that $\|\sigma F_p\|_{R',k',p} \le \|F_p\|_{R,k,p}$. Since we obviously have $\|\overline{\sigma F_p}\|_{S_{\overline{p}}'} \le \|\overline{F_p}\|_{S_{\overline{p}}}$, it follows
that $\sigma F \in \mathbb{C}\{(X')^*\}_{\tau'}$ and $\overline{\sigma F} = \overline{F} \circ \tilde{\sigma}$.

Moreover, since the above inequalities hold for all choices of $F_p$ such that $\overline{F} =_{\tau} \sum_p F_p$, it follows that $\|\sigma F\|_{\tau'} \le \|F\|_{\tau}$. ■

## 4.4 Translations

Let $\sigma : \{X, Y\} \to \mathbb{R}\{(X, X')^*, Y'\}_G$ be the translation by a point $(a, b) \in [0, \infty)^m \times \mathbb{R}^n$.

**Proposition 4.6** Let $\tau \in \mathcal{T}_m$ and $\rho \in (0, \infty)^n$ be such that $\|(a, b)\| \le (R, \rho)$. Then, for every $F \in \mathbb{C}\{X^*, Y\}_{\tau, \rho}$, we have $\sigma F \in \mathbb{C}\{(X')^*, Y'\}_G$ and $\overline{\sigma F} = \overline{F} \circ \tilde{\sigma}$.

*Proof* We may assume that all but one of the coordinates of $(a, b)$ are zero. If $a_i \ne 0$ for some $i$, we may assume, after permuting the Gevrey variables if necessary, that $i = m$. This case is handled similarly to the proof of Proposition 4.4, except that we use [11, Lemma 6.6] instead of [11, Lemma 6.5] (with a similar weakening of hypotheses for the former as used above for the latter). So we assume that $b_j \ne 0$ for some $j$; in this case, we adapt the usual Taylor expansion argument for convergent series to obtain the conclusion (details are left to the reader). ■

## 4.5 Infinitesimal substitutions

We recall the following observations.

**Lemma 4.7** Let $\tau = (K, R, r, \theta, \Delta) \in \mathcal{T}_m$ and $X' = (X'_1, \dots, X'_n)$, and let $F \in \mathbb{C}\{X^*\}_{\tau}$.

(1) For any $\tau' \ge \tau$, we have $F \in \mathbb{C}\{X^*\}_{\tau'}$ with $\|F\|_{\tau'} \le \|F\|_{\tau}$.

(2) The series $G(X, X') := F(X)$ belongs to $\mathbb{C}\{(X, X')^*\}_{\tau'}$ with $\|G\|_{\tau'} = \|F\|_{\tau}$, where $\tau' = (K', R', r, \theta, \Delta')$ with $K' := \{(k, 0) : k \in K\}$, $R' := (R, S)$ for any $S > 0$, and $\Delta' := \Delta \times \Gamma$ for any natural $\Gamma \subseteq [0, \infty)^n$.

*Proof* Part (1) is a consequence of the discussion in Section 3.6. For part (2), note that $S^{\tau'} = S^{\tau} \times H(\log S)$ and $S_p^{\tau'} = S_p^{\tau} \times H(\log S)$, so that $\|G\|_{\tau'} = \|F\|_{\tau}$. ■

**Remark 4.8** We also need the following (crude) estimates from combinatorics. Let $n \in \mathbb{N}$.

(1) Given $k \in \mathbb{N}$, the number of elements $\beta \in \mathbb{N}^n$ such that $\sum \beta = k$ is bounded above by $k^n$, because $\{\beta \in \mathbb{N}^n : \sum \beta = k\} \subseteq \{1, \dots, k\}^n$.

(2) For $\beta \in \mathbb{N}^n$ and $k \in \mathbb{N}$, the number of ways to write $\beta$ as the sum of at most $\sum \beta$ many nonzero elements in $\mathbb{N}^n$ is bounded above by $2^n \sum \beta$. To see this, note that if $n=1$, then each such sum corresponds to a strictly increasing $k$-tuple $0 < a_1 < \dots < a_k = \beta$ with $k \le \beta = \sum \beta$; so there are at most $2^\beta = 1 + \sum_{k=1}^\beta (\beta^k)$ many ways to write $\beta$ as the sum of at most $\beta$ many nonzero natural numbers. The claim for general $n$ follows.

(3) For $\gamma \in \mathbb{N}^{n'}$ and $k \in \mathbb{N}$, denote by $N(\gamma, k)$ the number of ways to write $\gamma$ as the sum of exactly $k$ many nonzero elements in $\mathbb{N}^{n'}$. Then, for any nonzero $\beta \in \mathbb{N}^n$,
we have

$$N(y, \sum \beta) = \sum_{y^{1} + \cdots + y^{n} = y} \left( \prod_{j=1}^{n} N(y^{j}, \beta_{j}) \right).$$

Let $\sigma$ be an infinitesimal substitution as in Example 4.1(5); in particular, we may assume that $n > 0$. Let $\tau = (K, R, r, \theta, \Delta) \in \mathcal{T}_m$ and $\rho \in (0, \infty)^n$, and let $F = \sum_{\beta \in \mathbb{N}^n} F_\beta Y^\beta \in \mathbb{C}\{X^*, Y\}_{\tau, \rho}$. By Lemma 3.8(2), there exist $\tau' \in \mathcal{T}_{m'}$ and $\rho' \in (0, \infty)^{n'}$ such that

$$(4.1) \qquad \|\sigma(Y_j)\|_{\tau', 2^{n'+1}\rho'} \le \frac{\rho_j}{2} \quad \text{for } j = 1, \dots, n;$$

in particular, we have $\tilde{\sigma}(S^\tau \times S^{\tau'} \times D(\rho')) \subseteq S^\tau \times D(\rho)$. For each $j$, we write $\sigma(Y_j) = \sum_{y \in \mathbb{N}^{n'}} G_{j,y}(X')(Y')^\gamma$ with $G_{j,y} \in \mathbb{C}\{(X')^*\}_{\tau'}$, and we write $\tau' = (K', R', r, \theta, \Delta')$ (we can always reduce to the case where $r$ and $\theta$ are the same for both $\tau$ and $\tau'$).

