$\mathcal{D}({\boldsymbol{\xi}},r_{l})\subset\mathcal{D}({\boldsymbol{\xi}},r_{2})$.
In addition,
$F({\boldsymbol{\Theta}}_{l_{0}}({\boldsymbol{z}}),{\boldsymbol{z}})$ is
analytic on $\mathcal{D}({\boldsymbol{\xi}},r_{2})$. Indeed, the functions
$f_{1,1}({\boldsymbol{z}}),\ldots,f_{r,m_{r}}({\boldsymbol{z}})$ are analytic
on
$\mathcal{D}(\boldsymbol{0},r_{1})\supset\mathcal{D}({\boldsymbol{\xi}},r_{2})$,
while our choice of $l_{0}$ ensures that the coordinates of
${\boldsymbol{\Theta}}_{l_{0}}({\boldsymbol{z}})$ are analytic on
$\mathcal{D}({\boldsymbol{\xi}},r_{2})$.

###  8.3. The key lemma

In this section, we prove the following result from which we will deduce
easily Theorem 6.2 in Section 8.4.

######  Lemma 8.3.

One has
$F({\boldsymbol{\Theta}}_{l_{0}}({\boldsymbol{z}}),{\boldsymbol{z}})=0$.

Let us first briefly describe the general strategy we use for proving this key
lemma. The scheme of the proof is classical and takes its source in the early
work of Mahler [42]. It was gradually improved and refined by Kubota [31],
Loxton and van der Poorten [39], and Ku. Nishioka [50, 51, 52]. It takes here
a more complicate shape, involving the matrices ${\boldsymbol{\Theta}}_{l}$.
This is due to the fact that we have to deal with a bunch of Mahler systems of
the form (6.1.$i$) without any restriction on the matrices
$A_{i}({\boldsymbol{z}}_{i})$.

Assuming by contradiction that
$F({\boldsymbol{\Theta}}_{l_{0}}({\boldsymbol{z}}),{\boldsymbol{z}})\not=0$,
we construct, for every triple of nonnegative integers
$(\delta_{1},\delta_{2},l)$, $l\geq l_{0}$, an auxiliary function of the form

$E({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})=\sum_{j=0}^{\delta_{1}}P_{j}({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})F({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})^{j}\,,$

where $P_{j}(\boldsymbol{Y},{\boldsymbol{z}})$ is a polynomial of degree at
most $\delta_{1}$ in each indeterminate $y_{i,j}$ and of total degree at most
$\delta_{2}$ in the indeterminates $z_{i,j}$. Recall that
$E({\boldsymbol{\Theta}}_{l}({\boldsymbol{\xi}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\xi}})=E(R_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$
and that

$F({\boldsymbol{\Theta}}_{l}({\boldsymbol{\xi}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\xi}})=F(R_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=0\,.$

Hence
$E(R_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=P_{0}(R_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$.
As our construction ensures that $P_{0}\not\in\mathcal{I}$, we have
$P_{0}(R_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})\not=0$
for many integers $l$, and we can use Liouville’s inequality to find a lower
bound for
$E(R_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$.
On the other hand, we have many choices for the polynomials $P_{i}$ in the
construction of our auxiliary function. This level of freedom is used to show
that, for a good choice of such polynomials, the quantity
$E(R_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$
is small enough. More precisely, we obtain a contradiction between the upper
and the lower bounds when the parameter $\delta_{1}$ is sufficiently large,
the parameter $\delta_{2}$ is sufficiently large with respect to $\delta_{1}$,
and the parameter $l$ is sufficiently large with respect to $\delta_{1}$ and
$\delta_{2}$.

### Warning

The auxiliary function
$E({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$
can be though of as a simultaneous Padé approximant of type I for the first
$\delta_{1}$th powers of
$F({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$.
However, we have to be careful:
$F({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$
it is not necessarily a power series in ${\boldsymbol{z}}$. It is a linear
combination of the power series
$f_{1,1}(T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}}),\ldots,f_{r,m_{r}}(T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$
whose coefficients are only known to be algebraic over
$\overline{\mathbb{Q}}({\boldsymbol{z}})$. We only know that it is analytic at
${\boldsymbol{\xi}}$.

In what follows, we argue by contradiction, assuming that

(8.8)
$F({\boldsymbol{\Theta}}_{l_{0}}({\boldsymbol{z}}),{\boldsymbol{z}})\not=0\,.$

We divide the proof of Lemma 8.3 into four steps.

####  8.3.1. First step: construction of the auxiliary function

Given a formal power series
$E=\sum_{{\boldsymbol{\lambda}}}e_{\boldsymbol{\lambda}}(\boldsymbol{Y}){\boldsymbol{z}}^{\boldsymbol{\lambda}}\in\overline{\mathbb{Q}}[\boldsymbol{Y}][[{\boldsymbol{z}}]]$
and an integer $p>0$, we let

$E_{p}:=\sum_{|{\boldsymbol{\lambda}}|<p}e_{\boldsymbol{\lambda}}(\boldsymbol{Y}){\boldsymbol{z}}^{\boldsymbol{\lambda}}\in\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]$

denote the truncation of $E$ at order $p$ with respect to ${\boldsymbol{z}}$.
We recall that $\mathcal{I}^{\perp}(\delta_{1},\delta_{2})$ is a complement to
$\mathcal{I}(\delta_{1},\delta_{2})$ in
$\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{\delta_{1},\delta_{2}}$.

######  Lemma 8.4.

Let $\delta_{1}\geq 0$ be an integer. For all integers $\delta_{2}$,
$\delta_{2}\gg\delta_{1}$, there exist polynomials
$P_{i}\in\mathcal{I}^{\perp}(\delta_{1},\delta_{2})$, $0\leq i\leq\delta_{1}$,
not all zero, such that the formal power series

$E^{\prime}(\boldsymbol{Y},{\boldsymbol{z}}):=\displaystyle\sum_{j=0}^{\delta_{1}}P_{j}(\boldsymbol{Y},{\boldsymbol{z}})F(\boldsymbol{Y},{\boldsymbol{z}})^{j}\in\overline{\mathbb{Q}}[\boldsymbol{Y}][[{\boldsymbol{z}}]]$

satisfies
$E^{\prime}_{p}({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})=0$
for all $l\geq l_{0}$, where
$p=\left\lfloor\frac{\delta_{1}^{1/N}\delta_{2}}{2^{(t^{2}+2)/N}}\right\rfloor$.

###### Proof.

Set

$\mathcal{J}(\delta_{1},\delta_{2}):=\\{P\in\overline{\mathbb{Q}}[{\boldsymbol{z}},\boldsymbol{Y}]\
:\
P({\boldsymbol{R}}_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\alpha}}){\boldsymbol{\phi}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})^{-1}\boldsymbol{Y},{\boldsymbol{z}})\in\mathcal{I}(\delta_{1},\delta_{2})\\}\,.$

The $\overline{\mathbb{Q}}$-vector spaces $\mathcal{J}(\delta_{1},\delta_{2})$
and $\mathcal{I}(\delta_{1},\delta_{2})$ have same dimension. This follows
directly from the fact that the map

$\begin{array}[]{ccc}\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{\delta_{1},\delta_{2}}&\rightarrow&\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{\delta_{1},\delta_{2}}\\\
P(\boldsymbol{Y},{\boldsymbol{z}})&\mapsto&P({\boldsymbol{R}}_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\alpha}}){\boldsymbol{\phi}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})^{-1}\boldsymbol{Y},{\boldsymbol{z}})\end{array}$

is an isomorphism, the matrix
${\boldsymbol{R}}_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\alpha}}){\boldsymbol{\phi}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})^{-1}$
being invertible. Furthermore, we have

(8.9)
$P({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})=0\,,\;\forall
P\in\mathcal{J}(\delta_{1},\delta_{2}),\,\forall l\geq l_{0}\,.$

Indeed, if $P\in\mathcal{J}(\delta_{1},\delta_{2})$, then
$P({\boldsymbol{R}}_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\alpha}}){\boldsymbol{\phi}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})^{-1}\boldsymbol{Y},{\boldsymbol{z}})\in\mathcal{I}(\delta_{1},\delta_{2})$,
and Lemma 7.6 implies that

$P({\boldsymbol{R}}_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\alpha}}){\boldsymbol{\phi}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})^{-1}{\boldsymbol{\phi}}({\boldsymbol{z}}){\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{z}}),T_{\boldsymbol{k}}{\boldsymbol{z}})=0\,,\;\forall{\boldsymbol{k}}\in
V_{+}^{\perp}\,.$

For $l\geq l_{0}$, using the previous equality with
${\boldsymbol{k}}:={\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}$, we obtain
that

$P({\boldsymbol{R}}_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\alpha}}){\boldsymbol{\phi}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})^{-1}{\boldsymbol{\phi}}({\boldsymbol{z}}){\boldsymbol{R}}_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})=0\,.$

By (8.7), we thus have
$P({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})=0$.

