even if it would be more canonical to consider the image of $(\alpha, \rho)$ in $\mathbb{P}^k(\mathbb{C})$. The set of normal vectors $\alpha$ as well as the set of affine constants $\rho$ have a special structure which we want to take advantage of. For this reason, we will consider a set of parameter pairs rather than a set of hyperplanes. The ambiguity between different parameterizations of a given hyperplane will be taken care of by introducing the notion of *non-recurring hyperplanes* later on.

For any primitive vector $\nu \in \mathbb{N}^k$, we introduce $\Theta_\nu$ as the set of complex numbers $\theta$ inside the punctured unit disc $(0 < |\theta| < 1)$ such that $X^\nu - \theta$ divides $q^+$ or $q^-$. Notice that this set is finite, and the set $C'$ of primitive vector $\nu \in \mathbb{N}^k \cap \Gamma^\vee$ such that $\Theta_\nu$ is not empty is also finite. We also define $\Theta = \cup_{\nu \in C'} \Theta_\nu$.

The hyperplanes in the left-hand-side sum in (21) are parameterized by the parameter pairs $(\log p \nu, -\log \theta + 2\pi i \kappa)$, where $p$ is a prime, $\nu$ is a primitive vector in $C'$, $\theta$ is in $\Theta_\nu$, and $\kappa$ is in $\mathbb{Z}$. Notice that this set of parameter pairs does not depend of the determination of complex logarithm, and that the parameter pairs are all distinct, even if they may describe the same hyperplane.

The hyperplanes in the right-hand-side sum in (21) are parameterized by the parameter pairs $(\alpha, \gamma) \in C \times G'$, where $C$ is the exponential support of $h$ and $G'$ is the set of zeros and pole of the $\zeta$ function in the half-plane $\{\Re e(s) > 0\}$. Notice that these parameter pairs are distinct from the previous ones, simply because the chosen normal vector has integer coefficients.

We denote $\mathcal{P}$ the set of all these parameter pairs. The projection of $\mathcal{P}$ onto the second coordinate lies inside $G' \cup \bigcup_{\theta \in \Theta} G_\theta$, where

$$G_{\theta} = \{\rho \in \mathbb{C} \mid e^{-\rho} = \theta\} = \{-\log \theta + 2\pi i \kappa \mid \kappa \in \mathbb{Z}\}.$$

The elements of this set have bounded positive real parts (because of the finiteness and range of $-\log|\theta|$ for $\theta \in \Theta$) and this set is discrete since $\Theta$ is finite. The projection of $\mathcal{P}$ onto the first coordinate lies inside $C \cup \bigcup_p \log p C'$ and is also a discrete set.

Every hyperplane in the divisor $(f)$ is parameterized by one parameter pair $(\alpha, \rho) \in \mathcal{P}$. However the converse may be false. Two distinct parameter pairs $(\alpha, \rho)$ and $(\alpha', \rho') \in \mathcal{P}$ generate the same hyperplane if and only if $\rho'\alpha = \rho\alpha'$. In that case, the coefficients of the corresponding term in (21) may cancel and it is not possible to infer that the hyperplane $H_\alpha(\rho) = H_{\alpha'}(\rho')$ has a non-zero order in $(f)$ without knowing these coefficients.

We say that an hyperplane is **non-recurring** if it is parameterized by exactly one parameter pair $(\alpha, \rho) \in \mathcal{P}$. Any non-recurring hyperplane has a non-zero order in $(f)$, since no cancellation between coefficients is possible. These hyperplanes are unequivocally parameterized by the set of pairs

$$\mathcal{P}^* = \{(\alpha, \rho) \in \mathcal{P} \mid \forall (\alpha', \rho') \in \mathcal{P}, \rho'\alpha = \rho\alpha' \Rightarrow (\alpha, \rho) = (\alpha', \rho')\}.$$

Actually, we will study only subsets of $\mathcal{P}^*$, for which the normal vector lies in a halfline directed by a primitive vector $\nu \in \mathbb{N}^k$. We define

$$\mathcal{P}_{\nu} = \{(a, g) \in \mathbb{R}_{+}^* \times \mathbb{C} \mid (a\nu, g) \in \mathcal{P}\}$$

and $\mathcal{P}_{\nu}^*$ in a similar way. By extension, we also call elements of $\mathcal{P}_{\nu}$ parameter pairs. By setting $A' = \{\log p \mid p \text{ prime}\}$ and $A_{\nu} = \{a \in \mathbb{N} \mid a\nu \in C\}$, we can represent $\mathcal{P}_{\nu}$
as a disjoint union

$$P_{\nu} = \left( \bigcup_{\theta \in \Theta_{\nu}} A' \times G_{\theta} \right) \cup A_{\nu} \times G',$$

where the left-hand-side term of this union correspond to the left-hand-side term in the expression (21) of (f), and similarly for the right-hand-side terms.

In some cases, we will have to work with a different subset, for which the origin of the halfline directed by $\nu$ is not necessarily the origin. For $\mu \in \mathbb{Z}^k$, we define

$$P_{\nu, \mu} = \{(a, g) \in \mathbb{R}_+^* \times \mathbb{C} \mid (\mu + a\nu, g) \in \mathcal{P}\},$$

and $P_{\nu, \mu}^*$ in a similar way. For $\mu = 0$, we retrieve the former sets of parameter pairs $P_{\nu}$ and $P_{\nu}^*$. If $\mu \neq 0$ is collinear to $\nu$, then $\mu = d\nu$ for some $d \in \mathbb{Z}$, thus $P_{\nu, \mu}^*$ is the translation by $(-d, 0)$ of $P_{\nu}^* \cap (]d, +\infty[ \times \mathbb{C})$, and this case does not add something new.

If $\mu$ is not collinear to $\nu$, any parameter pair $(a, g) \in P_{\nu, \mu}$ is such that $(\mu + a\nu, g) \in \mathcal{P}$, thus $\mu + a\nu$ is written $a'\nu'$ for such primitive vector $\nu' \in \mathbb{N}^k$. The relation $\mu + a\nu = a'\nu'$ ensures that $a$ and $a'$ are rational. Therefore $a' \notin A'$ since $\log p$ is irrational for any prime $p$, which means that $(a', g)$ belongs to $A_{\nu'} \times G' \subseteq \mathbb{N} \times G'$. In particular $a\nu = a'\nu' - \mu \in \mathbb{Z}^k$; and since the coordinates of $\nu$ are coprime, we have $a \in \mathbb{N}$ also. Therefore $P_{\nu, \mu}$ is contained in $\mathbb{N} \times G'$ and verifies that its both projections onto a coordinate are discrete sets. Notice that $P_{\nu, \mu}^*$ may parameterize far fewer hyperplanes than the family of all non-recurring hyperplanes which admit a normal vector of the form $\mu + a\nu$. To compare, a similar parameterization of this superfamily would be $\{(a, a'g'/b) \mid \mu + a\nu = a'\nu', (b, g') \in P_{\nu'}^*\}$.

In order to prove the existence of singular points for $f$, we want to show that the family of non-recurring hyperplanes has many cluster points. We state this property through the set of parameter pairs $P_{\nu}^*$: we say that the set $P_{\nu}^*$ accumulates vertically if every pure imaginary number $it$ is a cluster point of the family of $g/a$ for $(g, a) \in P_{\nu}^*$.

More generally, any set $S$ of parameters $(a, g) \in \mathbb{R}_+^* \times \mathbb{C}$ with $\Re(e(g)) > 0$ is said to verify the **vertical accumulation property** if $i\mathbb{R} \subset \{g/a \mid (a, g) \in S\}$. Notice that if $S$ verifies the vertical accumulation property, any superset $S' \supseteq S$ does also.

Every considered set $S$ is such that the two projections of $S$ along one coordinate (resp. $A_S \subset \mathbb{R}_+$ and $G_S \subset \mathbb{C}$) are discrete sets. In this case, the vertical accumulation property requires $A_S$ to be unbounded, and the imaginary parts of elements of $G_S$ to be unbounded by above and by below. In particular, for any $M \in \mathbb{R}_+$, the set $\{(a, g) \in S \mid a \le M\}$ can be removed from $S$ while maintaining the vertical accumulation property, because $\{g/a \mid (a, g) \in S, a \le M\}$ is discrete.

We now assert a general result for subsets of $P_{\nu, \mu}^*$ with vertical accumulation property.

**Proposition 4.2.** — Let $\nu \in \mathbb{N}^k$ be a primitive vector such that $\Gamma$ is inside the half-space $\{s \in \mathbb{C}^k \mid \Re\langle\nu, s\rangle > 0\}$, and $\mu \in \mathbb{N}^k$. Let $S \subseteq P_{\nu, \mu}^*$ be a set of parameter pairs with vertical accumulation property, and let $r$ and $r' \in \mathbb{R}_+$ be such that $0 \le r < \Re(e(g)) \le r'$ for every $(a, g) \in S$. Then, every point $s \in \bar{\Gamma}$ such that $\Re\langle\nu, s\rangle = 0$ and $\Re\langle\mu, s\rangle \le r$ is a singular point for $f$.
The condition $\Re(e) \le r'$ is actually always verified, because the second coordinate of any element of $S$ belongs to $G' \cup \bigcup_{\theta \in \Theta} G_\theta$, and its real part is in $]0, r']$ where $r'$ is 1 or $\max_{\theta \in \Theta} (-\log |\theta|)$.

If $\mu$ is collinear to $\nu$ (in particular when $\mu = 0$), the restriction $\Re\langle\mu, s\rangle \le r$ is
unsignificant, even for $r = 0$. If $\mu$ is not collinear to $\nu$ and $r > 0$, this restriction may
be removed as well by using our result on the structure of singular points in faces of
a tubular domain of meromorphy (cf. Theorem 4.14). Since its proof is independent
and requires a different setting, we postpone it to the end of this section and state
without proof the immediate corollary.

**Corollary 4.3.** — Let $\nu \in \mathbb{N}^k$ be a primitive vector such that $\Gamma$ is inside the half-space $\{s \in \mathbb{C}^k \mid \Re\langle\nu, s\rangle > 0\}$, and $\mu \in \mathbb{N}^k$. Let $S \subseteq \mathcal{P}_{\nu, \mu}^*$ be a set of parameter pairs with vertical accumulation property, and let $r$ and $r' \in \mathbb{R}_+$ be such that $0 \le r < \Re(e)$, $r' < e$ for every $(a, g) \in S$. If $\mu$ is collinear to $\nu$ or if $r > 0$, then every point $s \in \bar{\Gamma}$ such that $\Re\langle\nu, s\rangle = 0$ is a singular point for $f$.

*Proof of Proposition 4.2 and its Corollary.* — Since the normal vector of every considered hyperplane has integer (therefore real) coefficients, we can write any hyperplane $H_\alpha(\rho)$ as $H_\alpha^R(\rho) \oplus iH_\alpha^I(\rho)$, where $H_\alpha^R(\rho) = \{\sigma \in \Gamma \cap \mathbb{R}^k \mid \langle\alpha, \sigma\rangle = \Re(e(\rho))\}$ and $H_\alpha^I(\rho) = \{\tau \in \mathbb{R}^k \mid \langle\alpha, \tau\rangle = \Im(m(\rho))\}$.