By Lemma 4.7(2), we have $F_\beta \in \mathbb{C}\{(X, X')^*\}_{\eta_1}$ for each $\beta$, where

$$\eta_1 = (K_1, (R, R'), r, \theta, \Delta \times \Delta')$$

with $K_1 := \{(k, 0) : k \in K\}$. Again by Lemma 4.7(2), and by Lemma 4.3, we have $G_{j,y} \in \mathbb{C}\{(X, X')^*\}_{\eta_2}$ for each $j$ and each $\gamma$, where

$$\eta_2 = (K_2, (R, R'), r, \theta, \Delta \times \Delta')$$

with $K_1 := \{(0, k) : k \in K'\}$. So from Lemma 4.7(1), we get that each $F_\beta$ and each $G_{j,y}$ belongs to $\mathbb{C}\{(X, X')^*\}_{\eta}$, where $\eta = (L, (R, R'), r, \theta, \Delta \times \Delta')$ with $L := K_1 \cup K_2$. Moreover, Lemma 4.7 implies that $\|F_\beta\|_\eta \le \|F_\beta\|_{\tau}$ and $\|G_{j,y}\|_\eta \le \|G_{j,y}\|_{\tau'}$ for each $\beta$, $j$, and $\gamma$.

**Proposition 4.9** For $F \in \mathbb{C}\{X^*, Y\}_{\tau, \rho}$, we have $\sigma F \in \mathbb{C}\{(X, X')^*, Y'\}_{\eta, \rho'}$ such that $\overline{\sigma F} = \overline{F} \circ \tilde{\sigma}$ and $\|\sigma F\|_{\eta, \rho'} \le A \|F\|_{\tau, \rho}$, for some absolute constant $A > 0$.

**Proof** A first computation (left to the reader) shows that, for $\beta \in \mathbb{N}^n$, we have

$$(\sigma(Y))^\beta = \sum_{y \in \mathbb{N}^{n'}} H_{\beta, y}(X')(Y')^\gamma,$$

where

$$H_{\beta, y}(X') = \sum_{y^{1} + \cdots + y^{n} = y} \left( \prod_{j=1}^{n} \left( \sum_{\delta^{1} + \cdots + \delta^{\beta_j} = y_j} \left( \prod_{\rho=1}^{\beta_j} G_{j, \delta^\rho}(X') \right) \right) \right)$$

and each $y^j$ and $\delta^p$ belongs to $\mathbb{N}^{n'}$ and is nonzero. Therefore,

$$\sigma F(X, X', Y') = \sum_{\beta} F_{\beta}(X) \sigma(Y)^{\beta} = \sum_{\gamma} \left( \sum_{\beta} F_{\beta}(X) H_{\beta, \gamma}(X') \right) (Y')^{\gamma}.$$

By the above, we have

$$(4.2) \qquad \|F_{\beta}\|_{\eta} \le \frac{\|F\|_{\tau, \rho}}{\rho^{\beta}}$$
we have

$$N(y, \sum \beta) = \sum_{y^{1} + \cdots + y^{n} = y} \left( \prod_{j=1}^{n} N(y^{j}, \beta_{j}) \right).$$

Let $\sigma$ be an infinitesimal substitution as in Example 4.1(5); in particular, we may assume that $n > 0$. Let $\tau = (K, R, r, \theta, \Delta) \in \mathcal{T}_m$ and $\rho \in (0, \infty)^n$, and let $F = \sum_{\beta \in \mathbb{N}^n} F_\beta Y^\beta \in \mathbb{C}\{X^*, Y\}_{\tau, \rho}$. By Lemma 3.8(2), there exist $\tau' \in \mathcal{T}_{m'}$ and $\rho' \in (0, \infty)^{n'}$ such that

$$(4.1) \qquad \|\sigma(Y_j)\|_{\tau', 2^{n'+1}\rho'} \le \frac{\rho_j}{2} \quad \text{for } j = 1, \dots, n;$$

in particular, we have $\tilde{\sigma}(S^\tau \times S^{\tau'} \times D(\rho')) \subseteq S^\tau \times D(\rho)$. For each $j$, we write $\sigma(Y_j) = \sum_{y \in \mathbb{N}^{n'}} G_{j,y}(X')(Y')^\gamma$ with $G_{j,y} \in \mathbb{C}\{(X')^*\}_{\tau'}$, and we write $\tau' = (K', R', r, \theta, \Delta')$ (we can always reduce to the case where $r$ and $\theta$ are the same for both $\tau$ and $\tau'$).