Let $p$ be as in the lemma and let us consider the three
$\overline{\mathbb{Q}}$-linear maps:

$\displaystyle\left\\{\begin{array}[]{cc}\prod_{j=0}^{\delta_{1}}\mathcal{I}^{\perp}(\delta_{1},\delta_{2})\\\
(P_{0}(\boldsymbol{Y},{\boldsymbol{z}}),\ldots,P_{\delta_{1}}(\boldsymbol{Y},{\boldsymbol{z}}))\end{array}\right.$
$\displaystyle\big{\downarrow}$
$\displaystyle\left\\{\begin{array}[]{c}\overline{\mathbb{Q}}[\boldsymbol{Y}]_{2\delta_{1}}[[{\boldsymbol{z}}]]\\\
E^{\prime}(\boldsymbol{Y},{\boldsymbol{z}}):=\sum_{j=0}^{\delta_{1}}P_{j}(\boldsymbol{Y},{\boldsymbol{z}})F(\boldsymbol{Y},{\boldsymbol{z}})^{j}\end{array}\right.$
$\displaystyle\big{\downarrow}$
$\displaystyle\left\\{\begin{array}[]{c}\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{2\delta_{1},p-1}\\\
E^{\prime}_{p}(\boldsymbol{Y},{\boldsymbol{z}})\end{array}\right.$
$\displaystyle\big{\downarrow}$
$\displaystyle\left\\{\begin{array}[]{c}\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{2\delta_{1},p-1}/\mathcal{J}(2\delta_{1},p-1)\\\
E^{\prime}_{p}(\boldsymbol{Y},{\boldsymbol{z}})\mod\mathcal{J}(2\delta_{1},p-1)\end{array}\right.$

Note that these maps are well-defined. By Lemma 7.1, the dimension of the
$\overline{\mathbb{Q}}$-vector space
$\mathcal{I}^{\perp}(\delta_{1},\delta_{2})$ is at least equal to
$\frac{c_{1}(\delta_{1})}{2}\delta_{2}^{N}$, assuming that $\delta_{2}$ is
large enough. It follows that

(8.14)
$\dim_{\overline{\mathbb{Q}}}\left(\prod_{j=0}^{\delta_{1}}\mathcal{I}^{\perp}(\delta_{1},\delta_{2})\right)\geq\frac{c_{1}(\delta_{1})}{2}(\delta_{1}+1)\delta_{2}^{N}\,.$

For every pair of nonnegative integers $(n,m)$, set

$\overline{\mathcal{J}}(n,m):=\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{n,m}/\mathcal{J}(n,m)\,.$

Since $\mathcal{J}(\delta_{1},\delta_{2})$ and
$\mathcal{I}(\delta_{1},\delta_{2})$ have same dimension, Lemma 7.2 implies
that

$\dim_{\overline{\mathbb{Q}}}\overline{\mathcal{J}}(2\delta_{1},p-1)\leq
2^{t^{2}}\dim_{\overline{\mathbb{Q}}}\overline{\mathcal{J}}(\delta_{1},p-1)\,.$

Now, if $\delta_{2}$ is sufficiently large, Lemma 7.1 ensures that

$\dim_{\overline{\mathbb{Q}}}\overline{\mathcal{J}}(\delta_{1},p-1)\leq
2c_{1}(\delta_{1})p^{N}\,.$

On the other hand, the choice of $p$ ensures that

$2^{t^{2}}\left(2c_{1}(\delta_{1})p^{N}\right)<\frac{c_{1}(\delta_{1})}{2}(\delta_{1}+1)\delta_{2}^{N}$

and (8.14) implies that

$\dim_{\overline{\mathbb{Q}}}\left(\prod_{j=0}^{\delta_{1}}\mathcal{I}^{\perp}(\delta_{1},\delta_{2})\right)>\dim_{\overline{\mathbb{Q}}}\left(\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{2\delta_{1},p-1}/\mathcal{J}(2\delta_{1},p-1)\right)\,.$

Hence the $\overline{\mathbb{Q}}$-linear map defined by

$(P_{0}(\boldsymbol{Y},{\boldsymbol{z}}),\ldots,P_{\delta_{1}}(\boldsymbol{Y},{\boldsymbol{z}}))\mapsto
E^{\prime}_{p}(\boldsymbol{Y},{\boldsymbol{z}})\mod\mathcal{J}(2\delta_{1},p-1)$

has a nontrivial kernel. We deduce the existence of polynomials
$P_{0},\ldots,P_{\delta_{1}}$ in $\mathcal{I}^{\perp}(\delta_{1},\delta_{2})$,
not all zero, such that $E^{\prime}_{p}\in\mathcal{J}(2\delta_{1},p-1)$. By
(8.9), we obtain that
$E^{\prime}_{p}({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})=0$
for all $l\geq l_{0}$. This ends the proof. ∎

Let $E^{\prime}$ be a formal power series satisfying the properties of Lemma
8.4 and let $v_{0}$ be the smallest index such that the polynomial $P_{v_{0}}$
is nonzero. Then the formal power series

(8.15) $E(\boldsymbol{Y},{\boldsymbol{z}}):=\sum_{j\geq
v_{0}}P_{j}(\boldsymbol{Y},{\boldsymbol{z}})F(\boldsymbol{Y},{\boldsymbol{z}})^{j-v_{0}}\in\overline{\mathbb{Q}}[\boldsymbol{Y}][[{\boldsymbol{z}}]]\,$

is the auxiliary function that we were looking for. Note that we have

(8.16)
$E(\boldsymbol{Y},{\boldsymbol{z}})F^{v_{0}}(\boldsymbol{Y},{\boldsymbol{z}})=E^{\prime}(\boldsymbol{Y},{\boldsymbol{z}})\,.$

####  8.3.2. Second step: upper bound for
$|E({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})|$

The aim of this section is to prove that there exists a real number $c_{2}>0$
such that

(8.17)
$|E({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})|\leq
e^{-c_{2}e^{l}\delta_{1}^{1/N}\delta_{2}}\,,\;\;\forall
l\gg\delta_{2}\gg\delta_{1}.$

Let us first observe that

$b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})^{(t^{2}+1)\delta_{1}}E^{\prime}({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$

defines an analytic function on $\mathcal{D}({\boldsymbol{\xi}},r_{2})$. Then,
on this polydisc, it has a power series expansion

(8.18)
$b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})^{(t^{2}+1)\delta_{1}}E^{\prime}({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})=\sum_{{\boldsymbol{\lambda}}\in\mathbb{N}^{N}}\epsilon_{{\boldsymbol{\lambda}},l}({\boldsymbol{z}}-{\boldsymbol{\xi}})^{{\boldsymbol{\lambda}}}\,,$

where $\epsilon_{{\boldsymbol{\lambda}},l}\in\mathbb{C}$. Let $p$ be defined
as in Lemma 8.4. We first prove the following result.

######  Lemma 8.5.

There exists a positive real number $\gamma$ that does not depend on
$\delta_{1},\delta_{2}$, ${\boldsymbol{\lambda}}$, and $l$, such that

$|\epsilon_{{\boldsymbol{\lambda}},l}|\leq e^{-\gamma e^{l}p}\,,\;\forall
l\gg\delta_{1},\delta_{2},{\boldsymbol{\lambda}}\,.$

###### Proof.

Set

$G(\boldsymbol{Y},{\boldsymbol{z}}):=E^{\prime}(\boldsymbol{Y},{\boldsymbol{z}})-E^{\prime}_{p}(\boldsymbol{Y},{\boldsymbol{z}})\in\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}[\boldsymbol{Y}]\,.$

By Lemma 8.4, we have

(8.19)
$G({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})=E^{\prime}({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})\,,\;\forall
l\geq l_{0}\,.$

Let ${\boldsymbol{\nu}}_{1},\ldots,{\boldsymbol{\nu}}_{s}$ denote an
enumeration of the $t\times t$ matrices with nonnegative integer coefficients
that can be decomposed as

${\boldsymbol{\nu}}_{l}=M_{l}+N_{l}\,,$

where $M_{l}$ has coefficients in the set $\\{0,1,\ldots,\delta_{1}\\}$ and
$N_{l}$ is a matrix with nonnegative integer coefficients whose sum is at most
$\delta_{1}$. The sum of the coefficients of each matrix
${\boldsymbol{\nu}}_{i}$ is at most equal to $(t^{2}+1)\delta_{1}$. There
exists a unique decomposition of the form

$G(\boldsymbol{Y},{\boldsymbol{z}})=\sum_{i=1}^{s}\sum_{|{\boldsymbol{\lambda}}|\geq
p}g_{{\boldsymbol{\lambda}},i}{\boldsymbol{z}}^{\boldsymbol{\lambda}}\boldsymbol{Y}^{{\boldsymbol{\nu}}_{i}}\,,$

where $g_{{\boldsymbol{\lambda}},i}\in\mathbb{C}$. For every $i$, $1\leq i\leq
s$, we define the formal power series

$G_{i}({\boldsymbol{z}}):=\sum_{|{\boldsymbol{\lambda}}|\geq
p}g_{{\boldsymbol{\lambda}},i}{\boldsymbol{z}}^{\boldsymbol{\lambda}}\in\mathbb{C}[[{\boldsymbol{z}}]]\,.$

By definition of $F(\boldsymbol{Y},{\boldsymbol{z}})$, these series belong to
$\overline{\mathbb{Q}}[{\boldsymbol{z}},{\boldsymbol{f}}({\boldsymbol{z}})]$.
In particular, they are analytic on some polydisc
$\mathcal{D}(\boldsymbol{0},r)$ with $r>r_{1}$. By (3.1), there exists a
positive real number $\gamma_{1}(\delta_{1},\delta_{2})$ such that

(8.20)
$|g_{{\boldsymbol{\lambda}},i}|\leq\gamma_{1}(\delta_{1},\delta_{2})r_{1}^{-|{\boldsymbol{\lambda}}|}\,,\;\forall{\boldsymbol{\lambda}}\in\mathbb{N}^{N}\,.$

On the other hand, Lemma 5.5 ensures the existence of two positive real
numbers $\kappa_{1}$ and $\kappa_{2}$ such that

(8.21)
$\kappa_{1}e^{l}|{\boldsymbol{\lambda}}|\leq|{\boldsymbol{\lambda}}T_{{\boldsymbol{k}}_{l}}|\leq\kappa_{2}e^{l}|{\boldsymbol{\lambda}}|\,\;\;\forall
l,\forall{\boldsymbol{\lambda}}.$

For every $l\geq l_{0}$,
$G_{i}(T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$ can
thus be written as

(8.22)
$G_{i}(T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})=\sum_{|{\boldsymbol{\lambda}}|\geq\kappa_{1}e^{l}p}g_{{\boldsymbol{\lambda}},i,l}{\boldsymbol{z}}^{\boldsymbol{\lambda}}\,,$

with $g_{{\boldsymbol{\lambda}},i,l}\in\mathbb{C}$. Furthermore, this power
series is absolutely convergent on the polydisc
$\mathcal{D}(\boldsymbol{0},r_{1})$. Since the matrix
$T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}$ is invertible, has
nonnegative coefficients, and $r_{1}\leq 1$, we deduce from (8.20) that