Let $s \in \bar{\Gamma}$ verifying $\Re\langle\nu, s\rangle = 0$ and $\Re\langle\mu, s\rangle \le r$. Notice that $s \notin \Gamma$.

Let us start with the real parts. We set $\sigma = (\Re(e(s_1)), \dots, \Re(e(s_k)))$ and $U = \Gamma \cap \mathbb{R}^k$ so that $\Gamma = U + i\mathbb{R}^k$. We shall show that for any (convex) neighborhood $V_R$ of $\sigma$, there is a positive number $M_1$ such that for any $(a,g) \in S$ with $a \ge M_1$, the intersection $U \cap V_R \cap H_{\mu+a\nu}^R(\rho)$ is not empty. $^{(\dagger)}$

Let $V_R$ a convex neighborhood of $\sigma$ in $\mathbb{R}^k$. Since $\sigma \in \bar{U} = \bar{\Gamma} \cap \mathbb{R}^k$, the intersection $U \cap V_R$ is not empty and we fix an element $\sigma'$ in this set. Since $\sigma' \in U \subseteq \Gamma$, we have that $\langle \nu, \sigma' \rangle > 0$, hence there exists $M_1 > 0$ with $\langle \mu + M_1\nu, \sigma' \rangle > r'$. By our assumptions on $s$, we have that $\langle \mu + a\nu, \sigma \rangle = \langle \mu, \sigma \rangle \le r$ for any $a \in \mathbb{R}$. Therefore, for any $(a,g) \in S$ with $a \ge M_1$, we have that

$$
\langle \mu + a\nu, \sigma \rangle \le r < \Re(e) \le r' < \langle \mu + a\nu, \sigma' \rangle,
$$

which implies that there exists $\lambda \in ]0, 1[$ such that $\langle \mu + a\nu, \lambda\sigma' + (1-\lambda)\sigma \rangle = \Re(e(g))$.
By convexity of $U$ and $V_R$, the element $\lambda\sigma' + (1-\lambda)\sigma$ is in the intersection $U \cap V_R \cap H_{\mu+a\nu}^R(g)$.

We now consider the imaginary parts. We set $\tau = (\Im(m(s_1), \dots, \Im(m(s_k)))$. We shall show that any neighborhood of $\tau$ in $\mathbb{R}^k$ intersects with the hyperplane $H_{\mu+a\nu}^I(g)$ for infinitely many $(a,g) \in S$.

Let $\varepsilon > 0$. Choose $M_2 \ge M_1$ such that $2|\langle\mu, \tau\rangle| < \varepsilon M_2$ and $2\|\mu\| < M_2\|\nu\|$. The set $S' = \{(a,g) \in S \mid a \ge M_2\}$ has also the vertical accumulation property, and there are infinitely many $(a,g) \in S'$ with $\Im(m(g)/a$ in the interval $[\langle\nu, \tau\rangle - \varepsilon/2, \langle\nu, \tau\rangle + \varepsilon/2]$.

<footnote>$^{(\dagger)}$The redundant intersection $U \cap H_{\mu+a\nu}^{R}(\rho)$ is to emphasize that intersection of the neighborhood and the hyperplane has to be over $U$.</footnote>
For any one of those couples $(a, g)$, we have

$$ \frac{|\langle \boldsymbol{\mu} + a\boldsymbol{\nu}, \boldsymbol{\tau} \rangle - \Im(g)|}{a} = \left|\langle \boldsymbol{\nu}, \boldsymbol{\tau} \rangle - \frac{\Im(g)}{a} + \frac{\langle \boldsymbol{\mu}, \boldsymbol{\tau} \rangle}{a}\right| \le \left|\langle \boldsymbol{\nu}, \boldsymbol{\tau} \rangle - \frac{\Im(g)}{a}\right| + \frac{|\langle \boldsymbol{\mu}, \boldsymbol{\tau} \rangle|}{M_2} \le \varepsilon. $$

Therefore, for any of those $(a, g)$, and taking $\eta \in \mathbb{R}^k$ defined by

$$ \eta = \left( \Im(m(g)) - \langle \boldsymbol{\mu} + a\boldsymbol{\nu}, \boldsymbol{\tau} \rangle \right) \frac{\boldsymbol{\mu} + a\boldsymbol{\nu}}{\|\boldsymbol{\mu} + a\boldsymbol{\nu}\|^2} $$

we obtain $\langle \boldsymbol{\mu}+a\boldsymbol{\nu}, \boldsymbol{\tau}+\eta \rangle = \Im(m(g))$ and $\|\eta\| \le |\langle \boldsymbol{\mu}+a\boldsymbol{\nu}, \boldsymbol{\tau} \rangle - \Im(m(g))|/\|a\boldsymbol{\nu}+\boldsymbol{\mu}\| \le 2\varepsilon/\|\boldsymbol{\nu}\|$ since $\|\boldsymbol{\nu} + \frac{1}{a}\boldsymbol{\mu}\| \ge \frac{1}{2}\|\boldsymbol{\nu}\|$.

Finally, for any neighborhood of the form $V_R + iV_I$ of $s$, there are infinitely many $(a, g) \in S'$ such that $V_I \cap H_{\mu+a\nu}^I(g)$ is not empty, and for all of them $V_R \cap H_{\mu+a\nu}^R(g)$ is not empty either. Therefore the neighborhood intersects infinitely many non-recurring hyperplanes, that is infinitely many primitive hypersurfaces of the divisor of $f$, and since this is true for any neighborhood, $s$ is singular for $f$.

Now, assume that $r > 0$, and that $\boldsymbol{\mu}$ is not collinear to $\boldsymbol{\nu}$. Let $\mathcal{F}_{\boldsymbol{\nu}}$ be the intersection of $\bar{\Gamma}$ and the hyperplane of equation $\Re\langle\boldsymbol{\nu}, s\rangle = 0$. When possible, consider $s$ in the relative interior $\mathcal{F}_{\boldsymbol{\nu}}$ verifying $\Re\langle\boldsymbol{\nu}, s\rangle = 0$ and $\Re\langle\boldsymbol{\mu}, s\rangle > r > 0$. Write $s$ as $x + iy$. Since $\Gamma$ is convex and tubular, we have for any $\lambda \in ]0, 1]$ that the element $s_\lambda$ defined as $\lambda x + iy$ is also in the relative interior $\mathcal{F}_{\boldsymbol{\nu}}$. We have $\Re\langle\boldsymbol{\mu}, s_\lambda\rangle = \lambda\Re\langle\boldsymbol{\mu}, s\rangle$, and we can choose $\lambda \in ]0, 1]$ such that $\Re\langle\boldsymbol{\mu}, s_\lambda\rangle \le r$. We have just proved that $s_\lambda$ is a singular point of $f$. By Theorem 4.14 (with the “meromorphy” wording), the point $s$ is also singular for $f$. And since the set of singular points of $f$ is closed, any point in $\mathcal{F}_{\boldsymbol{\nu}}$ is a singular point of $f$. $\square$

A direct application of this proposition gives our following maximality result.

**Theorem 4.4.** — Let $h \in \mathbb{Z}[[X]]$ be a rational power series with $h(0) = 1$. The domain $\Gamma$ is the maximal domain of meromorphy of $f$, in the sense that $f$ is meromorphic on $\Gamma$ and any point on the boundary of $\Gamma$ is a singular point of $f$.

Let us recall that $h$ is the quotient $q^+/q^-$ of two coprime polynomials in $\mathbb{Z}[X]$ with $q^+(0) = q^-(0) = 1$. We assume without loss of generality that neither $q^+$ nor $q^-$ admits any cyclotomic factors. The domain $\Gamma$ is described as

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

where $B^+$ and $B^-$ are the supports of $q^+$ and $q^-$, respectively. Every supporting hyperplane of $\bar{\Gamma}$ is defined by $\Re\langle\boldsymbol{\nu}, s\rangle = 0$, where $\boldsymbol{\nu}$ is a primitive vector of $\mathbb{N}^k$ directing an extreme ray of the closed convex cone generated by $B^+ \cup B^-$.

The theorem derives from Corollary 4.3 if for any such $\boldsymbol{\nu}$ we are able to find a vector $\boldsymbol{\mu} \in \mathbb{N}^k$ and a subset $S \subseteq \mathcal{P}_{\boldsymbol{\nu}, \boldsymbol{\mu}}^*$ that verifies the condition of the corollary. The construction of such subset $S$ of parameter pairs changes greatly according to the arithmetic properties of $q_\nu^+$ and $q_\nu^-$ and each case will be treated in a different subsection.

In section 4.1, we will assume that $q^+$ and $q_\nu^+$ (or: $q^-$ and $q_\nu^-$) have a non trivial common factor. This common factor will help to describe the part of $(f)_\nu$ stemming
from the $(h_p)_\nu$. Using the argument of Estermann, we will show in Lemma 4.6 that some subset $S \subseteq \mathcal{P}_\nu^*$ verifies the conditions of Corollary 4.3 (with $\mu = \mathbf{0}$ and $r > 0$).

In section 4.2, we assume that $q^+$ and $q_\nu^+$ (and: $q^-$ and $q_\nu^-$) are coprime, and that $h_\nu = q_\nu^+/q_\nu^-$ cannot be written as a quotient of products of cyclotomic polynomials. Using the argument of Dahlquist, we will show in Lemma 4.7 that some subset $S \subseteq \mathcal{P}_\nu^*$ verifies the conditions of Corollary 4.3 (with $\mu = \mathbf{0}$ and $r = 0$).

In section 4.3, we assume that $q^+$ and $q_\nu^+$ are coprime, that $q^-$ and $q_\nu^-$ too, and that $h_\nu = q_\nu^+/q_\nu^-$ can be written as a quotient of products of cyclotomic polynomials. Using Proposition 3.22, we will show in Lemma 4.7 that some subset $S \subseteq \mathcal{P}_{\nu,\mu}^*$ with $\mu \in \mathbb{N}^k$ verifies the conditions of Corollary 4.3 (with $\mu$ not collinear to $\nu$ and $r = \frac{1}{2}$).

The section 4.4 is devoted to the proof of Theorem 4.14, which describes the structure of singular points in the boundary of a tubular domain of holomorphy.

**4.1. Estermann's argument.** — Set in the original case [Est28, Theorem 4] of an univariate polynomial $h$, Estermann's argument consists in estimating the rate of accumulation of zeros/poles of $f$ associated to each root of $h$, as well as the rate of accumulation of zeros associated to the zeta factors. A direct comparison of their rates shows that these zeros/poles of $f$ cannot all cancel together as long as $h$ has a root inside the unit circle.