By Lemma 4.7(2), we have $F_\beta \in \mathbb{C}\{(X, X')^*\}_{\eta_1}$ for each $\beta$, where

$$\eta_1 = (K_1, (R, R'), r, \theta, \Delta \times \Delta')$$

with $K_1 := \{(k, 0) : k \in K\}$. Again by Lemma 4.7(2), and by Lemma 4.3, we have $G_{j,y} \in \mathbb{C}\{(X, X')^*\}_{\eta_2}$ for each $j$ and each $\gamma$, where

$$\eta_2 = (K_2, (R, R'), r, \theta, \Delta \times \Delta')$$

with $K_1 := \{(0, k) : k \in K'\}$. So from Lemma 4.7(1), we get that each $F_\beta$ and each $G_{j,y}$ belongs to $\mathbb{C}\{(X, X')^*\}_{\eta}$, where $\eta = (L, (R, R'), r, \theta, \Delta \times \Delta')$ with $L := K_1 \cup K_2$. Moreover, Lemma 4.7 implies that $\|F_\beta\|_\eta \le \|F_\beta\|_{\tau}$ and $\|G_{j,y}\|_\eta \le \|G_{j,y}\|_{\tau'}$ for each $\beta$, $j$, and $\gamma$.

**Proposition 4.9** For $F \in \mathbb{C}\{X^*, Y\}_{\tau, \rho}$, we have $\sigma F \in \mathbb{C}\{(X, X')^*, Y'\}_{\eta, \rho'}$ such that $\overline{\sigma F} = \overline{F} \circ \tilde{\sigma}$ and $\|\sigma F\|_{\eta, \rho'} \le A \|F\|_{\tau, \rho}$, for some absolute constant $A > 0$.

**Proof** A first computation (left to the reader) shows that, for $\beta \in \mathbb{N}^n$, we have

$$(\sigma(Y))^\beta = \sum_{y \in \mathbb{N}^{n'}} H_{\beta, y}(X')(Y')^\gamma,$$

where

$$H_{\beta, y}(X') = \sum_{y^{1} + \cdots + y^{n} = y} \left( \prod_{j=1}^{n} \left( \sum_{\delta^{1} + \cdots + \delta^{\beta_j} = y_j} \left( \prod_{\rho=1}^{\beta_j} G_{j, \delta^\rho}(X') \right) \right) \right)$$

and each $y^j$ and $\delta^p$ belongs to $\mathbb{N}^{n'}$ and is nonzero. Therefore,

$$\sigma F(X, X', Y') = \sum_{\beta} F_{\beta}(X) \sigma(Y)^{\beta} = \sum_{\gamma} \left( \sum_{\beta} F_{\beta}(X) H_{\beta, \gamma}(X') \right) (Y')^{\gamma}.$$

By the above, we have

$$(4.2) \qquad \|F_{\beta}\|_{\eta} \le \frac{\|F\|_{\tau, \rho}}{\rho^{\beta}}$$
for each $\beta$, and setting $\tilde{\rho} := 2^{n'+1}\rho'$,

$$ (4.3) \qquad \|G_{j,\gamma}\|_{\eta} \le \frac{\|\sigma(Y_j)\|_{\tau',\tilde{\rho}}}{(\tilde{\rho})^{\gamma}} \le \frac{\rho_j/2}{(\tilde{\rho})^{\gamma}} $$

for each $j$ and $\gamma$. Therefore, denoting by $N(\gamma, k)$ the number of ways to write $\gamma$ as the sum of $k$ many nonzero elements in $\mathbb{N}^{n'}$, we get for each $\beta$ and $\gamma$ from Remark 4.8(3) that

$$ \begin{aligned} \|H_{\beta,\gamma}\|_{\eta} &\le \sum_{y^{1}+\cdots+y^{n}=\gamma} \left( \prod_{j=1}^{n} \left( \sum_{\delta^{1}+\cdots+\delta^{\beta_{j}}=\gamma j} \left( \prod_{\rho=1}^{\beta_{j}} \frac{\rho_{j}/2}{(\tilde{\rho})^{\delta^{\rho}}} \right) \right) \right) \\ &= \sum_{y^{1}+\cdots+y^{n}=\gamma} \left( \prod_{j=1}^{n} \left( \sum_{\delta^{1}+\cdots+\delta^{\beta_{j}}=\gamma j} \frac{(\rho_{j}/2)^{\beta_{j}}}{(\tilde{\rho})^{y^{j}}} \right) \right) \\ &= \sum_{y^{1}+\cdots+y^{n}=\gamma} \left( \prod_{j=1}^{n} \left( \frac{(\rho_{j}/2)^{\beta_{j}}}{(\tilde{\rho})^{y^{j}}} N(y^{j}, \beta_{j}) \right) \right) \\ &= \sum_{y^{1}+\cdots+y^{n}=\gamma} \left( \frac{(\rho/2)^{\beta}}{(\tilde{\rho})^{\gamma}} \left( \prod_{j=1}^{n} N(y^{j}, \beta_{j}) \right) \right) \\ &= \frac{(\rho/2)^{\beta}}{(\tilde{\rho})^{\gamma}} N(\gamma, \sum \beta). \end{aligned} $$

Also, since $N(\gamma, k) = 0$ for $k > \sum \gamma$, it follows from Remark 4.8(2) that

$$ (4.4) \qquad \|H_{\beta,\gamma}\|_{\eta} \le 2^{n' \cdot \sum \gamma} \cdot \frac{(\rho/2)^{\beta}}{(\tilde{\rho})^{\gamma}}. $$

So for each $\gamma$, we get

$$ \left\| \sum_{\beta} (F_{\beta} H_{\beta, \gamma}) \right\|_{\eta} \le \|F\|_{\tau, \rho} \frac{2^{n' \cdot \sum \gamma}}{(\tilde{\rho})^{\gamma}} \cdot \sum_{\beta} \left(\frac{1}{2}\right)^{\sum \beta}. $$