(8.23)
$|g_{{\boldsymbol{\lambda}},i,l}|\leq\gamma_{1}(\delta_{1},\delta_{2})r_{1}^{-|{\boldsymbol{\lambda}}|}\,,$

for all ${\boldsymbol{\lambda}}$, $i\in\\{1,\ldots,s\\}$, and $l\geq l_{0}$.
On the other hand, every function
$G_{i}(T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$,
$1\leq i\leq s$, $l\geq l_{0}$, is analytic on the polydisc
$\mathcal{D}({\boldsymbol{\xi}},r_{2})$. The power series expansion of
$G_{i}(T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$ at
${\boldsymbol{\xi}}$ is thus absolutely convergent on
$\mathcal{D}({\boldsymbol{\xi}},r_{2})$, and we can write

(8.24)
$G_{i}(T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})=\sum_{{\boldsymbol{\lambda}}\in\mathbb{N}^{N}}h_{{\boldsymbol{\lambda}},i,l}({\boldsymbol{z}}-{\boldsymbol{\xi}})^{\boldsymbol{\lambda}}\,,$

where $h_{{\boldsymbol{\lambda}},i,l}\in\mathbb{C}$. Since by assumption
$\mathcal{D}({\boldsymbol{\xi}},r_{2})\subset\mathcal{D}(\boldsymbol{0},r_{1})$,
the two power series expansions (8.22) and (8.24) match on
$\mathcal{D}({\boldsymbol{\xi}},r_{2})$ . Using the equality
${\boldsymbol{z}}^{{\boldsymbol{\lambda}}}=(({\boldsymbol{z}}-{\boldsymbol{\xi}})+{\boldsymbol{\xi}})^{\boldsymbol{\lambda}}$
and identifying, for every ${\boldsymbol{\lambda}}\in\mathbb{N}^{N}$, the
terms in $({\boldsymbol{z}}-{\boldsymbol{\xi}})^{{\boldsymbol{\lambda}}}$ in
(8.22) and (8.24), we obtain that

(8.25)
$h_{{\boldsymbol{\lambda}},i,l}=\sum_{\begin{subarray}{c}|{\boldsymbol{\gamma}}|\geq\kappa_{1}e^{l}p\\\
{\boldsymbol{\gamma}}\geq{\boldsymbol{\lambda}}\end{subarray}}\binom{{\boldsymbol{\gamma}}}{{\boldsymbol{\lambda}}}g_{{\boldsymbol{\gamma}},i,l}{\boldsymbol{\xi}}^{{\boldsymbol{\gamma}}-{\boldsymbol{\lambda}}}\,.$

For ${\boldsymbol{\gamma}}\geq{\boldsymbol{\lambda}}$, one has

(8.26)
$\binom{{\boldsymbol{\gamma}}}{{\boldsymbol{\lambda}}}=\prod_{i=1}^{N}\frac{\gamma_{i}!}{(\gamma_{i}-\lambda_{i})!\lambda_{i}!}\leq\prod_{i=1}^{N}\gamma_{i}^{\lambda_{i}}\leq|{\boldsymbol{\gamma}}|^{|{\boldsymbol{\lambda}}|}\,.$

Given ${\boldsymbol{\lambda}}\in\mathbb{N}^{N}$,
$|{\boldsymbol{\lambda}}|<\kappa_{1}e^{l}p$ as soon as $l$ is large enough,
and since $\|{\boldsymbol{\xi}}\|<r_{1}$, we infer from (8.23) and (8.25) the
existence of a real number $\gamma_{2}>0$ that does not depend on
$\delta_{1}$, $\delta_{2}$, ${\boldsymbol{\lambda}}$, and $l$, such that

(8.27) $|h_{{\boldsymbol{\lambda}},i,l}|\leq e^{-\gamma_{2}e^{l}p}\,,\;\forall
l\gg\delta_{1},\delta_{2},{\boldsymbol{\lambda}}\,.$

By Remark 8.2, the monomial
$(b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}}){\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}))^{{\boldsymbol{\nu}}_{i}}$
is analytic on $\mathcal{D}({\boldsymbol{\xi}},r_{2})$ for every $i$, $1\leq
i\leq s$, and every $l\geq l_{0}$. Multiplying by
$b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})^{(t^{2}+1)\delta_{1}-|{\boldsymbol{\nu}}_{i}|}$,
we obtain on $\mathcal{D}({\boldsymbol{\xi}},r_{2})$ a power series expansion
of the form

(8.28)
$b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})^{(t^{2}+1)\delta_{1}-|{\boldsymbol{\nu}}_{i}|}(b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}}){\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}))^{{\boldsymbol{\nu}}_{i}}=\sum_{{\boldsymbol{\lambda}}\in\mathbb{N}^{N}}\theta_{{\boldsymbol{\lambda}},i,l}({\boldsymbol{z}}-{\boldsymbol{\xi}})^{{\boldsymbol{\lambda}}}\,,$

where $\theta_{{\boldsymbol{\lambda}},i,l}\in\mathbb{C}$.

Given $Q({\boldsymbol{z}})\in\overline{\mathbb{Q}}[{\boldsymbol{z}}]$ and
$l\geq 0$, the polynomial
$Q_{l}({\boldsymbol{z}}):=Q(T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$
can be converted into a polynomial in $({\boldsymbol{z}}-{\boldsymbol{\xi}})$:

$Q_{l}({\boldsymbol{z}})=\sum_{{\boldsymbol{\lambda}}}q_{{\boldsymbol{\lambda}},l}({\boldsymbol{z}}-{\boldsymbol{\xi}})^{\boldsymbol{\lambda}}\,.$

Using the relation

${\boldsymbol{z}}^{\boldsymbol{\gamma}}=\sum_{{\boldsymbol{\lambda}}\leq{\boldsymbol{\gamma}}}\binom{{\boldsymbol{\gamma}}}{{\boldsymbol{\lambda}}}{\boldsymbol{\xi}}^{{\boldsymbol{\lambda}}-{\boldsymbol{\gamma}}}({\boldsymbol{z}}-{\boldsymbol{\xi}})^{\boldsymbol{\lambda}}\,,$

(8.21) and (8.26), we obtain that, for every ${\boldsymbol{\lambda}}$,
$|q_{{\boldsymbol{\lambda}},l}|=\mathcal{O}(e^{l})$, where the underlying
constant in the $\mathcal{O}$ notation depends both on $Q({\boldsymbol{z}})$
and ${\boldsymbol{\lambda}}$. Set
$\Gamma_{\boldsymbol{k}}({\boldsymbol{z}}):=b_{{\boldsymbol{k}}}({\boldsymbol{z}}){\boldsymbol{R}}_{{\boldsymbol{k}}}({\boldsymbol{z}})\in\overline{\mathbb{Q}}[{\boldsymbol{z}}]$
and

$\Gamma_{{\boldsymbol{k}},l}({\boldsymbol{z}}):=b_{{\boldsymbol{k}}}(T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}}){\boldsymbol{R}}_{{\boldsymbol{k}}}(T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})\,.$

Since there are only finitely many differences
${\boldsymbol{k}}_{l+1}-{\boldsymbol{k}}_{l}$, the coefficients in
$({\boldsymbol{z}}-{\boldsymbol{\xi}})^{\boldsymbol{\lambda}}$ for the entries
of $\Gamma_{{\boldsymbol{k}}_{l+1}-{\boldsymbol{k}}_{l},l}({\boldsymbol{z}})$,
seen as polynomials in ${\boldsymbol{z}}-{\boldsymbol{\xi}}$, belong to
$\mathcal{O}(e^{l})$. Using the recurrence relation

$b_{{\boldsymbol{k}}_{l+1}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}}){\boldsymbol{\Theta}}_{l+1}({\boldsymbol{z}})=b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}}){\boldsymbol{\Theta}}_{l}({\boldsymbol{z}})\Gamma_{{\boldsymbol{k}}_{l+1}-{\boldsymbol{k}}_{l},l}({\boldsymbol{z}})\,,$

we infer from (6.6) that, for every ${\boldsymbol{\lambda}}$, there exists a
real number $\gamma_{3}(\delta_{1},{\boldsymbol{\lambda}})>0$ that does not
depend on $i$, $\delta_{2}$, and $l$, and such that

(8.29)
$|\theta_{{\boldsymbol{\lambda}},i,l}|<e^{\gamma_{3}(\delta_{1},{\boldsymbol{\lambda}}){l}}\,,\;\;\forall
i,\,1\leq i\leq s,\,\forall l\geq l_{0}\,.$

From (8.18), (8.19), (8.24), and (8.28), we deduce that

(8.30)
$\sum_{i=1}^{s}\left(\sum_{{\boldsymbol{\lambda}}\in\mathbb{N}^{N}}h_{{\boldsymbol{\lambda}},i,l}({\boldsymbol{z}}-{\boldsymbol{\xi}})^{\boldsymbol{\lambda}}\right)\left(\sum_{{\boldsymbol{\lambda}}\in\mathbb{N}^{N}}\theta_{{\boldsymbol{\lambda}},i,l}({\boldsymbol{z}}-{\boldsymbol{\xi}})^{\boldsymbol{\lambda}}\right)=\sum_{{\boldsymbol{\lambda}}\in\mathbb{N}^{N}}\epsilon_{{\boldsymbol{\lambda}},l}({\boldsymbol{z}}-{\boldsymbol{\xi}})^{{\boldsymbol{\lambda}}}\,.$

Moreover, since the power series involved in (8.24) and (8.28) are absolutely
convergent on $\mathcal{D}({\boldsymbol{\xi}},r_{2})$, we obtain

$\displaystyle\sum_{i=1}^{s}\left(\sum_{{\boldsymbol{\lambda}}\in\mathbb{N}^{N}}h_{{\boldsymbol{\lambda}},i,l}({\boldsymbol{z}}-{\boldsymbol{\xi}})^{\boldsymbol{\lambda}}\right)\left(\sum_{{\boldsymbol{\lambda}}\in\mathbb{N}^{N}}\theta_{{\boldsymbol{\lambda}},i,l}({\boldsymbol{z}}-{\boldsymbol{\xi}})^{\boldsymbol{\lambda}}\right)$
(8.31)
$\displaystyle=\sum_{{\boldsymbol{\lambda}}\in\mathbb{N}^{N}}\sum_{i=1}^{s}\sum_{{\boldsymbol{\gamma}}\leq{\boldsymbol{\lambda}}}h_{{\boldsymbol{\gamma}},i,l}\theta_{{\boldsymbol{\lambda}}-{\boldsymbol{\gamma}},i,l}({\boldsymbol{z}}-{\boldsymbol{\xi}})^{\boldsymbol{\lambda}}\,.$