In our setting, we have seen that $\mathcal{P}_\nu$ is the disjoint union of several families of parameter pairs $A_\nu \times G' \cup \bigcup_{\theta \in \Theta_\nu} A' \times G_\theta$. The rate of accumulation associated to each family is obtained by computing the size of the intersection of the family with $B_{M,I} = \{(a,g) \in \mathbb{R}_+^* \times \mathbb{C} \mid g/a \in [1/M, +\infty[ \cup [iI, +\infty)\},$ where $M$ is a large real number and $I$ is an interval of $\mathbb{R}$ of length $|I|$. But this argument will work as long as $\Theta_\nu$ is finite and not empty. The following lemma translates this condition in term of factors of $q^+$ and $q^-$.

**Lemma 4.5.** — For both choices of the sign $\pm$, we define $s^\pm \in \mathbb{Z}[T]$, where $T = X^\nu$, as the $\gcd$ of $q^+$ and $q_\nu^+$. Then, the elements of the set $\Theta_\nu$ are the roots $\theta \in \mathbb{C}$ of $s^+(T)$ and those of $s^-(T)$ which lie inside the unit circle.

In particular, $\Theta_\nu$ is always finite, and it is empty if and only if $s^+$ and $s^-$ are the constant polynomial equal to 1.

Note that when $k \ge 2$, it is not a generic property for a polynomial to have a factor of the form $X^\nu - \theta$ for some fixed $\nu \in \mathbb{N}^k$. Dahlquist's argument is much more needed in the multivariate setting.

*Proof.* — Let $\theta \in \Theta_\nu$. Therefore $X^\nu - \theta$ divides either $q^+$ or $q^-$. Without loss of generality, we can assume that $X^\nu - \theta$ divides $q^+$. By Lemma 3.3, it also divides $q_\nu^+$. We deduce that $T - \theta$ divides $s^+$. Conversely, if $\theta$ is a root of $s^+$ with $|\theta| < 1$, we have that $X^\nu - \theta$ divides $s^+$, which is a divisor of $q^+$. Therefore $\theta \in \Theta_\nu$.

If $s^+ = 1$ and $s^- = 1$, then $\Theta_\nu$ is empty. Otherwise, $s^+s^-$ is a non-constant polynomial of $\mathbb{Z}[T]$ with $s^+(0)s^-(0) = 1$, and without cyclotomic factors (since $s^+s^-$ divides $q^+q^-$). Therefore $s^+s^-$ has at least a root $\theta$ with $|\theta| < 1$ and $\Theta_\nu$ is not empty. The set $\Theta_\nu$ is always finite as the zero set of the polynomial $s^+s^-$. $\square$

For $\theta \in \Theta_\nu$, the family $A' \times G_\theta$ consists of all pairs $(\log p, -\log \theta + 2\pi i\kappa)$ where $p$ is a prime number and $\kappa$ is an integer.
**Fact 1.** Any parameter pair of $A' \times G_\theta$ cannot cancel another one.

That is: for two pairs $(a, g)$ and $(a', g')$ in $A' \times G_\theta$, if $g/a = g'/a'$ then $(a, g) = (a', g')$. In fact, since all $g$ have the same real part, the equality $g/a = g'/a'$ implies that $a = a'$ and therefore $g = g'$.

The same argument works as long as the real part of the second coordinates are all the same.

**Fact 1'.** Let $\theta \neq \theta'$ with $|\theta| = |\theta'|$. Any parameter pair of $A' \times G_\theta$ cannot cancel any parameter pair of $A' \times G_{\theta'}$.

**Fact 2.** Let $\theta \neq 0$ with $|\theta| < 1$. We have card $((A' \times G_\theta) \cap B_{M,I}) \sim \frac{|I|}{2\pi}|\theta|^{-M}$ as $M$ goes to $+\infty$.

The condition $\Re(e(g/a)) \geq 1/M$ translates to $p \leq |\theta|^{-M}$. And for each $p$, there are $\frac{|I|}{2\pi} \log p + O(1)$ possible choices for the imaginary part of $g$. The prime number theorem states that $\sum_{p \leq x} \log p \sim x$ as $x$ goes to $+\infty$.

**Fact 3.** We have card $((A_\nu \times G') \cap B_{M,I}) \ll M^2 \log M$ as $M$ goes to $+\infty$.

Since no zero or pole of the Riemann $\zeta$ function have a real part larger than 1, the condition $\Re(e(g/a)) \geq 1/M$ implies that $a \leq M$. For such $a$, the Riemann-von Mangoldt formula indicates that there are at most $\frac{|I|}{2\pi}a(\log a + O(1))$ possible choices for $g$ as zero of $\zeta$ with an imaginary part in $aI$.

These simple facts are sufficient to carry the argument of Estermann in the multi-variate setting.

**Lemma 4.6.** — If $q^+$ and $q_\nu^+$ are not coprime, or if $q^-$ and $q_\nu^-$ are not coprime, then there exist $\theta \in \Theta_\nu$ and $S \subset A' \times G_\theta$ such that $S \subseteq \mathcal{P}_\nu^*$, $S$ verifies the vertical accumulation property, and for any $(a, g) \in S$, $\Re(e(g)) = -\log|\theta| \in \mathbb{R}_+^*$.

*Proof.* — By Lemma 4.5, the set $\Theta_\nu$ is finite and not empty. Since $q^+(0) = q^-(0) = 1$, this set does not contain $0$, and by its finiteness, we choose a $\theta$ in $\Theta_\nu$ with minimal modulus.

The third part of the assertion derives directly from $G_\theta = \{-\log \theta + 2\pi i \kappa \mid \kappa \in \mathbb{Z}\}$.

We set $S \subseteq A' \times G_\theta \subseteq \mathcal{P}_\nu$ as

$$S = A' \times G_\theta \setminus (A_\nu \times G' \cup \bigcup_{\substack{\theta' \in \Theta_\nu \\ |\theta'| > |\theta|}} A' \times G_{\theta'}).$$

Any element of this set does not cancel with another parameter pairs of $A' \times G_\theta$ by Fact 1, with another parameter pairs of $A' \times G_{\theta'}$ with $|\theta'| = |\theta|$ by Fact 1', with another parameter pairs of $A' \times G_{\theta'}$ with $|\theta'| \neq |\theta|$ or of $A_\nu \times G'$ by construction. Therefore it is subset of $\mathcal{P}_\nu^*$.

It remains to prove the second part of the assertion: any point of the imaginary line $i\mathbb{R}$ is a cluster point of the family $\{g/a \mid (a, g) \in S\}$. We first show that for any bounded interval $I$, $\text{card}(S \cap B_{M,I}) \to +\infty$ as $M$ goes to $+\infty$.

A crude estimate shows that $\text{card}(S \cap B_{M,I})$ is larger than

$$\text{card}((A' \times G_\theta) \cap B_{M,I}) - \text{card}((A_\nu \times G') \cap B_{M,I}) - \sum_{\substack{\theta' \in \Theta_\nu \\ |\theta'| > |\theta|}} \text{card}((A' \times G_{\theta'}) \cap B_{M,I}).$$
The first term is $\sim \frac{|I|}{2\pi}|\theta|^{-M}$ as $M$ goes to $+\infty$ by Fact 2. The second and third terms are negligible by Facts 2 and 3, and by the finiteness of $\Theta_\nu$. Altogether, we have proved that $\text{card}(S \cap B_{M,I}) \sim \frac{|I|}{2\pi}|\theta|^{-M}$ as $M$ goes to $+\infty$.

Therefore any interval $iI$ of $i\mathbb{R}$ contains at least one cluster point of $\{g/a \mid (a,g) \in S\}$. This means that the set of cluster points is dense in $i\mathbb{R}$, and since it is a closed set, we obtain the vertical accumulation property for $S$. $\square$

By Corollary 4.3, any point of the facet $F_\nu$ of $\bar{\Gamma}$ is a singular point of $f$, at least if $q^+$ and $q_\nu^+$ are not coprime, or if $q^-$ and $q_\nu^-$ are not coprime.

**4.2. Dahlquist's argument.** — Estermann's argument is efficient as long as $\Theta_\nu$ is not empty. For a rational function $h$ of one variable, this emptiness is equivalent to the fact that $h$ is a quotient of products of cyclotomic polynomials. But in several variables, this is no longer the case: it is frequent that $\Theta_\nu$ is empty but $h_\nu$ is not a quotient of products of cyclotomic polynomials, which is exactly when Dahlquist's argument is useful.

Let us assume in that subsection that $q^+$ and $q_\nu^+$ (and: $q^-$ and $q_\nu^-$) are coprime, and that $h_\nu$ is not a quotient of products of cyclotomic polynomials. We deduce that $P_\nu = A_\nu \times G'$ by Lemma 4.5, and that $A_\nu$ is an infinite subset of $\mathbb{N}$ by Lemma 3.1'. In that case, the vertical accumulation property is easily verified for $P_\nu$, but $P_\nu^*$ may differ from $P_\nu$. If the elements of $G' \setminus \{1\}$ have the same real part (which is Riemann Hypothesis), then $P_\nu^* = P_\nu$. The linear independence (over $\mathbb{Q}$) of the positive ordinates of non trivial zeros of the Riemann $\zeta$ function would also provide this equality.

Without this knowledge about the zeros of the Riemann $\zeta$ function, Dahlquist constructs a subset $A^* \subseteq A_\nu$ and for each $a \in A^*$ a subset $G_a \subseteq G'$ such that:

I- for any $a \in A^*$ and any $g \in G_a$, the parameter pair $(a,g)$ is in $P_\nu^*$.

II- for any $\varepsilon > 0$, and for any $T$ larger than some $T_0$ depending on $\varepsilon$, each set $G_a$ with $a \in A^*$ has at least one element in the interval $[(1-\varepsilon)T, T]$ (and one in the interval $[-T, -T(1-\varepsilon)$]),

III- the set $A^*$ is infinite.

We will follow the argument of Dahlquist, but in modern and geometric terms.

**Lemma 4.7.** — If $q^+$ and $q_\nu^+$ (and: $q^-$ and $q_\nu^-$) are coprime, and $h_\nu$ is not a quotient of products of cyclotomic polynomials, then there exists $S \subset A_\nu \times G'$ such that $S \subseteq P_\nu^*$, $S$ verifies the vertical accumulation property, and for any $(a,g) \in S$, $\text{Re}(g) \in [0, 1]$.

*Proof.* — The third part of the assertion comes directly from the fact that non-trivial zeros of $\zeta$ are located in the critical strip.

For the first two parts of the assertion, it suffices to prove that $S \subset \mathbb{N} \times G'$ verifies the three properties I, II and III above.

Property I asserts that $S \subseteq P_\nu^*$, which is the first part of the assertion. It remains to show that $S$ has the vertical accumulation property.