However, since $\sum_{\beta} (\frac{1}{2})^{\sum \beta} = \sum_{k=0}^{\infty} (\sum_{\beta=k} 1) (\frac{1}{2})^{k} \le C := \sum_{k} \frac{k^n}{2^k}$ by Remark 4.8(1), we conclude that

$$ \left\| \sum_{\beta} (F_{\beta} H_{\beta, \gamma}) \right\|_{\eta} \le C \cdot \|F\|_{\tau, \rho} \frac{2^{n' \cdot \sum \gamma}}{(\tilde{\rho})^{\gamma}}, $$

where $C > 0$ is an absolute constant. Finally, since $(\tilde{\rho})^{\gamma} = 2^{(n'+1)\sum \gamma} \cdot (\rho')^{\gamma}$, we obtain

$$ (4.5) \qquad \left\| \sum_{\beta} (F_{\beta} H_{\beta, \gamma}) \right\|_{\eta} \le \frac{C \cdot \|F\|_{\tau, \rho}}{2^{\sum \gamma} \cdot (\rho')^{\gamma}}. $$
for each $\beta$, and setting $\tilde{\rho} := 2^{n'+1}\rho'$,

$$ (4.3) \qquad \|G_{j,\gamma}\|_{\eta} \le \frac{\|\sigma(Y_j)\|_{\tau',\tilde{\rho}}}{(\tilde{\rho})^{\gamma}} \le \frac{\rho_j/2}{(\tilde{\rho})^{\gamma}} $$

for each $j$ and $\gamma$. Therefore, denoting by $N(\gamma, k)$ the number of ways to write $\gamma$ as the sum of $k$ many nonzero elements in $\mathbb{N}^{n'}$, we get for each $\beta$ and $\gamma$ from Remark 4.8(3) that

$$ \begin{aligned} \|H_{\beta, \gamma}\|_{\eta} &\le \sum_{y^{1}+\cdots+y^{n}=y} \left( \prod_{j=1}^{n} \left( \sum_{\delta^{1}+\cdots+\delta^{\beta_{j}}=y} \left( \prod_{\rho=1}^{\beta_{j}} \frac{\rho_{j}/2}{(\tilde{\rho})^{\delta^{\rho}}} \right) \right) \right) \\ &= \sum_{y^{1}+\cdots+y^{n}=y} \left( \prod_{j=1}^{n} \left( \sum_{\delta^{1}+\cdots+\delta^{\beta_{j}}=y} \frac{(\rho_{j}/2)^{\beta_{j}}}{(\tilde{\rho})^{y^{j}}} \right) \right) \\ &= \sum_{y^{1}+\cdots+y^{n}=y} \left( \prod_{j=1}^{n} \left( \frac{(\rho_{j}/2)^{\beta_{j}}}{(\tilde{\rho})^{y^{j}}} N(y^{j}, \beta_{j}) \right) \right) \\ &= \sum_{y^{1}+\cdots+y^{n}=y} \left( \frac{(\rho/2)^{\beta}}{(\tilde{\rho})^{\gamma}} \left( \prod_{j=1}^{n} N(y^{j}, \beta_{j}) \right) \right) \\ &= \frac{(\rho/2)^{\beta}}{(\tilde{\rho})^{\gamma}} N(\gamma, \sum \beta). \end{aligned} $$

Also, since $N(\gamma, k) = 0$ for $k > \sum \gamma$, it follows from Remark 4.8(2) that

$$ (4.4) \qquad \|H_{\beta, \gamma}\|_{\eta} \le 2^{n' \cdot \sum \gamma} \cdot \frac{(\rho/2)^{\beta}}{(\tilde{\rho})^{\gamma}}. $$

So for each $\gamma$, we get

$$ \left\| \sum_{\beta} (F_{\beta} H_{\beta, \gamma}) \right\|_{\eta} \le \|F\|_{\tau, \rho} \frac{2^{n' \cdot \sum \gamma}}{(\tilde{\rho})^{\gamma}} \cdot \sum_{\beta} \left(\frac{1}{2}\right)^{\sum \beta}. $$

However, since $\sum_{\beta} (\frac{1}{2})^{\sum \beta} = \sum_{k=0}^{\infty} (\sum_{\beta=k} 1) (\frac{1}{2})^k \le C := \sum_k \frac{k^n}{2^k}$ by Remark 4.8(1), we conclude that

$$ \left\| \sum_{\beta} (F_{\beta} H_{\beta, \gamma}) \right\|_{\eta} \le C \cdot \|F\|_{\tau, \rho} \frac{2^{n' \cdot \sum \gamma}}{(\tilde{\rho})^{\gamma}}, $$

where $C > 0$ is an absolute constant. Finally, since $(\tilde{\rho})^{\gamma} = 2^{(n'+1)\sum \gamma} \cdot (\rho')^{\gamma}$, we obtain

$$ (4.5) \qquad \left\| \sum_{\beta} (F_{\beta} H_{\beta, \gamma}) \right\|_{\eta} \le \frac{C \cdot \|F\|_{\tau, \rho}}{2^{\sum \gamma} \cdot (\rho')^{\gamma}}. $$
Multiplying by $(\rho')^\gamma$ and summing over $\gamma$, therefore, yields

$$ \|\sigma F\|_{\eta, \rho'} \le C \cdot \|F\|_{\tau, \rho} \left( \sum_{\gamma} \left(\frac{1}{2}\right)^{\sum \gamma} \right) \le CD \|F\|_{\tau, \rho} $$

for some absolute constant $D > 0$, as required.