Finally, identifying the terms in
$({\boldsymbol{z}}-{\boldsymbol{\xi}})^{{\boldsymbol{\lambda}}}$ in (8.30)
thanks to (8.3.2), we get that

$\epsilon_{{\boldsymbol{\lambda}},l}=\sum_{i=1}^{s}\sum_{{\boldsymbol{\gamma}}\leq{\boldsymbol{\lambda}}}h_{{\boldsymbol{\gamma}},i,l}\theta_{{\boldsymbol{\lambda}}-{\boldsymbol{\gamma}},i,l}\,.$

Now, we fix ${\boldsymbol{\lambda}}\in\mathbb{N}^{N}$. By (8.27) and (8.29),
there exists a real number $\gamma_{4}>0$ that does not depend on
$\delta_{1}$, $\delta_{2}$, ${\boldsymbol{\lambda}}$, and $l$, such that

$|\epsilon_{{\boldsymbol{\lambda}},l}|\leq e^{-\gamma_{4}e^{l}p}\,,\;\;\forall
l\gg\delta_{1},\delta_{2},{\boldsymbol{\lambda}}\,.$

Setting $\gamma=\gamma_{4}$, this ends the proof. ∎

Now, let us observe that the function
$b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})^{(t^{2}+1)\delta_{1}}E({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$
is analytic on $\mathcal{D}({\boldsymbol{\xi}},\boldsymbol{r}_{2})$. Hence it
has a power series expansion of the form

(8.32)
$b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})^{(t^{2}+1)\delta_{1}}E({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})=\sum_{{\boldsymbol{\lambda}}\in\mathbb{N}^{N}}e_{{\boldsymbol{\lambda}},l}({\boldsymbol{z}}-{\boldsymbol{\xi}})^{{\boldsymbol{\lambda}}}\,,$

with $e_{{\boldsymbol{\lambda}},l}\in\mathbb{C}$. Also,
$F({\boldsymbol{\Theta}}_{l_{0}}({\boldsymbol{z}}),{\boldsymbol{z}})^{v_{0}}$
is analytic on $\mathcal{D}({\boldsymbol{\xi}},r_{2})$, so that

(8.33)
$F({\boldsymbol{\Theta}}_{l_{0}}({\boldsymbol{z}}),{\boldsymbol{z}})^{v_{0}}=\sum_{{\boldsymbol{\lambda}}\in\mathbb{N}^{N}}a_{{\boldsymbol{\lambda}}}({\boldsymbol{z}}-{\boldsymbol{\xi}})^{\boldsymbol{\lambda}}\,,$

with $a_{{\boldsymbol{\lambda}}}\in\mathbb{C}$. Using (8.1), we get that

$F({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})=F({\boldsymbol{\Theta}}_{l_{0}}({\boldsymbol{z}}),{\boldsymbol{z}})\,,\;\;\forall
l\geq l_{0}\,.$

This equality is a priori valid for ${\boldsymbol{z}}$ in
$\mathcal{D}({\boldsymbol{\xi}},r_{l})$ (see Remark 8.2), but it extends to
$\mathcal{D}({\boldsymbol{\xi}},r_{2})$ by analytic continuation. By (8.16),
we thus have

(8.34)
$E({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})F({\boldsymbol{\Theta}}_{l_{0}}({\boldsymbol{z}}),{\boldsymbol{z}})^{v_{0}}=E^{\prime}({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})\,,$

for all $l\geq l_{0}$ and all
${\boldsymbol{z}}\in\mathcal{D}({\boldsymbol{\xi}},r_{l})$. By (8.8),
$F({\boldsymbol{\Theta}}_{l_{0}}({\boldsymbol{z}}),{\boldsymbol{z}})$ is
nonzero and there thus exists at least one nonzero coefficient
$a_{{\boldsymbol{\lambda}}}$ in (8.33). Let us consider an index
${\boldsymbol{\lambda}}_{0}$ such that $a_{{\boldsymbol{\lambda}}_{0}}\not=0$
with $|{\boldsymbol{\lambda}}_{0}|$ minimal. Multiplying both sides of (8.34)
by
$b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})^{(t^{2}+1)\delta_{1}}$,
and identifying the coefficients in
$({\boldsymbol{z}}-{\boldsymbol{\xi}})^{{\boldsymbol{\lambda}}_{0}}$ in their
power series expansion on $\mathcal{D}({\boldsymbol{\xi}},r_{2})$ with the
help of (8.18), (8.32), and (8.33), we obtain that

$e_{0,l}a_{{\boldsymbol{\lambda}}_{0}}=\epsilon_{{\boldsymbol{\lambda}}_{0},l}\,,\;\;\forall
l\geq l_{0}\,.$

Since
${\boldsymbol{\Theta}}_{l}({\boldsymbol{\xi}})={\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}})$,
we infer from Lemma 8.5 and the definition of $p$ (see Lemma 8.4), the
existence of a real number $\beta_{1}>0$ that does not depend on $\delta_{1}$,
$\delta_{2}$, and $l$, such that

$\displaystyle\left|b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\xi}})^{(t^{2}+1)\delta_{1}}E({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})\right|$
$\displaystyle=$
$\displaystyle\left|b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\xi}})^{(t^{2}+1)\delta_{1}}E({\boldsymbol{\Theta}}_{l}({\boldsymbol{\xi}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\xi}})\right|$
$\displaystyle=$ $\displaystyle|e_{0,l}|$ $\displaystyle\leq$ $\displaystyle
e^{-\beta_{1}e^{l}\delta_{1}^{1/N}\delta_{2}}\,,\;\;\forall
l\gg\delta_{1},\delta_{2}\,.$

Now, it just remains to find a lower bound for
$|b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\xi}})^{(t^{2}+1)\delta_{1}}|$.
By (8.21), there exists a positive real number $\beta_{2}$ that does not
depend on $l$, $\delta_{1}$, and $\delta_{2}$, such that the degree of the
polynomial
$b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})^{(t^{2}+1)\delta_{1}}$
is at most equal to $\beta_{2}e^{l}\delta_{1}$. Each
${\boldsymbol{\alpha}}_{i}$ being by assumption regular w.r.t. (6.1.$i$),
$b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\xi}})\neq 0$
for all $l\geq l_{0}$. Since the numbers
$b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\xi}})$,
$l\geq l_{0}$, belong to some fix number field, we infer from Liouville’s
inequality the existence of a real number $\beta_{3}>0$ that does not depend
on $l$, $\delta_{1}$, and $\delta_{2}$, such that

(8.36)
$|b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\xi}})|^{(t^{2}+1)\delta_{1}}\geq
e^{-\beta_{3}e^{l}\delta_{1}}\,,\;\;\forall l,\,\forall\delta_{1}\,.$

By (8.3.2) and (8.36), there exists a real number $\beta_{4}>0$ that does not
depend on $l$, $\delta_{1}$, and $\delta_{2}$, such that

$\displaystyle|E({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})|$
$\displaystyle\leq$ $\displaystyle
e^{\beta_{3}e^{l}\delta_{1}-\beta_{1}e^{l}\delta_{1}^{1/N}\delta_{2}}$
$\displaystyle\leq$ $\displaystyle
e^{-\beta_{4}e^{l}\delta_{1}^{1/N}\delta_{2}}\,,\;\;\forall
l\gg\delta_{2}\gg\delta_{1}\,.$

Setting $c_{2}=\beta_{4}$, this proves the upper bound (8.17).

####  8.3.3. Third step: lower bound for
$|E({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})|$

Let us first recall that, by (8.6), we have

$F({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=0\,,\;\;\forall
l\,.$

By construction of our auxiliary function, we deduce that

$E({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=P_{v_{0}}({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})\,.$

Furthermore, since this construction ensures that
$P_{v_{0}}\notin\mathcal{I}$, there exists an infinite set of positive
integers $\mathcal{E}$ such that

$P_{v_{0}}({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})\not=0\,,\;\;\forall
l\in\mathcal{E}\,.$

Since $P_{v_{0}}$ has degree at most $\delta_{1}$ in each indeterminate
$y_{i,j}$, and total degree at most $\delta_{2}$ in ${\boldsymbol{z}}$,
Liouville’s inequality and a computation similar to the previous one ensure
the existence of a real number $c_{3}>0$ that does not depend on $\delta_{1}$,
$\delta_{2}$, and $l$, such that

(8.37)
$|E({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})|=|P_{0}({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})|\geq
e^{-c_{3}e^{l}\delta_{2}}\,,\;\;\forall
l\in\mathcal{E},\delta_{2}\geq\delta_{1}\,.$

####  8.3.4. Fourth step: contradiction

By Inequalities (8.17) and (8.37), we obtain that

$e^{-c_{3}e^{l}\delta_{2}}\leq|E({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})|\leq
e^{-c_{2}e^{l}\delta_{1}^{1/N}\delta_{2}}\,,\;\;\forall
l\in\mathcal{E},\,l\gg\delta_{2}\gg\delta_{1}\,.$

Finally, we deduce that

$c_{3}\geq c_{2}\delta_{1}^{1/N}\,.$

Since $c_{2}$ and $c_{3}$ do not depend on $\delta_{1}$, this inequality
provides a contradiction, as soon as $\delta_{1}$ is large enough. This end
the proof of Lemma 8.3. ∎

###  8.4. End of the proof of Theorem 6.2

Let us recall that
$d({\boldsymbol{z}})\in\overline{\mathbb{Q}}[{\boldsymbol{z}}]$ stands for the
least common multiple of the denominators of the coefficients of the matrices
${\boldsymbol{\phi}}_{j}({\boldsymbol{z}})$ defined in (7.10). Hence
$d(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}}){\boldsymbol{\Theta}}_{l_{0}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$
has coefficients in
$\overline{\mathbb{Q}}[{\boldsymbol{z}},\varphi(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})]$.
Lemma 7.7 ensures that
$d(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})\neq 0$. Let
$q({\boldsymbol{z}})$ denote the least common multiple of the denominators of
the coefficients of the matrix
${\boldsymbol{R}}_{{\boldsymbol{k}}_{l_{0}}}^{-1}({\boldsymbol{z}})$. Since,
for every i, ${\boldsymbol{\alpha}}_{i}$ is assumed to be regular w.r.t.
(6.1.$i$), we have that $q({\boldsymbol{\alpha}})\not=0$. Similarly,
$b_{\boldsymbol{k}}({\boldsymbol{z}}){\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{z}})$
has coefficients in $\overline{\mathbb{Q}}[{\boldsymbol{z}}]$ and
$b_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\alpha}})\not=0$.