Let $x \in \mathbb{R}^*$ and $\varepsilon > 0$ and consider the intersection of $\{g/a \mid a \in A^*, g \in G_a\}$ and $\{s \in \mathbb{C} \mid \text{Im}(s) \in [(1-\varepsilon)x, x]\}$ (or $[x, (1-\varepsilon)x]$ if $x < 0$). By property I, all the ratios $g/a$ are distinct, and this intersection has the same cardinality as $\bigsqcup_{a \in A^*} \{g \in G_a|$
$\mathfrak{S}m(g) \in [(1 - \varepsilon)ax, ax]}$. Since $A^* \subseteq \mathbb{N}$ is infinite by property III, there are infinitely many $a \in A^*$ with $a|x| > T_0$ and for each one of them the set $\{g \in G_a | \mathfrak{S}m(g) \in [(1 - \varepsilon)ax, ax]}$ is not empty, by property II. Therefore $\{g/a | a \in A^*, g \in G_a\} \cap \{s \in \mathbb{C} | \mathfrak{S}m(s) \in [(1 - \varepsilon)x, x]\}$ is an infinite bounded set, thus it has at least one cluster point, which can only be on the imaginary line with imaginary part in $[(1 - \varepsilon)x, x]$. Since the result is true for every $x \in \mathbb{R}^*$ and any $\varepsilon > 0$, we conclude that each point of the imaginary line $i\mathbb{R}$ is a cluster point of $\{g/a | a \in A^*, g \in G_a\}$. Therefore, $\mathcal{S}$ verifies the vertical accumulation property.

It remains to prove that such a set of parameters $\mathcal{S}$ can be constructed. The details are presented in the lemmas below.
We define

$$A^* = \{a \in A_\nu \mid \forall r \ge 2, \forall (a_1, \dots, a_r) \in (A_\nu \setminus \{a\})^r, \prod_{i=1}^r a_i = a^r \Rightarrow \max_{1 \le i \le r} a_i \ge 2a\}$$

and we set $\mathcal{L}$ as the set of lines $L = \gamma\mathbb{R} \subset \mathbb{C}$ with $\gamma \in \mathbb{C}^*$ such that the intersection $L \cap G'$ is not empty and is a subset of a segment $]\frac{1}{2}\gamma t, \gamma t]$ with $t \in \mathbb{R}$.

Lemma 4.9 proves that for each $a \in A^*$ there is a choice function of $\mathcal{L}$ of image $G_a \subseteq G'$ such that the property I is verified for $S = \cup_{a \in A^*} \{a\} \times G_a$. The property II is a direct consequence of the Lemma 4.8. To prove property III, we apply Lemma 4.10 to the set $A_\nu$, which is infinite by Lemma 3.1'. $\square$

We recall Lemma 3.1 from [Dah52], which only relies on basic properties of the zeros of the Riemann zeta function.

**Lemma 4.8.** — For every $\varepsilon > 0$ there is a $T_0$ such that for each $T > T_0$ there is a line $L = \gamma\mathbb{R} \subset \mathbb{C}$ with $\gamma \in \mathbb{C}^*$ such that $L \cap G'$ is not empty and every point of $L \cap G'$ has its imaginary part in $[(1 - \varepsilon)T, T]$.

**Lemma 4.9.** — Consider a line $L = \gamma\mathbb{R} \subset \mathbb{C}$ with $\gamma \in \mathbb{C}^*$ such that $L \cap G'$ is not empty and every point of $L \cap G'$ has its imaginary part in $]T/2, T]$. Let $a \in A_\nu$ such that any representation of $a^r$ as a product of $r$ elements of $A_\nu \setminus \{a\}$ involves an element of $A_\nu$ greater than or equal to $2a$, for any $r \ge 2$.

Then, there is at least one element $g \in L \cap G'$ such that the hyperplane $H_{a\nu}(g)$ is non-recurring.

*Proof.* — Since $G'$ is discrete and the real parts of its elements are bounded, the set $L \cap G'$ is finite.

Assume that for $g \in L \cap G'$, the hyperplane $H_{a\nu}(g)$ is recurring. Then, there exists $(a', g') \in A_\nu \times G'$ with $(a', g') \ne (a, g)$ but $g'/a' = g/a$. Of course $g' \ne g$ and $g' = a'/a g \in L$ therefore $g' \in L \cap G'$.

If every hyperplane $H_{a\nu}(g)$ with $g \in L \cap G'$ is recurring, then there exist two functions $\sigma : L \cap G' \to L \cap G'$ and $\tau : L \cap G' \to A_\nu \setminus \{a\}$ such that for any $g \in L \cap G'$, we have $g/a = \sigma(g)/\tau(g)$. Notice that $\sigma$ has no fixed point. Since $L \cap G'$ is finite, there is a $g_0 \in L \cap G'$ and an integer $r \ge 2$ such that $\sigma^r(g_0) = g_0$. For any $i$ between 1 and $r$ we set $g_i = \sigma(g_{i-1})$ and $a_i = \tau(g_{i-1})$ so that $g_{i-1}/a = g_i/a_i$. In particular $g_r = g_0$ but also $a^r g_r = (\prod_{i=1}^r a_i)g_0$.

We deduce that $a$ should verify an equation of the form $a^r = \prod_{i=1}^r a_i$ with $r \ge 2$ and $a_i \ne a$. According to the assumption on $a$, any such identity should involve an integer
$a_i \ge 2a$ for some $i$. But since $g_{i-1}$ and $g_i$ are elements of $L \cap G'$, their imaginary parts are both in $]T/2, T]$. We deduce the contradiction

$$2 \le \frac{a_i}{a} = \frac{\Im(g_i)}{\Im(g_{i-1})} < \frac{T}{T/2} = 2.$$

We conclude that at least one $g \in L \cap G'$ is such that the hyperplane $H_{a\nu}(g)$ is non-recurring. □

**Lemma 4.10.** — Let $A \subset \mathbb{N}$ be an infinite set of positive integers. There exists an infinite subset $A^*$ of $A$ such that any relation $a^k = b_1 \cdots b_k$ with $a \in A^*$ and $b_1, \dots, b_k \in A$ implies either $b_1 = \cdots = b_k = a$, or there is a $b_i \ge 2a$.

In [Dah52, Lemma 3.2], a variation of the following statement (related to the concept of vertex number, developed therein) is given. We give a shorter proof using convex geometry, but relying on the same principles than the original proof. Doing so, we are able to generalize the previous lemma. Hereafter we denote as $\nu_p(a)$ the $p$-adic valuation of the integer $a$.

**Lemma 4.11.** — Let $A \subset \mathbb{N}$ be an infinite set of positive integers and $p_0$ a prime number such that $\prod_{p \ge p_0} p^{\nu_p(a)}$ is not bounded for $a \in A$.

There exists an infinite subset $A^*$ of $A$ such that any relation $a^k = b_1 \cdots b_k$ with $a \in A^*$ and $b_1, \dots, b_k \in A$ implies either $b_1 = \cdots = b_k = a$, or there is a $b_i \ge p_0a$.

The valuation map $a \mapsto (\nu_p(a))_{p \in \mathcal{P}}$ associates each positive integer to the sequence of its $p$-adic valuations (here $\mathcal{P}$ denotes the set of all prime numbers). This is a semigroup isomorphism between the multiplicative $\mathbb{N}^*$ and the additive semigroup $\mathbb{N}^{(\mathcal{P})}$. By this isomorphism, the relation $a^k = \prod_{i=1}^k b_i$ in $A$ is equivalent to the centroid relation $e = \frac{1}{k} \sum_{i=1}^k x_i$ between elements of $E = \{(\nu_p(a))_{p \in \mathcal{P}} | a \in A\}$.

We immerse the set $E$ in the $\mathbb{R}$-vector space $V = \mathbb{R}^{(\mathcal{P})}$. A linear form $\phi: V \to \mathbb{R}$ is defined by the family $(c_p)_{p \in \mathcal{P}}$ of its coefficients, that is the image of the basis vectors of $V$. Explicitly, $\phi(x) = \sum_{p \in \mathcal{P}} c_p x_p$ for $x = (x_p)_{p \in \mathcal{P}} \in V$. A particular meaningful linear form is the linear form $\Lambda$ of coefficients $(\log p)_{p \in \mathcal{P}}$. This form $\Lambda$ precomposed with the aforementioned valuation map corresponds to the logarithm over $\mathbb{N}^*$, since $\log a = \sum_{p \in \mathcal{P}} (\log p) \nu_p(a)$ for any $a \in \mathbb{N}^*$. Finally, for a given prime $p_0$, let $\pi_0$ the projection $(x_p)_{p \in \mathcal{P}} \mapsto (x_p)_{p \ge p_0}$ whose kernel is generated by the basis vectors of $\mathbb{R}^{(\mathcal{P})}$ corresponding to primes less than $p_0$. We are now able to state the previous lemma in this new setting.

**Lemma 4.11'.** — Let $E \subset \mathbb{N}^{(\mathcal{P})}$ be an infinite set, $p_0 \in \mathcal{P}$ and $\pi_0$ the associated projection. Assume that $\pi_0(E)$ is also infinite. Then, there exists an infinite subset $E^*$ of $E$ such that any centroid relation $e = \frac{1}{k} \sum_{i=1}^k x_i$ with $e \in E^*$ and $x_1, \dots, x_k \in E$ implies either $x_1 = \cdots = x_k = e$, or there is a $x_i$ with $\Lambda(x_i) \ge \log p_0 + \Lambda(e)$.

**4.2.1. Convex geometry.** — If we consider the convex hull of $E$ in $V$, any extreme point $e$ of it is guaranteed to be in $E^*$, since the centroid relation $e = \frac{1}{k} \sum_{i=1}^k x_i$ has no nontrivial solution $x_1, \dots, x_k \in E$. But the set of extreme points is not always infinite.

To get around this limitation, we soften our assumption of extreme points. Roughly speaking, an extreme point strictly minimizes a (well-chosen) linear form over $E$,
that is conv(E) meets the corresponding half-space only at this particular point. We consider weak extreme points(*), at which a given wedge (intersection of two half-spaces, that is defined by two linear forms $\phi_+$ and $\phi_-$) does not meet E. Choosing $\phi_+$ and $\phi_-$ wisely, we can ensure the infiniteness of these weak extreme points.

However, at the weak extreme points, the centroid equation may have non trivial solutions. Later, we will choose the forms $\phi_+$ and $\phi_-$ such that non trivial solutions of the centroid equation verify the property stated in Lemma 4.11'.

**Lemma 4.12.** — Let V be a $\mathbb{R}$-vector space and E an infinite subset of V. Assume that there are two linear forms $\phi_+$ and $\phi_-$ on V such that $\phi_+(E)$ is a discrete subset of $\mathbb{R}_+$, and $\phi_-(E)$ is not bounded by below. Set W as the wedge $\{x \in V \mid \phi_-(x) \le 0 \text{ and } \phi_+(x) < 0\}$. Then, there exists an infinite sequence $(e_n)_{n \in \mathbb{N}}$ of distinct elements of E such that $(e_n + W) \cap E$ is empty for every $n \in \mathbb{N}$.