# 5 Closure properties and o-minimality

The goal of this section is to verify, for those mixed series of Section 3.7 with only real coefficients, the closure properties listed in Paragraphs 1.8 and 1.15 of [7]. We will adopt the notations of the latter, and we need to define the real algebras $A_{m,n,r}$. So let $m, n \in \mathbb{N}$, and let $r = (s, t) = (s_1, \dots, s_m, t_1, \dots, t_n) \in (0, \infty)^{m+n}$ be a polyradius of type $(m, n)$. (While we used the letter $R$ for polyradii in the previous sections to mirror notations in [12], we use the letters $r, s$, and $t$ to mirror corresponding notations in [7].) We set

$$ \mathcal{T}_m^s := \{ \tau = (K, R, \rho, \theta, \Delta) \in \mathcal{T}_m : R > s \} $$

and define the $\mathbb{R}$-algebras

$$ \mathbb{R}\{X^*, Y\}_r := \bigcup_{\substack{\tau \in \mathcal{T}_m^s \\ u > t}} \mathbb{C}\{X^*, Y\}_{\tau, u} \cap \mathbb{R}[[X^*, Y]] $$

and

$$ \bar{\mathcal{A}}_{m,n,r} := \bigcup_{\tau \in \mathcal{T}_m^s} \{ \bar{F} \mid (-\infty, s) \times D(t) : F \in \mathbb{C}\{X^*, Y\}_{\tau, t} \cap \mathbb{R}[[X^*, Y]] \}. $$

Recall from [7, Notation 1.7] the following definitions:

$$ I_{m,n,r} := (0, s_1) \times \dots \times (0, s_m) \times (-t_1, t_1) \times \dots \times (-t_n, t_n), $$

$$ \hat{I}_{m,n,r} := [0, s_1) \times \dots \times [0, s_m) \times (-t_1, t_1) \times \dots \times (-t_n, t_n). $$

In particular, each $\bar{f} \in \bar{\mathcal{A}}_{m,n,r}$ defines a continuous function $f: \hat{I}_{m,n,r} \to \mathbb{R}$ by setting

$$ f(x, y) := \bar{f}(\log x, y); $$

this $f$ is real analytic on $I_{m,n,r}$. We let $A_{m,n,r}$ be the $\mathbb{R}$-algebra of all such functions obtained from $\mathbb{R}\{X^*, Y\}_r$, and we define the $\mathbb{R}$-algebra homomorphism $T_{m,n,r}: A_{m,n,r} \to \mathbb{R}\{X^*, Y\}_r$ by letting $T_{m,n,r}f$ be the (by quasianalyticity) unique $F \in \mathbb{R}\{X^*, Y\}_r$ such that $f(x, y) = \bar{F}(\log x, y)$. We leave it to the reader to verify Properties (1)–(5) and (8) of [7, Paragraph 1.8]. Properties (6) and (7) follow from Lemma 3.8(4,5).

Let $\{A_{m,n} : m, n \in \mathbb{N}\}$ be the corresponding family of algebras of germs, as defined in [7, Section 1.2]. By Proposition 4.6, every $f \in A_{m,n,r}$ is $\mathcal{A}$-analytic, as defined in [7, Definition 1.10].

It now remains to verify the properties listed in [7, Paragraph 1.15] for the corresponding family $\mathcal{A}$ of algebras of germs. Property (1) there is obvious here; Property (3) follows from Lemma 4.3; Property (5) follows from Proposition 4.9; and Property
Multiplying by $(\rho')^\gamma$ and summing over $\gamma$, therefore, yields

$$ \|\sigma F\|_{\eta, \rho'} \le C \cdot \|F\|_{\tau, \rho} \left( \sum_{\gamma} \left(\frac{1}{2}\right)^{\sum \gamma} \right) \le CD \|F\|_{\tau, \rho} $$

for some absolute constant $D > 0$, as required.

# 5 Closure properties and o-minimality

The goal of this section is to verify, for those mixed series of Section 3.7 with only real coefficients, the closure properties listed in Paragraphs 1.8 and 1.15 of [7]. We will adopt the notations of the latter, and we need to define the real algebras $A_{m,n,r}$. So let $m, n \in \mathbb{N}$, and let $r = (s, t) = (s_1, \dots, s_m, t_1, \dots, t_n) \in (0, \infty)^{m+n}$ be a polyradius of type $(m, n)$. (While we used the letter $R$ for polyradii in the previous sections to mirror notations in [12], we use the letters $r, s$, and $t$ to mirror corresponding notations in [7].) We set

$$ \mathcal{T}_m^s := \{ \tau = (K, R, \rho, \theta, \Delta) \in \mathcal{T}_m : R > s \} $$

and define the $\mathbb{R}$-algebras

$$ \mathbb{R}\{X^*, Y\}_r := \bigcup_{\substack{\tau \in \mathcal{T}_m^s \\ u > t}} \mathbb{C}\{X^*, Y\}_{\tau, u} \cap \mathbb{R}[[X^*, Y]] $$

and

$$ \bar{\mathcal{A}}_{m,n,r} := \bigcup_{\tau \in \mathcal{T}_m^s} \{ \bar{F} |_{(-\infty,s) \times D(t)} : F \in \mathbb{C}\{X^*, Y\}_{\tau,t} \cap \mathbb{R}[[X^*, Y]] \}. $$

Recall from [7, Notation 1.7] the following definitions:

$$ I_{m,n,r} := (0, s_1) \times \dots \times (0, s_m) \times (-t_1, t_1) \times \dots \times (-t_n, t_n), $$

$$ \hat{I}_{m,n,r} := [0, s_1) \times \dots \times [0, s_m) \times (-t_1, t_1) \times \dots \times (-t_n, t_n). $$