By Lemma 8.3, we know that
$F({\boldsymbol{\Theta}}_{l_{0}}({\boldsymbol{z}}),{\boldsymbol{z}})=0$, and
substituting $T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}}$ to
${\boldsymbol{z}}$, we obtain that
$F({\boldsymbol{\Theta}}_{l_{0}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})=0$.
The function $F(\boldsymbol{Y},{\boldsymbol{z}})$ being linear in
$\boldsymbol{Y}$, we deduce that

$F\left(\frac{b_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})d(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})q({\boldsymbol{z}})}{b_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\alpha}})d(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})q({\boldsymbol{\alpha}})}{\boldsymbol{\Theta}}_{l_{0}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}}\right)=0\,.$

Writing
${\boldsymbol{\Theta}}_{l_{0}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})={\boldsymbol{\Theta}}_{l_{0}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}}){\boldsymbol{R}}_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})^{-1}{\boldsymbol{R}}_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})$,
and using (8.1), we get that

(8.38)
$F\left(\frac{b_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})d(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})q({\boldsymbol{z}})}{b_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\alpha}})d(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})q({\boldsymbol{\alpha}})}{\boldsymbol{\Theta}}_{l_{0}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}}){\boldsymbol{R}}_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})^{-1},{\boldsymbol{z}}\right)=0\,.$

Set

$Q({\boldsymbol{z}},\boldsymbol{X}):={\boldsymbol{\tau}}\left(\frac{b_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})d(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})q({\boldsymbol{z}})}{b_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\alpha}})d(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})q({\boldsymbol{\alpha}})}{\boldsymbol{\Theta}}_{l_{0}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}}){\boldsymbol{R}}_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})^{-1}\right)\boldsymbol{X}^{\boldsymbol{\mu}}\,,$

where we let $\boldsymbol{X}^{\boldsymbol{\mu}}$ denote the column vector with
coordinates
$\boldsymbol{X}^{{\boldsymbol{\mu}}_{1}},\ldots,\boldsymbol{X}^{{\boldsymbol{\mu}}_{t}}$.
We thus get that
$Q({\boldsymbol{z}},\boldsymbol{X})\in\overline{\mathbb{Q}}[{\boldsymbol{z}},\varphi(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}}),\boldsymbol{X}]$.
Since $\varphi\circ T_{{\boldsymbol{k}}_{l_{0}}}$ is analytic at
${\boldsymbol{\alpha}}$, it follows that
$Q({\boldsymbol{z}},\boldsymbol{X})\in\overline{Q({\boldsymbol{z}})}_{{\boldsymbol{\alpha}}}[\boldsymbol{X}]$.
Moreover, since
${\boldsymbol{\Theta}}_{l_{0}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})={\boldsymbol{\Theta}}_{l_{0}}({\boldsymbol{\xi}})={\boldsymbol{R}}_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\alpha}})$,
we deduce that

$Q({\boldsymbol{\alpha}},\boldsymbol{X})={\boldsymbol{\tau}}\boldsymbol{X}^{\boldsymbol{\mu}}=P(\boldsymbol{X})\,.$

Finally, it follows from (8.38) that

$Q({\boldsymbol{z}},{\boldsymbol{f}}({\boldsymbol{z}}))=0\,.$

This ends the first part of the proof of Theorem 6.2.

Now, let us assume that
$\overline{\mathbb{Q}}({\boldsymbol{z}})({\boldsymbol{f}}({\boldsymbol{z}}))$
is a regular extension of $\overline{\mathbb{Q}}({\boldsymbol{z}})$. Let
$\mathbb{K}$ be an algebraic closure of
$\overline{\mathbb{Q}}({\boldsymbol{z}})$ containing
$\varphi(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$. By [33, Chapter
VIII],
$\overline{\mathbb{Q}}({\boldsymbol{z}})({\boldsymbol{f}}({\boldsymbol{z}}))$
and $\mathbb{K}$ are linearly disjoint over
$\overline{\mathbb{Q}}({\boldsymbol{z}})$. Let $\delta$ denote the degree of
$\varphi(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$ over
$\overline{\mathbb{Q}}({\boldsymbol{z}})$. Since the functions
$\varphi(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})^{j}$, $0\leq
j\leq\delta-1$, are linearly independent over
$\overline{\mathbb{Q}}({\boldsymbol{z}})$, they remain linearly independent
over
$\overline{\mathbb{Q}}({\boldsymbol{z}})({\boldsymbol{f}}({\boldsymbol{z}}))$.
Splitting the polynomial $Q$ as

$Q=\sum_{j=0}^{\delta-1}Q_{j}({\boldsymbol{z}},\boldsymbol{X})\varphi(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})^{j}\,,$

where
$Q_{j}({\boldsymbol{z}},\boldsymbol{X})\in\overline{\mathbb{Q}}[{\boldsymbol{z}},\boldsymbol{X}]$,
we thus deduce that
$Q_{j}({\boldsymbol{z}},{\boldsymbol{f}}({\boldsymbol{z}}))=0$ for all $j$,
$0\leq j\leq\delta-1$. Finally, setting

$R({\boldsymbol{z}},\boldsymbol{X}):=\sum_{j=0}^{\delta-1}Q_{j}({\boldsymbol{z}},\boldsymbol{X})\varphi(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})^{j}\in\overline{\mathbb{Q}}[{\boldsymbol{z}},\boldsymbol{X}]\,,$

we obtain that $R({\boldsymbol{z}},{\boldsymbol{f}}({\boldsymbol{z}}))=0$ and
$R({\boldsymbol{\alpha}},\boldsymbol{X})=P(\boldsymbol{X})$, as wanted. ∎

##  9\. Proofs of Theorems 2.3, 2.6, 2.8, and of Corollaries 2.5 and 2.9

In this section, we complete the proof of our main results. Note that we
establish Corollary 2.9 before Theorems 2.6 and 2.8. The latter are in fact
deduced from Corollary 2.9.

###  9.1. Proof of Theorem 2.3

There is nothing more to do. Theorem 2.3 just corresponds to the case $r=1$ of
Theorem 6.2. ∎

###  9.2. Proof of Corollary 2.5

We first note that the inequality

${\rm
tr.deg}_{\overline{\mathbb{Q}}}(f_{1}({\boldsymbol{\alpha}}),\ldots,f_{m}({\boldsymbol{\alpha}}))\leq{\rm
tr.deg}_{\overline{\mathbb{Q}}({\boldsymbol{z}})}(f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}}))$

always holds. Let $\mathbb{K}$ denote the field of fractions of the ring
$\overline{\mathbb{Q}({\boldsymbol{z}})}_{\boldsymbol{\alpha}}$. Since
$\mathbb{K}$ is algebraic over $\overline{\mathbb{Q}}(z)$, we have

${\rm
tr.deg}_{\overline{\mathbb{Q}}({\boldsymbol{z}})}(f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}}))={\rm
tr.deg}_{\mathbb{K}}(f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}}))\,.$

It thus remains to prove that

(9.1) ${\rm
tr.deg}_{\overline{\mathbb{Q}}}(f_{1}({\boldsymbol{\alpha}}),\ldots,f_{m}({\boldsymbol{\alpha}}))\geq{\rm
tr.deg}_{\mathbb{K}}(f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}}))\,.$

We follow the same strategy as the one used for proving the Siegel–Shidlovskii
theorem in the framework of $E$-functions (see [25, Theorem 5.23, p. 230]). We
first replace Proposition 5.1 in [25] by the following result.

######  Lemma 9.1.

Let
$g_{1}({\boldsymbol{z}}),\ldots,g_{\ell}({\boldsymbol{z}})\in\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}$
be related by a linear $T$-Mahler system. Let
${\boldsymbol{\alpha}}\in(\overline{\mathbb{Q}}^{\star})^{n}$ be a regular
point w.r.t. this system and let us assume that the pair
$(T,{\boldsymbol{\alpha}})$ is admissible. Let $s$ be the maximum number of
functions among $g_{1}({\boldsymbol{z}}),\ldots,g_{\ell}({\boldsymbol{z}})$
that are linearly independent over $\mathbb{K}$. Then at least $s$ of the
numbers $g_{1}({\boldsymbol{\alpha}}),\ldots,g_{\ell}({\boldsymbol{\alpha}})$
are linearly independent over $\overline{\mathbb{Q}}$.