Indeed, we can soften the conditions on $\phi_+(E)$ and $\phi_-(E)$ to $\phi_+(E)$ is well-ordered (namely, any non-empty subset has a minimum) and $\phi_-(E)$ doesn't have a minimum. These conditions only use the intrinsic order of $\phi_+(E)$ and $\phi_-(E)$, and not the topology of $\mathbb{R}$, but the construction of the proof still holds.

*Proof.* — We will construct, for any $e \in E$, an element $e' \in E$ with $\phi_-(e') < \phi_-(e)$ and $(e' + W) \cap E = \emptyset$. By iteration of this process, the sequence $(e_n)_{n \in \mathbb{N}}$ can be produced, with the property that $\phi_-(e_{n+1}) < \phi_-(e_n)$ for all $n \in \mathbb{N}$. In particular, the terms of this sequence are all distinct.

Choose $e \in E$ arbitrarily. Since $\phi_-(E)$ is not bounded by below, the set $E_0$ of elements $x \in E$ with $\phi_-(x) < \phi_-(e)$ is infinite, therefore not empty. The set $\phi_+(E_0) \subseteq \phi_+(E)$ is a non-empty discrete subset of $\mathbb{R}_+$, thus it has a minimum, and there is at least an element $e' \in E_0$ realizing this minimum. By construction, no element $x \in E_0$ verifies $\phi_+(x) < \phi_+(e')$, thus no element $x \in E$ verifies both $\phi_-(x) < \phi_-(e)$ and $\phi_+(x) < \phi_+(e')$. This means that $(e' + W) \cap E$ is empty. $\square$

Actually, this construction (see Figure 5) gives a slightly wider region without element of E: For any $n \in \mathbb{N}$, no element $x$ of E verifies both $\phi_-(x) < \phi_-(e_n)$ and $\phi_+(x) < \phi_+(e_{n+1})$.

**4.2.2. Diophantine approximation.** — While the geometric intuition suggests to pick $\phi_+$ near $\phi_-$ to widen the wedge, it turns out that the quality of the inequality depends more on the choice of $\phi_+$.

The linear forms $\phi_+$ and $\phi_-$ are chosen as linear combination of $\Lambda$ (since we want an inequality involving $\Lambda$) and a well-chosen form with integer coefficients, allowing to use the basic trick of diophantine approximation: if $x > 0$ and $x \in \mathbb{Z}$, then $x \ge 1$. *Proof of Lemma 4.11'.* — Let $N_0$ be the linear form for which the family of coefficients $(c_p)_{p \in \mathcal{P}}$ is defined by:

$$c_p = \begin{cases} 0, & \text{if } p < p_0, \\ 1, & \text{if } p = p_0, \\ \lfloor \log p / \log(p_0 + 1) \rfloor, & \text{if } p > p_0. \end{cases}$$

<footnote>(*)which correspond to the vertex numbers in [Dah52, Definition 3.2]</footnote>
<figure><img src="image_10.png" /><figcaption>FIGURE 5. Linear projection of E on a plane (complementary to $\ker \phi_+ \cap \ker \phi_-$) and construction of the sequence $(e_n)_{n \in \mathbb{N}}$.</figcaption></figure>

Consider $\phi = \Lambda - (\log p_0)N_0$. The coefficients $(c'_p)_{p \in \mathcal{P}}$ of this linear form verify $c'_p = \log p$ if $p < p_0$, $c'_{p_0} = 0$ and

$$c'_{p} = \log p - \log p_{0} \left\lfloor \frac{\log p}{\log(p_{0} + 1)} \right\rfloor \geq \left(1 - \frac{\log p_{0}}{\log(p_{0} + 1)}\right) \log p,$$

if $p > p_0$. These coefficients are non-negative, so that $\phi(E) \subseteq \phi(N^{(\mathcal{P})}) \subseteq \mathbb{R}_+$, and they tend to $+\infty$ when $p$ goes to $+\infty$. In particular, for any $M > 0$, there exists $P$ such that $c'_p \geq M$ for any $p > P$. We deduce that for any $M > 0$

$$\phi(N^{(\mathcal{P})}) \cap [0, M] = \left( \sum_{p \in \mathcal{P}} c'_{p} \mathbb{N} \right) \cap [0, M] \subseteq \left( \sum_{p \leq P} c'_{p} \mathbb{N} \right) \cap [0, M] \subseteq \sum_{p \leq P} (c'_{p} \mathbb{N} \cap [0, M])$$

which is a finite sum of finite sets. This implies that $\phi(N^{(\mathcal{P})})$ and thus $\phi(E)$ are discrete subsets of $\mathbb{R}_+$.

The form $\Lambda - \log p_0 N_0$ satisfies the condition required for $\phi_+$ in Lemma 4.12.

Since $\lfloor t \rfloor \geq t/2$ for any $t \geq 1$, we have that

$$N_0(\mathbf{x}) = x_{p_0} + \sum_{p > p_0} c_p x_p > \frac{1}{2 \log(p_0 + 1)} \sum_{p \ge p_0} \log p x_p = \frac{1}{2 \log(p_0 + 1)} \Lambda(\pi_0(\mathbf{x})),$$
for any $x \in \mathbb{N}^{(\mathcal{P})}$. Since $\pi_0(E)$ is not finite, $\Lambda(\pi_0(x))$ is not bounded by above, and $-N_0(E)$ is not bounded by below.

Therefore $-N_0$ satisfies the condition for $\phi_-$ in Lemma 4.12.

Set $\phi_+ = \Lambda - (\log p_0)N_0$ and $\phi_- = -N_0$. Using these two linear forms in Lemma 4.12, we get a sequence of distincts points $e_n \in E$ such that for every $x$ of $E$ and every $n \in \mathbb{N}$, if $\phi_-(x) \le \phi_-(e_n)$ then $\phi_+(x) \ge \phi_+(e_n)$.

Let $(x_i)_{i \in I}$ be a finite family of elements of $E$ whose centroid is $e_n$. Then either $\phi_-(x_i) = \phi_-(e_n)$ for every $i \in I$, or there exists some $i \in I$ with $\phi_-(x_i) < \phi_-(e_n)$.

In the first case, we also have that $\phi_+(x_i) \ge \phi_+(e_n)$ for any $i \in I$, therefore $\phi_+(x_i) = \phi_+(e_n)$ for $i \in I$. Since $\Lambda$ can be written as $\phi_+ - \log p_0 \phi_-$, we get $\Lambda(x_i) = \Lambda(e_n)$ and since the restriction of $\Lambda$ on $\mathbb{N}^{(\mathcal{P})}$ is injective by the fundamental theorem of arithmetic, we get $x_i = e_n$ for every $i \in I$.

In the latter case, since $\phi_-(x_i) < \phi_-(e_n)$, we also have that $\phi_+(x_i) \ge \phi_+(e_n)$. Since $N_0(x_i - e_n) = \phi_-(e_n - x_i) > 0$ and $N_0(x_i - e_n) \in \mathbb{Z}$, we deduce that $N_0(x_i - e_n) \ge 1$. Then

$$ \Lambda(x_i) - \Lambda(e_n) = \phi_+(x_i) - \phi_+(e_n) + (\log p_0)N_0(x_i - e_n) \ge 0 + (\log p_0) \cdot 1 = \log p_0 $$

as stated. $\square$

<figure><img src="image_11.png" /></figure>

**4.3. The remaining case.** — In the previous cases, the set $\mathcal{P}_{\nu}^*$ verified the conditions of Corollary 4.3, because the set $\Theta_{\nu}$ of zeros and poles inside the unit circle was not empty or because the set of non-zero exponents (indexed by $A_{\nu}$) in $h_{\nu} = \prod_{n \ge 1} (1 - T^n)^{c(n\nu)}$ was not finite.

In the remaining case, we assume that $q^+$ and $q_{\nu}^+$ are coprime, that $q^-$ and $q_{\nu}^-$ are coprime, and that $h_{\nu}$ is a quotient of products of cyclotomic factors. This implies that $\Theta_{\nu}$ is empty by Lemma 4.5, and that $A_{\nu}$ is finite by Lemma 3.1'. In that case, the set $\mathcal{P}_{\nu} = A_{\nu} \times G'$ parameterizes a discrete set of hyperplanes. Hence, there cannot be an accumulation of parallel hyperplanes in $(f)$ along the face of $\Gamma$ of normal vector $\nu$. Of course, one can try to find an accumulation of analytic surfaces (in the divisor $(f)$) of larger complexity, and to some extent this is the strategy in [Del10, Del13].

Instead, we consider a family of hyperplanes $H_{\alpha}(\rho)$ with $\rho \in G'$ and where the normal vector $\alpha$ is in an affine line directed by $\nu$, that is $\alpha \in C \cap (\mu + \nu \mathbb{R}_+)$ with $C = \text{xsupp } h$ and $\mu$ non-collinear with $\nu$. Any pair of distinct normal vectors in this family are non-parallel. This will correspond to an accumulation of hyperplanes on the points of the corresponding face whose real part is sufficiently near the origin (see Figure 1, page 15).

**Lemma 4.13.** — If $q^+$ and $q_{\nu}^+$ are coprime, $q^-$ and $q_{\nu}^-$ are coprime, and $h_{\nu}$ is a quotient of products of cyclotomic factors, then there exist $\mu \in \mathbb{N}^k$ and $S = A_0 \times G_0$ with $A_0 \subseteq \{a \in \mathbb{N} \mid \mu + a\nu \in C\}$ and $G_0 \subseteq G'$ such that $S \subseteq \mathcal{P}_{\mu, \nu}^*$, $S$ verifies the vertical accumulation property, and for any $(a, g) \in S$, $\Re(e(g)) \in [\frac{1}{2}, 1]$.

Notice that the conditions on $q_{\nu}^+$ and $q_{\nu}^-$ do not mean that $q_{\nu}^+$ and $q_{\nu}^-$ are products of cyclotomic factors: they could have a non trivial common factor, which is not a product of cyclotomic polynomials, and which is coprime to $q^+$ and $q^-$.
Proof. — By Lemma 3.1', $C \cap \nu\mathbb{N}$ is finite, and the assertion (i) of Proposition 3.22 is not verified. Therefore, its assertion (ii) holds, and there exists $\mu \in \mathbb{N}^k$ such that

- the set $A_1 = \{a \in \mathbb{N} \mid \mu + a\nu \in C\}$ is infinite,

- the set $A_1$ is syndetic, that is there exist $a_0 \in \mathbb{N}$ and $d \ge 1$ such that for any $a \ge a_0$ the intersection $A_1 \cap ]a, a+d]$ is not empty,

- $\mu$ can be thought as “minimal” in the sense that there are only finite many elements of $C$ of the form $t\mu + x\nu$, with $t \in [0, 1[$ and $x \in \mathbb{R}_+$.

Notice that $\mu$ cannot be collinear to $\nu$.