In particular, each $\bar{f} \in \bar{\mathcal{A}}_{m,n,r}$ defines a continuous function $f: \hat{I}_{m,n,r} \to \mathbb{R}$ by setting

$$ f(x, y) := \bar{f}(\log x, y); $$

this $f$ is real analytic on $I_{m,n,r}$. We let $A_{m,n,r}$ be the $\mathbb{R}$-algebra of all such functions obtained from $\mathbb{R}\{X^*, Y\}_r$, and we define the $\mathbb{R}$-algebra homomorphism $T_{m,n,r}: A_{m,n,r} \to \mathbb{R}\{X^*, Y\}_r$ by letting $T_{m,n,r}f$ be the (by quasianalyticity) unique $F \in \mathbb{R}\{X^*, Y\}_r$ such that $f(x, y) = \bar{F}(\log x, y)$. We leave it to the reader to verify Properties (1)–(5) and (8) of [7, Paragraph 1.8]. Properties (6) and (7) follow from Lemma 3.8(4,5).

Let $\{A_{m,n} : m, n \in \mathbb{N}\}$ be the corresponding family of algebras of germs, as defined in [7, Section 1.2]. By Proposition 4.6, every $f \in A_{m,n,r}$ is $\mathcal{A}$-analytic, as defined in [7, Definition 1.10].

It now remains to verify the properties listed in [7, Paragraph 1.15] for the corresponding family $\mathcal{A}$ of algebras of germs. Property (1) there is obvious here; Property (3) follows from Lemma 4.3; Property (5) follows from Proposition 4.9; and Property
(6) follows from Proposition 3.9. The remaining properties are handled below; we fix arbitrary $\tau = (K, R, r, \theta, \Delta) \in \mathcal{T}_m$ and $\rho \in (0, \infty)^n$.

## 5.1 Monomial division

Let $F \in \mathbb{C}\{X^*, Y\}_{\tau, \rho}$.

First, we let $\alpha > 0$ and assume that $F = X_1^\alpha G$ with $G \in \mathbb{C}[[X^*, Y]]$. We write $F = \sum_{\beta \in \mathbb{N}^n} F_\beta(X)Y^n$ and $G = \sum_{\beta \in \mathbb{N}} G_\beta(X)Y^n$ with each $F_\beta \in \mathbb{C}\{X^*\}_\tau$ and each $G_\beta \in \mathbb{C}[[X^*]]$, and we set $\tau' := (K, R, s, \theta, \Delta)$ for some fixed (but arbitrary) $s \in (1, r)$. Then, by Lemma 4.3 and Lemma 3.6, there exist $C > 0$ (depending only on $s$) such that $\|G_\beta\|_\tau' \le C\|F_\beta\|_\tau$, for each $\beta$. It follows that $\sum_\beta \|G_\beta\|_{\tau'} \rho^n \le C\|F\|_{\tau, \rho}$, so that $G \in \mathbb{C}\{X^*, Y\}_{\tau', \rho}$.

Second, we let $n \in \mathbb{N}$ and assume that $F = Y_1^n G$ with $G \in \mathbb{C}[[X^*, Y]]$. Then by definition, we have $\|F\|_{\tau, \rho} = \rho_1^n \|G\|_{\tau, \rho}$, so that $G \in \mathbb{C}\{X^*, Y\}_{\tau, \rho}$.

Putting together the two cases discussed here proves Property (2) of [7, Paragraph 1.15].

## 5.2 Setting a variable equal to 0

Let $F \in \mathbb{C}\{X^*, Y\}_{\tau, \rho}$, and write $F = \sum_{\beta \in \mathbb{N}^n} F_\beta(X)Y^n$ with each $F_\beta \in \mathbb{C}\{X^*\}_\tau$. Applying Lemma 3.3 with $\nu = 1$ and $a = -\infty = \log 0$ followed by Lemma 4.3, we get that $F_{\beta,0} := F_\beta(X_1, \dots, X_{n-1}, 0) \in \mathbb{C}\{(X_1, \dots, X_{n-1})^*, Y\}_{\mu(a)}$ with $\|F_{\beta,0}\|_{\mu(a)} \le \|F_\beta\|_\tau$, for each $\beta \in \mathbb{N}^n$. Hence, $F_0 := F(X_1, \dots, X_{n-1}, 0, Y)$ belongs to $\mathbb{C}\{(X_1, \dots, X_{n-1})^*, Y\}_{\mu(a), \rho}$; this proves Property (4) of [7, Paragraph 1.15].

## 5.3 Blow-up charts

Here we refer to the blow-up charts (1)–(5) of [7, Definition 1.13]. Closure under blow-up charts (1) (regular blow-ups) is proved by Proposition 4.4; closure under blow-up charts (2) (singular blow-ups) is similar, but easier and left to the reader. Blow-up charts (3) are infinitesimal substitutions and thus are handled by Proposition 4.9. For blow-up charts (4), note that $\mathbb{C}\{X^*, Y\}_G \subseteq \mathbb{C}\{(X_1, \dots, X_{m+1})^*, Y\}_G$ by Lemma 3.8(6); so closure under these blow-up charts follows from closure under blow-up charts (2) and Proposition 4.9. Closure under blow-up charts (5) follows again from Proposition 4.9. This proves Property (7) of [7, Paragraph 1.15].

As the results of [7] do not make use of the Weierstrass Preparation Theorem (which is in general not available in the quasianalytic setting), we may dispense with proving this property here.