###### Proof.

Let $t$ denote the dimension of the $\overline{\mathbb{Q}}$-vector space
spanned by the numbers $g_{i}({\boldsymbol{\alpha}})$, $1\leq i\leq\ell$.
Without any loss of generality, we assume that
$g_{1}({\boldsymbol{\alpha}}),\ldots,g_{t}({\boldsymbol{\alpha}})$ are
linearly independent over $\overline{\mathbb{Q}}$. Then, for every $i$,
$t<i\leq\ell$, there exist algebraic numbers
$\gamma_{i,1},\ldots,\gamma_{i,t}$ such that

$g_{i}({\boldsymbol{\alpha}})=\gamma_{i,1}g_{1}({\boldsymbol{\alpha}})+\cdots+\gamma_{i,t}g_{t}({\boldsymbol{\alpha}})\,.$

By Theorem 2.3, there exist
$p_{i,1}({\boldsymbol{z}}),\ldots,p_{i,\ell}({\boldsymbol{z}})\in\overline{\mathbb{Q}({\boldsymbol{z}})}_{\boldsymbol{\alpha}}$
such that

$p_{i,1}({\boldsymbol{z}})g_{1}({\boldsymbol{z}})+\cdots+p_{i,\ell}({\boldsymbol{z}})g_{\ell}({\boldsymbol{z}})=0\,,$

with $p_{i,i}({\boldsymbol{\alpha}})=1$ and $p_{i,j}({\boldsymbol{\alpha}})=0$
when $t<j\leq\ell$ and $j\neq i$. Set

$L_{i}({\boldsymbol{z}},X_{1},\ldots,X_{\ell}):=p_{i,1}({\boldsymbol{z}})X_{1}+\cdots+p_{i,\ell}({\boldsymbol{z}})X_{\ell}=0\,,\;t<i\leq\ell\,,.$

Note that the linear form $L_{i}({\boldsymbol{\alpha}},X_{1},\ldots,X_{m})$ is
equal to

$X_{i}+\sum_{j=1}^{t}\gamma_{i,j}X_{j}\,.$

Hence the linear forms $L_{i}({\boldsymbol{\alpha}},X_{1},\ldots,X_{m})$,
$t<i\leq\ell$, are linearly independent over $\overline{\mathbb{Q}}$ and a
fortiori the corresponding linear forms
$L_{i}({\boldsymbol{z}},X_{1},\ldots,X_{\ell})$ are linearly independent over
$\mathbb{K}$. It follows that $t\geq s$, as wanted. ∎

###### Proof of Corollary 2.5.

Let $D\geq 0$ be an integer. As in [25, p. 231], we let
$\varphi_{\boldsymbol{\alpha}}(D)$ denote the dimension of the
$\overline{\mathbb{Q}}$-vector space spanned by the monomials of total degree
at most $D$ in
$f_{1}({\boldsymbol{\alpha}}),\ldots,f_{m}({\boldsymbol{\alpha}})$. We also
let $\varphi_{\boldsymbol{z}}(D)$ denote the dimension of the
$\mathbb{K}$-vector space spanned by the monomials of total degree at most $D$
in $f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}})$. Now, we observe
that the monomials of total degree at most $D$ in
$f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}})$ are also related by a
linear $T$-Mahler system for which ${\boldsymbol{\alpha}}$ remains regular.
Indeed, such a system can be obtained by first taking the system associated
with the matrix $((1)\oplus A({\boldsymbol{z}}))^{\otimes d}$, that is, the
$D$th power of Kronecker of the matrix $(1)\oplus A({\boldsymbol{z}})$, where
$A({\boldsymbol{z}})$ is defined as in (2.2), and then applying some standard
reduction procedure (see [24, Lemme 5]). Then, we infer from Lemma 9.1 that

(9.2)
$\varphi_{\boldsymbol{\alpha}}(D)\geq\varphi_{\boldsymbol{z}}(D),\,\forall
D\in\mathbb{N}\,.$

By the Hilbert-Serre Theorem (see for example [56, Theorem 42, p. 235]), for
sufficiently large $D$, $\varphi_{\boldsymbol{\alpha}}(D)$ and
$\varphi_{\boldsymbol{z}}(D)$ are polynomials in $D$ whose degree are
respectively ${\rm
tr.deg}_{\overline{\mathbb{Q}}}(f_{1}({\boldsymbol{\alpha}}),\ldots,f_{m}({\boldsymbol{\alpha}}))$
and ${\rm
tr.deg}_{\mathbb{K}}(f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}}))$.
Hence (9.2) implies that Inequality (9.1) holds, as wanted. ∎

###  9.3. Proof of Corollary 2.9

We first need the two following simple results.

######  Lemma 9.2.

Let ${\boldsymbol{z}}=(z_{1,1},\ldots,z_{1,n_{1}},z_{2,1},\ldots,z_{r,n_{r}})$
be a tuple of $n_{1}+\cdots+n_{r}$ distinct variables. For every $i$, $1\leq
i\leq r$, let
$f_{i,1}({\boldsymbol{z}}_{i}),\ldots,f_{i,m_{i}}({\boldsymbol{z}}_{i})\in\overline{\mathbb{Q}}[[{\boldsymbol{z}}_{i}]]$
be some power series, where
${\boldsymbol{z}}_{i}=(z_{i,1},\ldots,z_{i,n_{i}})$. Then

$\displaystyle{\rm
tr.deg}_{\overline{\mathbb{Q}}({\boldsymbol{z}})}\\{f_{i,j}({\boldsymbol{z}}_{i}):1\leq
i\leq r,\,1\leq j\leq m_{i}\\}=\hskip 99.58464pt$ $\displaystyle\hskip
156.49014pt\sum_{i=1}^{r}{\rm
tr.deg}_{\overline{\mathbb{Q}}({\boldsymbol{z}}_{i})}\\{f_{i,j}({\boldsymbol{z}}_{i}):1\leq
j\leq m_{i}\\}\,.$

###### Proof.

The result follows directly from the fact that the sets of variables
$\\{z_{1,1},\ldots,z_{1,n_{1}}\\},\ldots,\\{z_{r,1},\ldots,z_{r,n_{r}}\\}$ are
disjoint. ∎

######  Lemma 9.3.

Let
$\mathcal{E}_{1},\ldots,\mathcal{E}_{r},\mathcal{F}_{1},\ldots,\mathcal{F}_{r}$
be nonempty finite sets of complex numbers such that
$\mathcal{E}_{i}\subset\mathcal{F}_{i}$ for every $i$, $1\leq i\leq r$. Let us
assume that

${\rm
tr.deg}_{\overline{\mathbb{Q}}}\left(\bigcup_{i=1}^{r}\mathcal{F}_{i}\right)=\sum_{i=1}^{r}{\rm
tr.deg}_{\overline{\mathbb{Q}}}(\mathcal{F}_{i})\,.$

Then

${\rm
tr.deg}_{\overline{\mathbb{Q}}}\left(\bigcup_{i=1}^{r}\mathcal{E}_{i}\right)=\sum_{i=1}^{r}{\rm
tr.deg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i})\,.$

###### Proof.

Suppose first that all elements of each $\mathcal{F}_{i}$, $1\leq i\leq r$,
are algebraically independent. By assumption, all elements of the set
$\bigcup_{i=1}^{r}\mathcal{F}_{i}$ are algebraically independent. Hence, all
elements of the set $\bigcup_{i=1}^{r}\mathcal{E}_{i}$ are also algebraically
independent, and the lemma is proved. Let us assume now that some elements of
the family $\mathcal{F}_{i}$, $1\leq i\leq r$, are algebraically dependent.
For every $i$, we choose a subset of algebraically independent elements
$\mathcal{E}^{\prime}_{i}\subset\mathcal{E}_{i}$ such that ${\rm
tr.deg}_{\overline{\mathbb{Q}}}(\mathcal{E}^{\prime}_{i})={\rm
tr.deg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i})$. Then, we complete the set
$\mathcal{E}^{\prime}_{i}$ in a set of algebraically independent elements
$\mathcal{F}^{\prime}_{i}\subset\mathcal{F}_{i}$ such that ${\rm
tr.deg}_{\overline{\mathbb{Q}}}(\mathcal{F}^{\prime}_{i})={\rm
tr.deg}_{\overline{\mathbb{Q}}}(\mathcal{F}_{i})$. From the first part of the
proof, we have

${\rm
tr.deg}_{\overline{\mathbb{Q}}}\left(\bigcup_{i=1}^{r}\mathcal{E}^{\prime}_{i}\right)=\sum_{i=1}^{r}{\rm
tr.deg}_{\overline{\mathbb{Q}}}(\mathcal{E}^{\prime}_{i})\,.$

It follows that

${\rm
tr.deg}_{\overline{\mathbb{Q}}}\left(\bigcup_{i=1}^{r}\mathcal{E}_{i}\right)={\rm
tr.deg}_{\overline{\mathbb{Q}}}\left(\bigcup_{i=1}^{r}\mathcal{E}^{\prime}_{i}\right)=\sum_{i=1}^{r}{\rm
tr.deg}_{\overline{\mathbb{Q}}}(\mathcal{E}^{\prime}_{i})=\sum_{i=1}^{r}{\rm
tr.deg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i})\,,$

which ends the proof. ∎

We are now ready to prove Corollary 2.9.

###### Proof of Corollary 2.9.

We continue with the notation of Theorem 2.6 and 2.8.

Let us first assume that the assumptions of Theorem 2.6 are satisfied. We can
gather all the linear Mahler systems (2.8.$i$) into a big Mahler system of the
form (2.2), where
$A({\boldsymbol{z}})=A_{1}({\boldsymbol{z}}_{1})\oplus\cdots\oplus
A_{r}({\boldsymbol{z}}_{r})$, ${\boldsymbol{z}}=(z_{1,1},\ldots,z_{r,m_{r}})$,
and $T:=T_{1}\oplus\cdots\oplus T_{r}$. Then, we infer from assumptions (i)
and (ii) of Theorem 2.6 and from Theorem 4.6 that the pair
$(T,{\boldsymbol{\alpha}})$ is admissible and that the point
${\boldsymbol{\alpha}}=({\boldsymbol{\alpha}}_{1},\ldots,{\boldsymbol{\alpha}}_{r})$
is regular with respect to this $T$-Mahler system. Hence we can apply
Corollary 2.5 to this larger system. We obtain that

(9.3) $\displaystyle{\rm
tr.deg}_{\overline{\mathbb{Q}}}\\{f_{i,j}({\boldsymbol{\alpha}}_{i}):1\leq
i\leq r,\,1\leq j\leq m_{i}\\}\hskip 113.81102pt$ $\displaystyle\hskip
113.81102pt={\rm
tr.deg}_{\overline{\mathbb{Q}}({\boldsymbol{z}})}\\{f_{i,j}({\boldsymbol{z}}_{i}):1\leq
i\leq r,\,1\leq j\leq m_{i}\\}\,.$

On the other hand, applying Corollary 2.5 to the system (2.8.$i$), for every
$i$, $1\leq i\leq r$, we deduce that

(9.4) ${\rm
tr.deg}_{\overline{\mathbb{Q}}}\\{f_{i,j}({\boldsymbol{\alpha}}_{i})\\}:1\leq
j\leq m_{i}\\}={\rm
tr.deg}_{\overline{\mathbb{Q}}({\boldsymbol{z}}_{i})}\\{f_{i,j}({\boldsymbol{z}}_{i}):1\leq
j\leq m_{i}\\}\,.$

It follows from (9.3), (9.4), and Lemma 9.2 that

(9.5) $\displaystyle{\rm
tr.deg}_{\overline{\mathbb{Q}}}\\{f_{i,j}({\boldsymbol{\alpha}}_{i}):1\leq
i\leq r,\,1\leq j\leq m_{i}\\}\hskip 85.35826pt$ $\displaystyle\hskip
113.81102pt=\sum_{i=1}^{r}{\rm
tr.deg}_{\overline{\mathbb{Q}}}\\{f_{i,j}({\boldsymbol{\alpha}}_{i}):1\leq
j\leq m_{i}\\}\,.$

For every $i$, set
$\mathcal{F}_{i}:=\\{f_{i,j}({\boldsymbol{\alpha}}_{i}):1\leq j\leq m_{i}\\}$.
Using (9.5), we can thus apply Lemma 9.3 to deduce that ${\rm
tr.deg}_{\overline{\mathbb{Q}}}(\mathcal{E})=\sum_{i=1}^{r}{\rm
tr.deg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i})$, as wanted.