To any parameter pair $(a,g) \in A_1 \times G'$ we associate the hyperplane $H_{\alpha_0}(g)$ with $\alpha_0 = \mu + a\nu$. By construction, this hyperplane appears in a term of our expression of the divisor of $f$

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

However it may occur in multiple terms. We distinguish three types of terms where the hyperplane may also appear(†):

I. Terms of the first sum $\sum_p (h_p)$.

II. Terms $H_{t\alpha_0}(tg)$ in the second sum with $t < 1$, $t\alpha_0 \in C$ and $tg \in G'$.

III. Terms $H_{t\alpha_0}(tg)$ in the second sum with $t > 1$, $t\alpha_0 \in C$ and $tg \in G'$.

We first construct a cofinite subset $A_0$ of $A_1$ such that no hyperplane corresponding to a parameter pair $(a,g) \in A_0 \times G'$ appears in (f) as a term of type I or II.

Let $(a,g) \in A_1 \times G'$, and assume that the hyperplane $H_{\alpha_0}(g)$ appears in the first sum $\sum_p (h_p)$. We denote by $d$ the gcd of the coordinates of $\alpha_0$, so that $\nu' = \frac{1}{d}\alpha_0$ is primitive. Thus the hyperplane $H_{\nu'}(g/d)$ appears in the divisor of $h_p$ for some $p$. By Lemma 4.1, the polynomial $q^+q^-$ can be factorized by the irreducible polynomial $X^{\nu' - p^{-g/d}}$. The set $C'$ of primitive vector $\nu'$ such that $q^+q^-$ is divisible by $X^{\nu' - \theta}$ for some $\theta \in \mathbb{C}$ is finite. And since $\mu$ and $\nu$ are not collinear, for each such $\nu' \in C'$, the relation $\alpha_0 = \mu + a\nu = d\nu'$ is verified for at most one couple $(a,d) \in A_1 \times \mathbb{N}$. Therefore only finitely many values of $a \in A_1$ are concerned with terms of type I.

Now assume that the hyperplane $H_{\alpha_0}(g)$ appears in the second sum with parameters $t\alpha_0 \in C$ and $tg \in G'$ with $t < 1$. By minimality of the chosen $\mu$, there are only finitely many elements of $C$ of the form $t\mu + x\nu$, with $t \in [0, 1[$ and $x \in \mathbb{R}_+$. For each such $t$ and $x$, there is at most one element $a \in A_1$ such that $\mu + a\nu$ is collinear to $t\mu + x\nu$ ($a$ must verified $ta = x$). Therefore only finitely many values of $a \in A_1$ are concerned with terms of type II.

Set $A_0$ as the set of elements of $A_1$ which are not in one of the two finite sets described above. Since $A_0$ is cofinite in $A_1$, $A_0$ is infinite and syndetic.

We now construct an infinite subset $G_0$ of $G'$ such that no hyperplane corresponding to a parameter pair $(a,g) \in A_1 \times G_0$ also appears in (f) as a term of type III.

We also want that for any $g \in G_0$, $\Re(e(g)) \ge \frac{1}{2}$. For this matter, we define $G''$ as $\{\gamma \in G' \mid \Re(e(\gamma)) \ge \frac{1}{2}\}$, and we construct $G_0$ as a subset of $G''$. Since there are

<footnote>(†)Obviously, types II and III occur only if the Riemann Hypothesis is not verified, which we cannot yet rule out.</footnote>
infinitely many zeros of $\zeta$ with real part $\ge 1/2$, $G''$ is infinite. It is also closed under complex conjugation.

The key argument is that there exists some $\eta > 1$ such that for any $a \in \mathbb{Z}$ and any $t > 1$ such that $t(\boldsymbol{\mu} + a\boldsymbol{\nu}) \in \mathbb{Z}^k$ then $t \ge \eta$.

Consider the 2-dimensional lattice $\mathbb{Z}^k \cap \text{vect}_{\mathbb{R}}\{\boldsymbol{\nu}, \boldsymbol{\mu}\}$. Since $\boldsymbol{\nu}$ is primitive, we can find $\boldsymbol{\nu}'$ in this lattice such that $(\boldsymbol{\nu}, \boldsymbol{\nu}')$ is a basis of it. The vector $\boldsymbol{\mu}$ can be written as $d\boldsymbol{\nu} + d'\boldsymbol{\nu}'$ with $d, d' \in \mathbb{Z}$. Notice that $|d'|$ does not depend on the choice of $\boldsymbol{\nu}',$ but only from the choice of $\boldsymbol{\nu}$ and $\boldsymbol{\mu}$. For any $a \in \mathbb{Z}$, if the vector $t(\boldsymbol{\mu} + a\boldsymbol{\nu})$ is in $\mathbb{Z}^k$, then it is in that lattice, and its coordinates with respect to that basis are $(ta + td, td')$. Therefore $td'$ is an integer, and since $t > 1$, we have $t \ge 1 + 1/|d'|$.

Setting $\eta = 1 + 1/|d'| > 1$, we partition the interval $[0, 1]$ into $\bigcup_{n \in \mathbb{N}} I_n$ with $I_n = ]\eta^{-(n+1)}, \eta^{-n}]$ and define $G_n''$ as $\{g \in G'' | \Re(e(g)) \in I_n\}$. If $\eta^n > 2$, $G_n''$ is empty. The infinite set $G''$ can be written as the finite union of $G_n''$ with $n$ such that $\eta^n \le 2$. There is at least one set $G_n''$ which is infinite. We set $n_1$ as the minimal $n$ such that $G_n''$ is infinite, and define

$$G_1 = G_{n_1}'' \quad \text{and} \quad G_2 = \bigcup_{n<n_1} G_n'' = \{g \in G' | \Re(e(g)) > \eta^{-n_1}\}.$$

By construction, the set $G_1$ is infinite, and by minimality of $n_1$, the set $G_2$ is finite. Both sets are closed under complex conjugation. Finally, we set

$$G_0 = G_1 \setminus \{rg | r \in [0, 1], g \in G_2\}.$$

As a subset of $G'$, $G_1$ is discrete, and the complemented term is compact, therefore $G_0$ is cofinite in $G_1$. We deduce that $G_0$ is infinite and closed under complex conjugation. Let $(a, g) \in A_1 \times G_0$, and assume that its associated hyperplane $H_{\alpha_0}(g)$ also appears in $(f)$ as $H_{t\alpha_0}(tg)$ with $t > 1$, $t\alpha_0 \in C$ and $tg \in G'$. We set $g' = tg$. Since $t\alpha_0 \subseteq \mathbb{Z}^k$, we have that $t \ge \eta$. Since $g \in G_0 \subseteq G_{n_1}'$, we have that $\Re(e(g)) > \eta^{-(n_1+1)}$. We deduce that $\Re(e(g')) > \eta^{-n_1}$, therefore $g' \in G_2$. Finally, $g$ can be written as $rg'$ with $r = 1/t \in [0, 1]$ and $g' \in G_2$, which is impossible by construction of $G_0$.

Therefore, no hyperplane $H_{\alpha_0}(g)$ corresponding to $(a, g) \in A_1 \times G_0$ also appears in $(f)$ as a term of type III.

Putting all these properties together, we have proved that $A_0 \times G_0$ is a set of non-recurring pairs in $\mathcal{P}_{\boldsymbol{\mu},\boldsymbol{\nu}}^*$, and that for any $(a, g) \in A_0 \times G_0$, $\Re(e(g)) \in [\frac{1}{2}, 1]$. It remains to prove that this set have the vertical accumulation property, namely that for any $x \in \mathbb{R}$, $ix$ is a cluster point of $\{g/a | a \in A_0, g \in G_0\}$.

Since $G_0$ is closed under complex conjugation, we may consider only $x > 0$ (the case $x = 0$ is trivial since $A_0$ is infinite).

Actually, we show the vertical accumulation property for a subset $A_0 \times G_0'$ where

$$G_0' = \{g \in G_0 | \forall g' \in G_0, \forall r \in \mathbb{R}_+, rg' = g \Rightarrow r \le 1\},$$

so that each real line $\gamma \mathbb{R} \subseteq \mathbb{C}$ which intersects $G_0$ intersects $G_0'$ in exactly one point, the furthest from the origin. Note that the $\gamma \mathbb{R} \cap G_0$ is finite for any $\gamma \in \mathbb{C}^*$, which implies that $G_0'$ is an infinite set. Just like $G_0$, it is also discrete and closed under complex conjugation. Since the real parts of its elements are $[0, 1]$, their imaginary
parts are not bounded by above. This set has been constructed so that the function $\rho : (a, g) \mapsto g/a$ defined over $A_0 \times G'_0$ is one-to-one.

Fix $x \in \mathbb{R}_+^*$ and $\varepsilon > 0$, and for any $a > 0$ set $R_a = [0, 1] + i[(1 - \varepsilon)ax, ax] \subseteq \mathbb{C}$. We have the following composition of one-to-one functions

$$ \bigcup_{a \in A_0} (G'_0 \cap R_a) \xrightarrow{\pi_2^{-1}} \bigcup_{a \in A_0} \{(a, g) \mid g \in G'_0 \cap R_a\} \xrightarrow{\rho} \{g/a \mid (a, g) \in A_0 \times G'_0\} \cap R_1, $$

where the first injection is a section of the projection $\pi_2$ onto the second coordinate.

Since $A_0$ is syndetic, there exist $a_0 \in \mathbb{N}$ and $d \ge 1$ such that for any $a \in A_0$ with $a \ge a_0$, there exists $a' \in A_0 \cap ]a, a+d]$. If moreover $a \ge d/\varepsilon$, then

$$ (1 - \varepsilon)a' \le (1 - \varepsilon)(a + d) \le (1 - \varepsilon)(a + a\varepsilon) < a. $$

Therefore, the union $\bigcup_{a \in A_0} R_a$ contains the half-strip $S = [0, 1] + i[T_0x, +\infty[$ with $T_0 = \max(a_0, d/\varepsilon)$. The set $G'_0 \cap S$ is infinite, and its image by the one-to-one function

$$ G'_0 \cap S \subseteq G'_0 \cap \left( \bigcup_{a \in A_0} R_a \right) \xrightarrow{\rho \circ \pi_2^{-1}} \{g/a \mid (a, g) \in A_0 \times G'_0\} \cap R_1, $$

is also infinite. Thus, the set $\{g/a \mid a \in A_0, g \in G_0\}$ admits a cluster point in $R_1$, which can only be on the imaginary line with imaginary part in $[(1 - \varepsilon)x, x]$. Since the result is true for any $x > 0$ and any $\varepsilon > 0$, we conclude that each point of the vertical half-line $i\mathbb{R}_+$ is such a cluster point.

Therefore, the set $S = A_0 \times G_0$ verifies the three aimed properties. $\square$

<figure><img src="image_12.png" /></figure>

This Lemma alone is insufficient to conclude that every point of the face of $\Gamma$ of normal vector $\nu$ is a singular point of $f$. In the proof of Corollary 4.3, we use a somewhat surprising property of the singular set of a meromorphic function defined on a tubular domain, which remains to be proven. The next subsection is dedicated to this proof. It can be seen as an extension of this subsection, since it completes the argument of Corollary 4.3 in this case. However, since this result may have an interest on its own, and since it is better stated in a general setting, we have set its treatment apart.