**Proof of Main Theorem** The previous discussion implies that our system $\mathcal{A}$ of algebras satisfies Conditions (1) and (4) of [7, Proviso 1.20]. Moreover, Condition (2) is implied by Proposition 4.6, while Condition (3) follows from Proposition 3.4. So the theorem follows from [7, Theorems A and B]. Finally, note that it suffices to add the reciprocal function to obtain quantifier elimination, as all real powers with nonnegative exponents are already in the language $\mathcal{L}_{G^*}$. ■
(6) follows from Proposition 3.9. The remaining properties are handled below; we fix arbitrary $\tau = (K, R, r, \theta, \Delta) \in \mathcal{T}_m$ and $\rho \in (0, \infty)^n$.

## 5.1 Monomial division

Let $F \in \mathbb{C}\{X^*, Y\}_{\tau, \rho}$.

First, we let $\alpha > 0$ and assume that $F = X_1^\alpha G$ with $G \in \mathbb{C}[[X^*, Y]]$. We write $F = \sum_{\beta \in \mathbb{N}^n} F_\beta(X)Y^n$ and $G = \sum_{\beta \in \mathbb{N}} G_\beta(X)Y^n$ with each $F_\beta \in \mathbb{C}\{X^*\}_\tau$ and each $G_\beta \in \mathbb{C}[[X^*]]$, and we set $\tau' := (K, R, s, \theta, \Delta)$ for some fixed (but arbitrary) $s \in (1, r)$. Then, by Lemma 4.3 and Lemma 3.6, there exist $C > 0$ (depending only on $s$) such that $\|G_\beta\|_\tau' \le C\|F_\beta\|_\tau$, for each $\beta$. It follows that $\sum_\beta \|G_\beta\|_{\tau'} \rho^n \le C\|F\|_{\tau, \rho}$, so that $G \in \mathbb{C}\{X^*, Y\}_{\tau', \rho}$.

Second, we let $n \in \mathbb{N}$ and assume that $F = Y_1^n G$ with $G \in \mathbb{C}[[X^*, Y]]$. Then by definition, we have $\|F\|_{\tau, \rho} = \rho_1^n \|G\|_{\tau, \rho}$, so that $G \in \mathbb{C}\{X^*, Y\}_{\tau, \rho}$.

Putting together the two cases discussed here proves Property (2) of [7, Paragraph 1.15].

## 5.2 Setting a variable equal to 0

Let $F \in \mathbb{C}\{X^*, Y\}_{\tau, \rho}$, and write $F = \sum_{\beta \in \mathbb{N}^n} F_\beta(X)Y^n$ with each $F_\beta \in \mathbb{C}\{X^*\}_\tau$. Applying Lemma 3.3 with $\nu = 1$ and $a = -\infty = \log 0$ followed by Lemma 4.3, we get that $F_{\beta,0} := F_\beta(X_1, \dots, X_{n-1}, 0) \in \mathbb{C}\{(X_1, \dots, X_{n-1})^*, Y\}_{\mu(a)}$ with $\|F_{\beta,0}\|_{\mu(a)} \le \|F_\beta\|_\tau$, for each $\beta \in \mathbb{N}^n$. Hence, $F_0 := F(X_1, \dots, X_{n-1}, 0, Y)$ belongs to $\mathbb{C}\{(X_1, \dots, X_{n-1})^*, Y\}_{\mu(a), \rho}$; this proves Property (4) of [7, Paragraph 1.15].

## 5.3 Blow-up charts

Here we refer to the blow-up charts (1)–(5) of [7, Definition 1.13]. Closure under blow-up charts (1) (regular blow-ups) is proved by Proposition 4.4; closure under blow-up charts (2) (singular blow-ups) is similar, but easier and left to the reader. Blow-up charts (3) are infinitesimal substitutions and thus are handled by Proposition 4.9. For blow-up charts (4), note that $\mathbb{C}\{X^*, Y\}_G \subseteq \mathbb{C}\{(X_1, \dots, X_{m+1})^*, Y\}_G$ by Lemma 3.8(6); so closure under these blow-up charts follows from closure under blow-up charts (2) and Proposition 4.9. Closure under blow-up charts (5) follows again from Proposition 4.9. This proves Property (7) of [7, Paragraph 1.15].

As the results of [7] do not make use of the Weierstrass Preparation Theorem (which is in general not available in the quasianalytic setting), we may dispense with proving this property here.

**Proof of Main Theorem** The previous discussion implies that our system $\mathcal{A}$ of algebras satisfies Conditions (1) and (4) of [7, Proviso 1.20]. Moreover, Condition (2) is implied by Proposition 4.6, while Condition (3) follows from Proposition 3.4. So the theorem follows from [7, Theorems A and B]. Finally, note that it suffices to add the reciprocal function to obtain quantifier elimination, as all real powers with nonnegative exponents are already in the language $\mathcal{L}_{G^*}$. ■
**Acknowledgment.** The authors would like to thank the Fields Institute for Research in Mathematical Sciences for its hospitality and financial support, as part of this work was done while at its Thematic Program on Tame Geometry and Applications. They would also like to thank the referee for many useful remarks.

**References**

[1] E. Artin, The gamma function, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, Toronto, and London, 1964.

[2] W. Balser, *Formal power series and linear systems of meromorphic ordinary differential equations*, Universitext, Springer, New York, 2000.

[3] Y. Il'yashenko, Finiteness theorems for limit cycles, Translations of Mathematical Monographs, 94, American Mathematical Society, Providence, RI, 1991.

[4] T. Kaiser, J.-P. Rolin, and P. Speissegger, *Transition maps at non-resonant hyperbolic singularities are $o$-minimal*. J. Reine Angew. Math. 636(2009), 1–45.