Now, let us assume that the assumptions of Theorem 2.8 are satisfied. The
proof is essentially the same. The only change occurs when establishing
Equality (9.3). We infer from assumptions (i) and (ii) of Theorem 2.8 that we
can apply Theorem 6.2. Then, using Theorem 6.2 and arguing as in the proof of
Corollary 2.5, we deduce that Equality (9.3) holds. The rest of the proof
remains unchanged. ∎

###  9.4. Proof of Theorems 2.6 and 2.8

We continue with the notation of Theorem 2.6 and 2.8. For every $i$, $1\leq
i\leq r$, we set

$\mathcal{E}_{i}:=(f_{i,\ell_{1}}({\boldsymbol{\alpha}}_{i}),\ldots,f_{i,\ell_{s_{i}}}({\boldsymbol{\alpha}}_{i}))\,,$

where $1\leq\ell_{1}<\ell_{2}<\cdots<\ell_{s_{i}}\leq m_{i}$. Note that the
inclusion

(9.6) $\sum_{i=1}^{r}{\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i}\mid\mathcal{E})\subset{\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E})\,$

is trivial. Suppose that the assumptions of either Theorem 2.6 or Theorem 2.8
hold. By Corollary 2.9, we have

(9.7) ${\rm
tr.deg}_{\overline{\mathbb{Q}}}\left(\mathcal{E}\right)=\sum_{i=1}^{r}{\rm
tr.deg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i})\,.$

Given an ideal $\mathcal{I}$, we let ${\rm ht}(\mathcal{I})$ denote its
height. Then, we have

(9.8) $\begin{array}[]{rcl}{\rm ht}\left({\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i})\right)&=&s_{i}-{\rm
tr.deg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i})\,,\;\;\forall i,\;1\leq i\leq
r,\\\ {\rm ht}\left({\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E})\right)&=&S-{\rm
tr.deg}_{\overline{\mathbb{Q}}}(\mathcal{E})\,,\end{array}$

where $S:=s_{1}+\cdots+s_{r}$. From (9.7) and (9.8) we deduce that

${\rm ht}\left({\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E})\right)=\sum_{i=1}^{r}{\rm
ht}\left({\rm Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i})\right)\,.$

Set $\mathcal{I}:=\sum_{i=1}^{r}{\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i}\mid\mathcal{E})$. Then the
isomorphism777See, for instance, [29, Chap. I, Exercise 3.15].

$\overline{\mathbb{Q}}[\boldsymbol{X}_{1}]\big{/}{\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{1})\otimes_{\overline{\mathbb{Q}}}\cdots\otimes_{\overline{\mathbb{Q}}}\overline{\mathbb{Q}}[\boldsymbol{X}_{r}]\big{/}{\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{r})\cong\overline{\mathbb{Q}}[\boldsymbol{X}]\big{/}{\mathcal{I}}\,$

implies that $\mathcal{I}$ is a prime ideal. Indeed, the tensor product of
integral domains, over an algebraically closed field, is an integral domain.
Furthermore, this isomorphism also gives that ${\rm
ht}(\mathcal{I})=\sum_{i=1}^{r}{\rm ht}({\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i}))$. It follows that ${\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E})$ and $\sum_{i=1}^{r}{\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i}\mid\mathcal{E})$ are both prime
ideals with the same height. By (9.6), these two ideals are equal. This ends
the proof. ∎

##  10\. Proof of Theorem 1.1

In this section, we show how to deduce Theorem 1.1 from the two purity
theorems. We first prove the following lemma.

######  Lemma 10.1.

Let $f(z)$ be a $q$-Mahler function and $\alpha$ be a nonzero algebraic number
such that $f(z)$ is well-defined at $\alpha$. Then there exists a $q$-Mahler
function $g(z)$ such that the following properties hold.

  * (a)

$g(\alpha)=f(\alpha)$.

  * (b)

The function $g(z)$ is the first coordinate of a vector solution to a
$q^{l}$-Mahler system, say

$\left(\begin{array}[]{c}g_{1}(z)=g(z)\\\ \vdots\\\
g_{m}(z)\end{array}\right)=B(z)\left(\begin{array}[]{c}g_{1}(z^{q^{l}})\\\
\vdots\\\ g_{m}(z^{q^{l}})\end{array}\right)\,.$

for some integer $l>0$.

  * (c)

The point $\alpha$ is regular with respect to this system.

###### Proof.

We first note that if $f(\alpha)$ is algebraic, the lemma is trivial for we
can choose $g(z):=f(\alpha)$ to be constant. We assume now that $f(\alpha)$ is
transcendental. Using a minimal $q$-Mahler equation for $f(z)$, we deduce that
$f(z)$ is the first coordinate of some $q$-Mahler system, say

(10.1) $\left(\begin{array}[]{c}f_{1}(z)\\\ \vdots\\\
f_{m}(z)\end{array}\right)=A(z)\left(\begin{array}[]{c}f_{1}(z^{q})\\\
\vdots\\\ f_{m}(z^{q})\end{array}\right)\,,$

where $f_{1}(z)=f(z),\ldots,f_{m}(z)$ are linearly independent over
$\overline{\mathbb{Q}}(z)$. Since the functions $f_{1}(z),\ldots,f_{m}(z)$ are
linearly independent, we infer from [8, Theorem 1.10] that there exists an
integer $l$ such that the numbers $\alpha^{q^{l}}$ is regular with respect to
the system

(10.2) $\left(\begin{array}[]{c}f_{1}(z)\\\ \vdots\\\
f_{m}(z)\end{array}\right)=A_{l}(z)\left(\begin{array}[]{c}f_{1}(z^{q^{l}})\\\
\vdots\\\ f_{m}(z^{q^{l}})\end{array}\right)\,,$

where

$A_{l}(z)=A(z)A(z^{q})\cdots A(z^{q^{l-1}})\,.$

Furthermore, this theorem ensures that $\alpha$ is not a pole of the matrix
$A_{l}(z)$. Let $(a_{1}(z),\ldots,a_{m}(z))$ denote the first row of
$A_{l}(z)$. Set

(10.3)
$g(z)=a_{1}(\alpha)f_{1}(z^{q^{l}})+\cdots+a_{m}(\alpha)f_{m}(z^{q^{l}})\,.$

Note that $g(z)$ is a $q$-Mahler function for it is obtained as a linear
combination over $\overline{\mathbb{Q}}$ of $q$-Mahler functions888Indeed, if
$f(z)$ is a $q$-Mahler function then $f(z^{q^{l}})$ is clearly a $q$-Mahler
function too.. Since $f(\alpha)$ is transcendental, the vector
$(a_{1}(\alpha),\ldots,a_{m}(\alpha))$ is nonzero. Then applying a suitable
constant gauge transformation to the Mahler system associated with the matrix
$A_{l}(z^{q^{l}})$, we can obtain a Mahler system which has a solution vector
with $g(z)$ as first coordinate. Furthermore, since the point $\alpha^{q^{l}}$
is regular w.r.t. (10.2), $\alpha$ is a regular point w.r.t. this new system.
On the other hand, we infer from (10.2) and (10.3) that $g(\alpha)=f(\alpha)$,
as wanted. ∎

###### Proof of Theorem 1.1.

We keep on with the notation of Theorem 1.1. We assume that no number among
$f_{1}(\alpha_{1}),\ldots,f_{r}(\alpha_{r})$ belongs to $\mathbb{K}$, so that
it remains to prove that $f_{1}(\alpha_{1}),\ldots,f_{r}(\alpha_{r})$ are
algebraically independent over $\overline{\mathbb{Q}}$. By [8, Corollaire
1.8], our assumption implies that the numbers
$f_{1}(\alpha_{1}),\ldots,f_{r}(\alpha_{r})$ are all transcendental. For every
$i$, $1\leq i\leq r$, we let $z_{i}$ denote an indeterminate.

By Lemma 10.1, with each pair $(f_{i},\alpha_{i})$, we can associate a
$q_{i}$-Mahler function $g_{i}(z_{i})$ such that

(10.4.$i$) $\left(\begin{array}[]{c}g_{i,1}(z_{i})=g_{i}(z_{i})\\\ \vdots\\\
g_{i,m_{i}}(z_{i})\end{array}\right)=B_{i}(z_{i})\left(\begin{array}[]{c}g_{i,1}\left(z_{i}^{q_{i}^{l_{i}}}\right)\\\
\vdots\\\ g_{i,m_{i}}\left(z_{i}^{q_{i}^{l_{i}}}\right)\end{array}\right)\,,$

$g_{i}(\alpha_{i})=f_{i}(\alpha_{i})$ and $\alpha_{i}$ is regular w.r.t.
(10.4.$i$).