**4.4. Singularities along faces of tubular domains.** — A celebrated theorem of Bochner [Boc38] asserts that any holomorphic function on a tubular domain can be holomorphically continued to the convex hull of this domain. Consider a tubular convex domain $\Omega = K + i\mathbb{R}^k$, where $K$ is an convex open subset of $\mathbb{R}^k$. As a convex domain, $\Omega$ is also a domain of holomorphy, which means that for any $b$ be on the boundary $\partial\Omega$, we can find a function which is holomorphic on $\Omega$ but not at $b$. And indeed, for convex tubular domain, finding such functions is straightforward.

Let $b$ be on the boundary $\partial\Omega$. We can write $b$ as $x + iy$, with $x \in \partial K$ and $y \in \mathbb{R}^k$. Since $K$ is convex, by the hyperplane separation theorem, there exists $c \in \mathbb{R}^k$ such that $\langle c, \sigma - x \rangle < 0$ for every $\sigma \in K$. Setting $V$ as $\{c\}^{\perp} \subseteq \mathbb{R}^k$, we get that $V$ is a linear subspace of codimension 1 in $\mathbb{R}^k$ such that $K \cap (x + V)$ is empty and the function $s \mapsto 1/\langle c, s - b \rangle$ is obviously holomorphic on $\Omega$, but not at $b$.

The choice of functions $\xi_b : s \mapsto \exp(1/\langle c, s - b \rangle)$ for any $b \in \partial\Omega$ proves that $\Omega$ is a domain of meromorphy. More generally, by a similar argument, a domain of
holomorphy $\Omega \subseteq \mathbb{C}^k$ is also a domain of meromorphy. The converse is also true, but
more elaborate(‡).

Observe that this elementary example can produce a large set of singular points in
$\partial\Omega$, with a particular structure. Set $F$ as the face of $\bar{K}$ defined by $\bar{K} \cap (\boldsymbol{x} + V)$, which
contains at least $\boldsymbol{x}$. The set of singular points of the function $\xi_b$ over $\partial\Omega$ is exactly
$F + i(\boldsymbol{y} + V)$.

A different choice of **c** would have lead to a different choice of $\xi_b$, and a different
set of singular points. But if **x** is in the relative interior of the face **F**, the relation
$F = \bar{K} \cap (\mathbf{x} + V)$ implies that **V** should contain the linear subspace **W** associated to
the affine hull of **F**, namely its direction. In that case, for any choice of **c**, thus of
$\xi_b$, the set of singular points in $\partial\Omega$ always contains the set $F + i(\mathbf{y} + W)$, which is
independent of **c** and is of dimension 2 dim **F**.

Even if this property seems very particular to the functions $\xi_b$, we show that for
any holomorphic function $\xi$ defined on $\Omega$, the set of singular points has a similar
property over the face $F + i\mathbb{R}^k$.

**Theorem 4.14.** — Let $\xi$ 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 denote $W$ the direction of the affine hull of $F$ in $\mathbb{R}^k$.

If there exist $\mathbf{x} \in F$ and $\mathbf{y} \in \mathbb{R}^k$ such that $\xi$ can be extended holomorphically around $\mathbf{x} + i\mathbf{y}$, then, for any $\mathbf{x}'$ in the relative interior of $F$, and for any $\mathbf{y}' \in \mathbf{y} + W$, the function $\xi$ can be extended holomorphically around $\mathbf{x}' + i\mathbf{y}'$.

Consequently, if there exist $\mathbf{x}$ in the relative interior of $F$ and $\mathbf{y} \in \mathbb{R}^k$ such that $\mathbf{x} + i\mathbf{y}$ is a singular point of $\xi$, then for any $\mathbf{x}' \in F$, and for any $\mathbf{y}' \in \mathbf{y} + W$, $\mathbf{x}' + i\mathbf{y}'$ is a singular point of $\xi$.

The Theorem is obviously true (but useless) if $F$ is a singleton.

Notice that the second assertion of the theorem just is the contrapositive of the
first one. We will only give a proof for the first statement.

**Remark.** — Observe also that this theorem can be stated *mutatis mutandis* for meromorphic functions, in the sense that every occurence of the term “holomorph-” in the theorem has to be replaced by “meromorph-”. This version is the one required in the proof of Corollary 4.3.

As we have already seen, any domain of meromorphy on $\mathbb{C}^k$ is a domain of holomorphy,
and conversely. Since the first statement of the theorem can be phrased in terms of
a domain of holomorphy: Any domain of holomorphy containing $\Omega \cup \{\mathbf{x} + i\mathbf{y}\}$ also
contains $ri(F) + i(\mathbf{y} + W)$, it can be written in terms of domains of meromorphy as
well.

To sum up both assertions of the theorem, we state a direct consequence, which
may be more useful in a large variety of contexts.

**Corollary 4.15.** — Let $\xi$ be a function holomorphic on a tubular domain $K + i\mathbb{R}^k$, with $K \subseteq \mathbb{R}^k$ convex, let $F$ be a proper face of $\bar{K}$ and denote $W$ the direction of the affine hull of $F$ in $\mathbb{R}^k$. Then, there is a closed subset $Y \subseteq \mathbb{R}^k$, invariant under

<footnote>(‡) Levi has shown that every domain of meromorphy is pseudoconvex (see for example [Nis01, §4.1], also for the equivalence of the different notions of pseudoconvexity) and Oka has succeeded in proving that every pseudoconvex domain is a domain of holomorphy (see [Sha92, 39.Theorem 4]).</footnote>
translation by elements of W, such that the set of singular points of $\xi$ inside $ri(F)+i\mathbb{R}^k$
is exactly $ri(F) + iY$.

To prove Theorem 4.14, we use a result of Siciak [Sic69, Theorem 5], which can be
seen as a local version of Bochner’s theorem. We have to introduce some geometric
objects first.

Let $c$ be a positive real number. For any $R \ge c$, set $D(R)$ as the compact region of $\mathbb{C}$ enclosed by the ellipse of foci $-ic$ and $ic$ and for which the sum the small half-axis and large half-axis is $R$. In that case, the small half-axis is $r = \frac{1}{2}(R - c^2/R)$, and the projection of $D(R)$ over $\mathbb{R}$ is the intersection of $D(R)$ with $\mathbb{R}$, i.e. $[-r, r]$. The large half-axis is $\frac{1}{2}(R + c^2/R) = \sqrt{r^2 + c^2}$ and $D(R)$ is inside the closed ball $\bar{B}_{\mathbb{C}}(0, \sqrt{r^2 + c^2})$.

This ellipse is described by the equation $|z + (z^2 + c^2)^{1/2}| = R$ where the chosen branch of $z \mapsto (z^2 + c^2)^{1/2}$ on $\mathbb{C} \setminus [-ic, ic]$ is determined by the requirement that $(z^2 + c^2)^{1/2}$ is a positive real number when $z$ is a positive real number (to avoid confusion, we will use the square root sign only for nonnegative arguments). For $z \in [-ic, ic]$ we have $|z + (z^2 + c^2)^{1/2}| = c$. Indeed, $D(c) = [-ic, ic]$.

Let $\{\mathbf{u}_1, \dots, \mathbf{u}_k\}$ be an orthonormal basis of $\mathbb{R}^k$ (as a $\mathbb{R}$-vector space), therefore of $\mathbb{C}^k$ (as a $\mathbb{C}$-vector space) and $R_1, \dots, R_k$ be $k$ real numbers larger than $c$. For any $m \in [1, k]$, we set $E_m$ as the set of $z$ of coordinates $(z_1, \dots, z_k)$ with respect to the basis $\{\mathbf{u}_1, \dots, \mathbf{u}_k\}$, with $z_i$ in the interval $D(c) = [-ic, ic]$ for all $i \ne m$ and with $z_m \in D(R_m)$. In particular, $E_m$ is inside the closed ball $\bar{B}_{\mathbb{C}^k}(0, \sqrt{kc^2 + r_m^2})$, where $r_m$ is the small half-axis of $D(R_m)$. And since $D(R_m)$ is contained in $[-r_m, r_m] + i\mathbb{R}$, we also have $E_m \subseteq [-r_m, r_m]\mathbf{u}_m + i\mathbb{R}^k$.

The Theorem of Siciak states that if a function $\xi$ is defined on $\bigcup_{m=1}^k E_m$ and holomorphic with respect to the $m^{th}$ coordinate on $E_m$ for any $m \in [1, k]$, then $\xi$ can be extended to an holomorphic function defined over

$$
\Xi = \left\{ z_1 \mathbf{u}_1 + \dots + z_k \mathbf{u}_k \in \mathbb{C}^k \left| \sum_{m=1}^k \frac{\log\left(\frac{|z_m + (z_m^2 + c^2)^{1/2}|}{c}\right)}{\log(R_m/c)} < 1 \right. \right\}
$$

We will only use a much weaker version, which focuses on the real parts and singles
the first coordinate out.

**Lemma 4.16.** — Let $\xi$ be a function holomorphic on $([\-r, r]\mathbf{u}_1 + i\mathbb{R}^k) \cup B_{\mathbb{C}^k}(0, \rho)$, then $\xi$ can be holomorphically extended on the polyhedron

$$
\Sigma = \left\{ x_1 \mathbf{u}_1 + \dots + x_k \mathbf{u}_k \in \mathbb{R}^k \left| \frac{r - |x_1|}{\sqrt{r^2 + c^2 \log(R_1/c)}} > \sum_{m=2}^{k} \frac{|x_m|}{c \log 2} \right\} \subseteq \mathbb{C}^k
$$

where $c = \frac{1}{\sqrt{k+1}}\rho$ and $R_1 = r + \sqrt{r^2 + c^2}$. It can also be holomorphically extended to
the polyhedron $\Sigma' = \{(iz_1, z_2, \dots, z_k) \mid (z_1, z_2, \dots, z_k) \in \Sigma\}$.

