\Phi^{-1}(U_i), \Phi^*g_i)_{i \in I}$, so that $s \in \Phi^*S \Leftrightarrow \Phi(s) \in S$.

If $(f) = \sum_S m_S S$, we cannot yet assert that $(\Phi^*f) = \sum_S m_S \Phi^*S$. However, if the Jacobian matrix of $\Phi$ at $s$ is invertible, then the order of vanishing of $\Phi^*f$ at $s$ is equal to the order of vanishing of $f$ at $\Phi(s)$. Therefore, $\sum_{S \ni \Phi(s)} m_S = \sum_{\Phi^*S \ni s} m_S$ is the order of vanishing of $\Phi^*f$ at $s$. If the Jacobian matrix of $\Phi$ at $s$ is invertible for any $s \in \Omega$ ($\Phi$ is then said to be nondegenerate), we do have that $(\Phi^*f) = \sum_S m_S \Phi^*S$, even if $\Phi^*S$ is not necessarily an irreducible hypersurface. Since the functions $g_i$ defining $S$ may be chosen with vanishing order at most 1 at any point, so are the functions $\Phi^*g_i$ defining $\Phi^*S$. We deduce that if $S$ is irreducible and $\Phi$ nondegenerate, then $\Phi^*S$ is a disjoint union of irreducible hypersurfaces, and each connex component of $\Phi^*S$ has order $m_S$ in the divisor $(\Phi^*f)$.

**Lemma 4.1.** — Let $\nu \in \mathbb{N}^k$ with coprime coordinates, $\rho \in \mathbb{C}$ such that $H_\nu(\rho) \cap \Gamma \neq \emptyset$ and $p$ a prime number. There is equality between:

(i) the order of $H_\nu(\rho)$ in the divisor of $h_p$ over $\Gamma$;

(ii) the order of the zero set of the irreducible polynomial $X^\nu - p^{-\rho}$ in the divisor of $h$ over $\Omega = \{s \in \mathbb{C}^{*k} | (-\log|s_1|, \dots, -\log|s_k|) \in \Gamma\}$;

(iii) the order of $H_\nu(\rho + m \frac{2\pi i}{\log p})$ in the divisor of $h_p$ over $\Gamma$ for any $m \in \mathbb{Z}$.

Moverover, if $h$ is rational, those numbers are also equal to:

(iv) the multiplicity of $X^\nu - p^{-\rho}$ in the factorization of $h$.
Proof. — Set $P = X^\nu - p^{-\rho}$. Its Newton polytope $\Delta(P)$ is the line segment $[\nu, 0]$. By Ostrowski's Theorem, any factor of $P$ has a Newton polytope of the form $[\lambda\nu, 0]$, with $\lambda \in [0, 1]$. Since $\nu$ has coprime coordinates, the fact that $\lambda\nu \in \mathbb{N}^k$ implies that $\lambda \in \{0, 1\}$, thus this factor of $P$ is either a nonzero constant or a multiple of $P$. We conclude that $P$ is an irreducible polynomial in $\mathbb{C}[X]$; its divisor $(P)$ over $\Omega$ is an irreducible hypersurface, which we denote by $S$.

The holomorphic mapping $\Phi : s \mapsto (p^{-s_1}, \dots, p^{-s_k})$ maps $\Gamma$ onto $\Omega$, its Jacobian matrix is diagonal and invertible at any point. And by definition, $h_p = \Phi^*h$. Since $S$ is an irreducible hypersurface and $\Phi$ is nondegenerate, $\Phi^*S$ is the disjoint union of irreducible hypersurfaces. Actually,

$$ \Phi^*S = \{s \in \Omega | \langle \nu, s \rangle \in \rho + \frac{2\pi i}{\log p} \mathbb{Z} \} = \bigcup_{m \in \mathbb{Z}} H_{\nu} \left( \rho + \frac{2\pi i}{\log p} m \right), $$

and the order of each hyperplane $H_\nu(\rho + \frac{2\pi i}{\log p}m)$ in the divisor $(\Phi^*h) = (h_p)$ is equal to the order of $S$ in $(h)$, proving the equality between (ii) and (iii), and thus (i).

If $h$ is rational, the order of $S$ in $(h)$ is equal to the algebraic multiplicity of the polynomial $P$ in $h$, proving the equality between (ii) and (iv). $\square$

Note that by hyperplanes we actually mean the intersection of an affine $\mathbb{C}$-hyperplane of $\mathbb{C}^k$ with $\Gamma$, so $H_\alpha(\rho) = \{s \in \Gamma | \langle \alpha, s \rangle = \rho\}$. As a matter of fact, if the parts of hyperplanes which are outside $\Gamma$ create cluster points on $\partial\Gamma$, we can not conclude that these cluster points are singularities of $f$ (see Figure 1, left, page 15).

For this reason, any term corresponding to an hyperplane which does not intersect $\Gamma$ should be removed from our expression of $(f)$. As noticed in section 2.3, any hyperplane $H_\alpha(\rho)$ with $\alpha \in \Gamma^\vee$ and $\Re(\rho) \le 0$ does not intersect $\Gamma$.

The situation becomes clearer when the power series $h$ is assumed rational, and neither its numerator $q^+$ nor its denominator $q^-$ has a cyclotomic factor. In this case, the exponential support $C$ of $h$ is included in $\Gamma^\vee$, and by Theorem 2.10, the expression of the divisor is simplified to

$$ (21) \qquad (f) = \sum_p (h_p) - \sum_{\alpha \in C} \sum_{\gamma \in G'} m_\gamma c(\alpha) H_\alpha(\gamma), $$

where $G' = \{1\} \cup \{\rho \in \mathbb{R}_+^* + i\mathbb{R} | \zeta(\rho) = 0\}$. Every hyperplane $H_\alpha(\gamma)$ in the rightmost sum has non zero order and intersects the domain $\Gamma$.

By Lemma 4.1, for $p$ prime and $\nu \in \mathbb{N}^k$ primitive, the hyperplanes $H_{\log_p \nu}(\rho)$ in $(h_p)$ correspond to the factors $X^\nu - e^{-\rho}$ of $q^+$ or $q^-$. By Ostrowski's Theorem (16), the support $\{0, \nu\}$ of this factor is included in $\Delta(q^+)$ (or $\Delta(q^-)$). Therefore, by Proposition 3.17, $\nu$ is contained in $\text{cone}(\text{supp } q^+ \cup \text{supp } q^-) = \Gamma^\vee$. This means that the hyperplane $H_\nu(\rho)$ intersects $\Gamma$ if and only if $\Re(\rho) > 0$, or equivalently, $|e^{-\rho}| < 1$.

From now on, we assume that the power series $h \in \mathbb{Z}[\mathbf{X}]$ is rational and that no factor of its numerator $q^+$ or its denominator $q^-$ is cyclotomic. We recall that $q^+$ and $q^-$ are two coprime polynomials in $\mathbb{Z}[\mathbf{X}]$ with $q^+(0) = q^-(0) = 1$, by Proposition 3.12.

We have identified every hyperplane $H_\alpha(\rho)$ that appears in the expression (21) of $(f)$, and we parameterize these hyperplanes by using the **parameter pair** $(\alpha, \rho) \in \mathbb{R}^k \times \mathbb{C}$,