[5] F. Loray, *Analyse des séries divergentes*. In: Quelques aspects des mathématiques actuelles, Éditions Ellipses, Paris, 1998, pp. 111–173.

[6] J.-P. Ramis and Y. Sibuya, *A new proof of multisummability of formal solutions of nonlinear meromorphic differential equations*. Ann. Inst. Fourier (Grenoble), 44(1994), no. 3, 811–848.

[7] J.-P. Rolin and T. Servi, *Quantifier elimination and rectilinearization theorem for generalized quasianalytic algebras*. Proc. Lond. Math. Soc. (3), 110(2015), no. 5, 1207–1247.

[8] J.-P. Rolin, P. Speissegger, and A. J. Wilkie, *Quasianalytic Denjoy-Carleman classes and $o$-minimality*. J. Amer. Math. Soc. 16(2003), no. 4, 751–777.

[9] J.-C. Tougeron, *Sur les ensembles semi-analytiques avec conditions Gevrey au bord*. Ann. Sci. École Norm. Sup. (4), 27(1994), no. 2, 173–208.

[10] L. Van den Dries, A. Macintyre, and D. Marker, *Logarithmic-exponential power series*. J. Lond. Math. Soc. (2), 56(1997), no. 3, 417–434.

[11] L. Van den Dries and P. Speissegger, *The real field with convergent generalized power series*. Trans. Amer. Math. Soc. 350(1998), no. 11, 4377–4421.

[12] L. Van den Dries and P. Speissegger, *The field of reals with multisummable series and the exponential function*. Proc. Lond. Math. Soc. 81(2000), no. 3, 513–565.

[13] W. Wasow, *Asymptotic expansions for ordinary differential equations*, Dover Publications, New York, 1987.

[14] E. T. Whittaker and G. N. Watson, *A course of modern analysis*, 4th ed., Cambridge University Press, New York, 1962.

Institut de Mathématiques de Bourgogne, Université de Bourgogne Franche-Comté, UMR 5584, CNRS, B.P. 47870, 21078 Dijon, France

e-mail: jean-philippe.rolin@u-bourgogne.fr

Institut de Mathématiques de Jussieu—Paris Rive Gauche, Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75013 Paris, France

e-mail: servi@math.univ-paris-diderot.fr

Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4K1, Canada

e-mail: speisseg@math.mcmaster.ca
**Acknowledgment.** The authors would like to thank the Fields Institute for Research in Mathematical Sciences for its hospitality and financial support, as part of this work was done while at its Thematic Program on Tame Geometry and Applications. They would also like to thank the referee for many useful remarks.

References

[1] E. Artin, The gamma function, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York, Toronto, and London, 1964.

[2] W. Balser, *Formal power series and linear systems of meromorphic ordinary differential equations*, Universitext, Springer, New York, 2000.

[3] Y. Il'yashenko, Finiteness theorems for limit cycles, Translations of Mathematical Monographs, 94, American Mathematical Society, Providence, RI, 1991.

[4] T. Kaiser, J.-P. Rolin, and P. Speissegger, *Transition maps at non-resonant hyperbolic singularities are $o$-minimal*. J. Reine Angew. Math. 636(2009), 1–45.

[5] F. Loray, *Analyse des séries divergentes*. In: Quelques aspects des mathématiques actuelles, Éditions Ellipses, Paris, 1998, pp. 111–173.

[6] J.-P. Ramis and Y. Sibuya, *A new proof of multisummability of formal solutions of nonlinear meromorphic differential equations*. Ann. Inst. Fourier (Grenoble), 44(1994), no. 3, 811–848.

[7] J.-P. Rolin and T. Servi, *Quantifier elimination and rectilinearization theorem for generalized quasianalytic algebras*. Proc. Lond. Math. Soc. (3), 110(2015), no. 5, 1207–1247.

[8] J.-P. Rolin, P. Speissegger, and A. J. Wilkie, *Quasianalytic Denjoy-Carleman classes and $o$-minimality*. J. Amer. Math. Soc. 16(2003), no. 4, 751–777.

[9] J.-C. Tougeron, *Sur les ensembles semi-analytiques avec conditions Gevrey au bord*. Ann. Sci. École Norm. Sup. (4), 27(1994), no. 2, 173–208.

[10] L. Van den Dries, A. Macintyre, and D. Marker, *Logarithmic-exponential power series*. J. Lond. Math. Soc. (2), 56(1997), no. 3, 417–434.

[11] L. Van den Dries and P. Speissegger, *The real field with convergent generalized power series*. Trans. Amer. Math. Soc. 350(1998), no. 11, 4377–4421.

[12] L. Van den Dries and P. Speissegger, *The field of reals with multisummable series and the exponential function*. Proc. Lond. Math. Soc. 81(2000), no. 3, 513–565.

[13] W. Wasow, *Asymptotic expansions for ordinary differential equations*, Dover Publications, New York, 1987.

[14] E. T. Whittaker and G. N. Watson, *A course of modern analysis*, 4th ed., Cambridge University Press, New York, 1962.

Institut de Mathématiques de Bourgogne, Université de Bourgogne Franche-Comté, UMR 5584, CNRS,
B.P. 47870, 21078 Dijon, France

e-mail: jean-philippe.rolin@u-bourgogne.fr

Institut de Mathématiques de Jussieu—Paris Rive Gauche, Université Paris Cité and Sorbonne Université,
CNRS, IMJ-PRG, F-75013 Paris, France

e-mail: servi@math.univ-paris-diderot.fr

Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, ON
L8S 4K1, Canada

e-mail: speisseg@math.mcmaster.ca