Proof. — We fix $R_m = 2c$ for $m \ge 2$ (we have already set $R_1 = r + \sqrt{r^2 + c^2}$), and we consider the sets $E_m$ as defined hereinabove. In that case, we have $r_1 = r$ and $E_1$ is inside $[-r, r]\mathbf{u}_1 + i\mathbb{R}^k$, and for any $m \ge 2$, we have $r_m < c$, so $E_m \subseteq B_{\mathbb{C}^k}(0, \sqrt{k+1}c) = B_{\mathbb{C}^k}(0, \rho)$. Therefore the function $\xi$ verifies the conditions of the Theorem of Siciak, and it can be extended to $\Xi$.
The projection of $\Xi$ onto $\mathbb{R}^k$ is its intersection with $\mathbb{R}^k$

$$ \Xi \cap \mathbb{R}^k = \left\{ x_1 \mathbf{u}_1 + \cdots + x_k \mathbf{u}_k \in \mathbb{R}^k \;\middle|\; \sum_{m=1}^k \frac{\log(|x_m/c| + \sqrt{(x_m/c)^2 + 1})}{\log R_m/c} < 1 \right\}. $$

The function $x \mapsto \log(x + \sqrt{x^2 + c^2})$ maps 0 to $c$ and $r$ to $R_1$, and its derivative $x \mapsto 1/\sqrt{x^2 + c^2}$ is decreasing on $\mathbb{R}_+$. Therefore, by concavity, we deduce that $\log(|x| + \sqrt{x^2 + c^2}) \le \log c + |x|/c$ and $\log(|x| + \sqrt{x^2 + c^2}) \le \log R_1 - (r - |x|)/\sqrt{r^2 + c^2}$ for any $x \in \mathbb{R}$. Therefore, any point in $\Sigma$ verifies

$$ \sum_{m=1}^{k} \frac{\log(|x_m/c| + \sqrt{(x_m/c)^2 + 1})}{\log R_m/c} \le \frac{\log(R_1/c) - (r - |x_1|)/\sqrt{r^2 + c^2}}{\log(R_1/c)} + \sum_{m=2}^{k} \frac{|x_m|/c}{\log(2c/c)} < 1 $$

and therefore is in $\Xi \cap \mathbb{R}^k$.

For $z = ix$ with $x \in \mathbb{R}$, we have

$$ \log |z + (z^2 + c^2)^{1/2}| = \begin{cases} \log c & \text{if } |x| \le c \\ \log(|x| + \sqrt{x^2 - c^2}) & \text{if } |x| \ge c \end{cases} \le \log(|x| + \sqrt{x^2 + c^2}), $$

and the same inequality is valid for elements of $\Sigma'$. $\square$

**Lemma 4.17.** — Let $L \subset \partial K$ be a line segment with midpoint $\mathbf{x}$. If there is $\mathbf{y} \in \mathbb{R}^k$ such that $\xi$ can be holomorphically extended around $\mathbf{x} + i\mathbf{y}$, then $\xi$ can be holomorphically extended around any point $\mathbf{x}' + i\mathbf{y}$ with $\mathbf{x}'$ in the relative interior of $L$, and around any point $\mathbf{x} + i\mathbf{y}'$ with $\mathbf{x} + \mathbf{y}' - \mathbf{y}$ in the relative interior of $L$.

*Proof.* — Let $\sigma \in K$; $\sigma$ is not aligned with the extremities $\mathbf{a}$ and $\mathbf{b}$ of $L$. For any $\lambda \in [0, 1]$ set $L_\lambda$ as the line segment $[\lambda\mathbf{a} + (1-\lambda)\sigma, \lambda\mathbf{b} + (1-\lambda)\sigma]$. In particular, since $K$ is an open convex set, $L_\lambda \subset K$ for any $\lambda < 1$. We set $\mathbf{x}_\lambda$ as $\lambda\mathbf{x} + (1-\lambda)\sigma$; it is the midpoint of $L_\lambda$. We denote $2r$ the length of $L$, therefore $L_\lambda$ has length $2\lambda r$.

Let $\rho > 0$ such that $\xi$ can be extended holomorphically to $B_{\mathbb{C}^k}(\mathbf{x} + i\mathbf{y}, \rho)$ with $\rho_\lambda = \rho - (1-\lambda)\|\mathbf{x} - \mathbf{\sigma}\|$. We will consider $\lambda$ larger than $1 - \frac{1}{2}\rho/\|\mathbf{x} - \mathbf{\sigma}\|$, so that $\rho_\lambda > \rho/2$.

Let $\mathbf{u}_1 = (\mathbf{b} - \mathbf{a})/\|\mathbf{b} - \mathbf{a}\|$ a unit vector in the direction of $L$. Let $\mathbf{u}_2$ a unit vector in the plane generated by $\mathbf{a} - \mathbf{\sigma}$ and $\mathbf{b} - \mathbf{\sigma}$, and orthogonal to $\mathbf{u}_1$. We complete this family into an orthonormal basis $\{\mathbf{u}_1, \dots, \mathbf{u}_k\}$ of $\mathbb{R}^k$ (and of $\mathbb{C}^k$). For $\lambda < 1$, the function $s \mapsto \xi(s + \mathbf{x}_\lambda + i\mathbf{y})$ is holomorphic on $[-\lambda r, \lambda r]\mathbf{u}_1 + i\mathbb{R}^k$, since $L_\lambda \subset K$, and holomorphic on $B_{\mathbb{C}^k}(0, \rho/2)$.

By Lemma 4.16, this function is also holomorphically extended to $\Sigma$, which contains the points $x_1\mathbf{u}_1 + \langle\mathbf{x} - \mathbf{x}_\lambda, \mathbf{u}_2\rangle\mathbf{u}_2$ with $x_1$ such that

$$ \frac{\lambda r - |x_1|}{\sqrt{\lambda^2 r^2 + c^2} \log(\lambda r + \sqrt{\lambda^2 r^2 + c^2})} > \frac{\langle \mathbf{x} - \mathbf{x}_\lambda, \mathbf{u}_2 \rangle}{c \log 2}, $$

with $c = \rho/2\sqrt{k+1}$. It is also extended to $\Sigma'$, which contains the points $ix_1\mathbf{u}_1 + \langle\mathbf{x} - \mathbf{x}_\lambda, \mathbf{u}_2\rangle\mathbf{u}_2$ with $x_1$ verifying the same inequality.
<figure><img src="image_13.png" /><figcaption>FIGURE 6. Getting a large intersection of $x_\lambda + \Sigma$ and of $L$</figcaption></figure>

This means that $\xi$ is holomorphic at any point $x_\lambda + iy + x_1u_1 + \langle x - x_\lambda, u_2 \rangle u_2 = x + iy + (x_1 + \langle x_\lambda - x, u_1 \rangle)u_1$, and any point $x + iy + (ix_1 + \langle x_\lambda - x, u_1 \rangle)u_1$, with $x_1$ verifying the previous inequality, that is

$$|x_1| < \lambda r - \langle x_\lambda - x, u_2 \rangle \frac{\sqrt{\lambda^2 r^2 + c^2} \log(\lambda r + \sqrt{\lambda^2 r^2 + c^2})}{c \log 2}.$$

Since this is verified for any $\lambda < 1$, and that $\|x_\lambda - x\|$ tends to 0 as $\lambda$ gets close to 1, we deduce that $\xi$ can be holomorphically continued at any point of

$$]x + iy - ru_1, x + iy + ru_1[ = ]a, b[$$

namely the relative interior of $L$, and at any point of $]x + i(y - ru_1), x + i(y + ru_1)[$.

□

*Proof of the Theorem 4.14.* — Let $x$ and $x'$ be two distinct points in the relative interior of the face $F$. Let $L$ be the intersection of $F$ with the affine line of $\mathbb{R}^k$ generated by $x$ and $x'$, which is a convex set. We assume that for a given $y \in \mathbb{R}^k$, $\xi$ can be holomorphically continued at the point $x + iy$. From the previous Lemma 4.17, $\xi$ can also be holomorphically continued at any point $x_1 + iy$, with $x_1$ in any open segment with midpoint $x$ included in $L$. Such a segment contains the point $x + (x' - x)/2^m$ for some integer $m \ge 0$. If $m = 0$, then it contains $x'$, and the statement is proved. If not, $\xi$ can also be holomorphically continued at $x + (x' - x)/2^m + iy$, and we use the Lemma 4.17 again for a segment with midpoint $x + (x' - x)/2^m$ containing $x$. This segment contains $x + (x' - x)/2^{m-1}$. And the argument can be reiterated until we can apply it to a segment with midpoint $x + (x' - x)/2$ containing $x$ and $x'$. Therefore, $\xi$ can be holomorphically continued at the point $x' + iy$.
If $\boldsymbol{x}$ is on the boundary of $F$ and $\xi$ can be holomorphically continued at the point $\boldsymbol{x} + i\boldsymbol{y}$, then $\xi$ can be holomorphically continued at any point of a neighborhood of $\boldsymbol{x} + i\boldsymbol{y}$ which contains an element $\tilde{\boldsymbol{x}} + i\boldsymbol{y}$ with $\tilde{\boldsymbol{x}}$ in the relative interior of $F$, and the previous argument can be used starting from $\tilde{\boldsymbol{x}}$ instead of $\boldsymbol{x}$.

We now choose $\boldsymbol{y}'$ such that $\boldsymbol{y}' - \boldsymbol{y} \in W$. Since $W$ is the affine hull of $F$ and $\boldsymbol{x}'$ is in the relative interior of $F$, there exists a positive integer $m \ge 2$, such that $L' = [\boldsymbol{x}' - \frac{1}{m-1}(\boldsymbol{y}' - \boldsymbol{y}), \boldsymbol{x}' + \frac{1}{m-1}(\boldsymbol{y}' - \boldsymbol{y})]$ is a subset of $F$ whose midpoint is $\boldsymbol{x}'$.
For every $j$ between 0 and $m$, we set $\boldsymbol{y}_j = \boldsymbol{y} + \frac{j}{m}(\boldsymbol{y}' - \boldsymbol{y})$, so that for every $j$ with $1 \le j \le m$ we have $\boldsymbol{x}' + \boldsymbol{y}_j - \boldsymbol{y}_{j-1} = \boldsymbol{x}' + \frac{1}{m}(\boldsymbol{y}' - \boldsymbol{y})$ is in the relative interior of $L'$. The second part of Lemma 4.17 induces that if $\xi$ is holomorphic at $\boldsymbol{x}' + i\boldsymbol{y}_{j-1}$ then $\xi$ is holomorphic at $\boldsymbol{x}' + i\boldsymbol{y}_j$. We have just proved that $\xi$ is holomorphic at $\boldsymbol{x}' + i\boldsymbol{y}_0 = \boldsymbol{x}' + i\boldsymbol{y}$, therefore $\xi$ is holomorphic at $\boldsymbol{x}' + i\boldsymbol{y}_m = \boldsymbol{x}' + i\boldsymbol{y}'$. $\square$

## Acknowledgements

This work was initiated and partially supported by the Math-Amsud Project MZFTTA. Oswaldo Velásquez was partially supported by Concytec Grant CG-176-2015-FONDECYT. Arnaud Chadozeau warmly thanks the staff and members of IMCA for their hospitality. We both would like to express our gratitude to Professor Driss Essouabri for useful discussions. We also want to thank the referee for useful suggestions.

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July 8, 2025

ARNAUD CHADOZEAU, CNRS UMR 7586, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG), Campus des Grands Moulins, Université de Paris – Boite Courrier 7012, 8 Place Aurélie Nemours, 75205 PARIS Cedex 13, France • E-mail : arnaud.chadozeau@imj-prg.fr

OSWALDO VELÁSQUEZ CASTAÑÓN, Instituto de Matemática y Ciencias Afines, Universidad Nacional de Ingeniería, Calle Los Biólogos 245, Lima 15012, Peru • E-mail : oswaldo@imca.edu.pe