Let us first prove Case (i) of Theorem 1.1. Let us divide the natural numbers
$1,\ldots,r$ into $s$ classes
$\mathcal{I}_{1}=\\{i_{1,1},\ldots,i_{1,\nu_{1}}\\},\ldots,\mathcal{I}_{s}=\\{i_{s,1},\ldots,i_{s,\nu_{s}}\\}$,
such that $i$ and $j$ belong to the same classe if and only if $q_{i}$ and
$q_{j}$ are multiplicatively dependent. Iterating each system (10.4.$i$) a
suitable number of times, we can assume without loss of generality that
$q_{i}^{l_{i}}=q_{j}^{l_{j}}:=\rho_{k}$ whenever $i$ and $j$ belong to
$\mathcal{I}_{k}$. Set
$\mathcal{E}:=(g_{1}(\alpha_{1}),\ldots,g_{r}(\alpha_{r}))$ and

$\mathcal{E}_{k}:=(g_{i_{k,1}}(\alpha_{i_{k,1}}),\ldots,g_{i_{k,\nu_{k}}}(\alpha_{i_{k,\nu_{k}}}))\,,\;\forall
k\in\\{1,\ldots,s\\}\,.$

Given $k\in\\{1,\ldots,s\\}$, we consider the Mahler system in the variables
$z_{i},i\in\mathcal{I}_{k}$, associated with the matrix
$\oplus_{i\in\mathcal{I}_{k}}B_{i}(z_{i})$ and the transformation
$T_{k}=\rho_{k}{\rm I}_{\nu_{k}}$, where we let ${\rm I}_{n}$ denote the
identity matrix of size $n$. In this way, we have converted our $r$ Mahler
systems in one variable into $s$ Mahler systems, each having respectively
$\nu_{1},\ldots,\nu_{s}$ variables. Furthermore, since by assumption the
algebraic numbers $\alpha_{1},\ldots,\alpha_{r}$ are multiplicatively
independent, we deduce that each pair

$(T_{k},{\boldsymbol{\alpha}}_{k}:=(\alpha_{i_{k,1}},\ldots,\alpha_{i_{k,\nu_{k}}}))\,,\;1\leq
k\leq s\,,$

is admissible. Finally, the point ${\boldsymbol{\alpha}}_{k}$ is regular for
each $\alpha_{i}$ is regular w.r.t. (10.4.$i$). Since, by construction, the
numbers $\rho(T_{1})=\rho_{1},\ldots,\rho(T_{s})=\rho_{s}$ are pairwise
multiplicatively independent, we can apply our second purity theorem, Theorem
2.8, to these $s$ Mahler systems. We deduce that

(10.5) ${\rm Alg}_{\overline{\mathbb{Q}}}(\mathcal{E})=\sum_{k=1}^{s}{\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{k}\mid\mathcal{E})\,.$

Now, let us fix $k\in\\{1,\ldots,s\\}$. Since the numbers
$\alpha_{i},i\in\mathcal{I}_{k}$, are multiplicatively independent, we can
apply our first purity theorem, Theorem 2.6, to the $\nu_{k}$ distinct Mahler
systems (10.4.$i$), with $i\in\mathcal{I}_{k}$. For every
$i\in\mathcal{I}_{k}$, set $\mathcal{E}_{k,i}:=(g_{i}(\alpha_{i}))$. Since
$g_{i}(\alpha_{i})=f_{i}(\alpha_{i})$ is transcendental, we have ${\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{k,i})=\\{0\\}$ for every
$i\in\mathcal{I}_{k}$. We thus deduce from Theorem 2.6 that

${\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{k})=\sum_{i\in\mathcal{I}_{k}}{\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{k,i}\mid\mathcal{E}_{k})=\\{0\\}\,.$

Since this holds for every $k$, $1\leq k\leq s$, it follows from (10.5), that
${\rm Alg}_{\overline{\mathbb{Q}}}(\mathcal{E})=\\{0\\}$. That is,
$f_{1}(\alpha_{1}),\ldots,f_{r}(\alpha_{r})$ are algebraically independent
over $\overline{\mathbb{Q}}$.

Now, let us prove Case (ii) of Theorem 1.1. As previously, we associate with
each pair $(f_{i}(z),\alpha_{i})$ a function $g_{i}(z)$ satisfying the
conditions of Lemma 10.1. Since the natural numbers $q_{i}$ are pairwise
multiplicatively independent, we can apply our second purity theorem, Theorem
2.8, to the Mahler systems associated with each $g_{i}(z)$ in Lemma 10.1.
Setting

$\mathcal{E}:=(g_{1}(\alpha_{1}),\ldots,g_{r}(\alpha_{r}))\,$

and $\mathcal{E}_{i}:=(g_{i}(\alpha_{i}))$, $1\leq i\leq r$, we deduce that

${\rm Alg}_{\overline{\mathbb{Q}}}(\mathcal{E})=\sum_{i=1}^{r}{\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i}\mid\mathcal{E})\,.$

Again, since by assumption $g_{i}(\alpha_{i})=f_{i}(\alpha_{i})$ is
transcendental, we get that ${\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i}\mid\mathcal{E})=0$ for every $i$,
$1\leq i\leq r$. This shows that ${\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E})=\\{0\\}$, and we conclude, as
previously, that $f_{1}(\alpha_{1}),\ldots,f_{r}(\alpha_{r})$ are
algebraically independent over $\overline{\mathbb{Q}}$. ∎

##  Appendix A Representing numbers in independent bases

In this appendix, we show how our main results apply to certain problems
concerning expansions of numbers in multiplicatively independent bases. In
particular, we state and prove Conjectures A.2, A.3, and Corollary A.4, which
were our initial goal. All these results are deduced from Theorem 1.1.

###  A.1. The dynamical point of view: Furstenberg’s conjecture

In the late 1960s, Furstenberg [26, 27] suggested a series of conjectures
whose aim is to capture the heuristic which has been alluded to only in very
vague terms at the beginning of this paper. These conjectures, which became
famous, take place in a dynamical setting. This does not come as a great
surprise for there is a well-known dictionary transferring combinatorial
properties of the expansion of a real number $x$ in an integer base $q$ in
terms of dynamical properties of the orbit of $\\{x\\}$ under the map $T_{q}$
defined on $\mathbb{R}/\mathbb{Z}$ by $x\mapsto qx$. We let
$\mathcal{O}_{q}(x)$ denote the forward orbit of $x$ under $T_{q}$, that is,

$\mathcal{O}_{q}(x):=\left\\{x,T_{q}(x),T^{2}_{q}(x),\ldots\right\\}\,.$

If $X\subset\mathbb{R}$, we let $\dim_{H}(X)$ denote the Hausdorff dimension
of $X$ and $\overline{X}$ its closure. One of Furstenberg’s conjecture [27]
reads as follows.

######  Conjecture A.1 (Furstenberg).

Let $p$ and $q$ be two multiplicatively independent natural numbers, and let
$x\in[0,1)$ be a real number. Then

$\dim_{H}\overline{\mathcal{O}_{p}(x)}+\dim_{H}\overline{\mathcal{O}_{q}(x)}\geq
1\,,$

unless $x$ is rational.

This conjecture has wonderful consequences about expansions of both real and
natural numbers. It beautifully expresses the expected balance between the
complexity of expansions of an irrational real number in two multiplicatively
independent bases:

_If $x$ has a simple expansion in base $p$, then it should have a complex
expansion in base $q$. _

It is easy to see that Conjecture A.1 holds true generically. Indeed, endowed
with the Haar measure, the topological dynamical system
$(T_{q},\mathbb{R}/\mathbb{Z})$ becomes ergodic, and it follows from the
ergodic theorem that

$\dim_{H}\overline{\mathcal{O}_{p}(x)}=\dim_{H}\overline{\mathcal{O}_{q}(x)}=1\,,$

for almost all real numbers $x$ in $[0,1)$. In fact, all the strength of
Conjecture A.1 takes shape when $x$ has a simple expansion in one of the two
bases. Defining the entropy of $x$ with respect to the base $q$ as the
topological entropy of the dynamical system
$(T_{q},\overline{\mathcal{O}_{q}(x)})$, Conjecture A.1 implies that if $x$
has zero entropy in base $p$, then it has a dense orbit under $T_{q}$.

Let us illustrate this with a concrete example. The binary Thue-Morse number
$\tau$ is defined as follows. Its $n$th binary digit is equal to $0$ if the
sum of digits in the binary expansion of $n$ is even, and to $1$ otherwise. It
is somewhat puzzling that its decimal expansion

$\langle\tau\rangle_{10}=0.412\,454\,033\,640\,107\,597\,783\,361\,368\,258\,455\,283\,089\cdots$

seems unpredictable, while its binary expansion

$\langle\tau\rangle_{2}=0.011\,010\,011\,001\,011\,010\,010\,110\,011\,010\,011\,001\,011\cdots\,$

is, by definition, so regular. This intriguing phenomenon would be nicely
explained by Conjecture A.1. Indeed, since $\tau$ has zero entropy in base
$2$, it should have a dense orbit under $T_{10}$, meaning that all blocks of
digits should occur in its decimal expansion.

Other astonishing consequences of Conjecture A.1 concern expansions of natural
numbers. For instance, using an elementary construction, Furstenberg shows in
[27] how to deduce from Conjecture A.1 that any finite block of digits occurs
in the decimal expansion of $2^{n}$, as soon as $n$ is large enough. Note
that, in the same vein, a conjecture of Erdös claims that the digit $2$ occurs
in the ternary expansion of $2^{n}$ for all $n>8$ (see, for instance, [32]).

Recently, Shmerkin [58] and Wu [61] proved that the set of exceptions to
Conjecture A.1 has Hausdorff dimension zero. Unfortunately, this remarkable
results does not tell us anything about expansions of real numbers with zero
entropy in some base. Indeed, the set of all such real numbers has Hausdorff
dimension zero [46]. Though the works of Shmerkin and Wu mark significant
progress towards Conjecture A.1, the latter remains far out of the reach of
current methods. Even worse, we are afraid that their result could be
essentially the best dynamical methods have to say about this conjecture.

###  A.2. The computational point of view: from finite automata to Mahler’s
method

From a computational point of view, there is another relevant notion of simple
number, namely the notion of _automatic real number_ (see [12, Chap. 13]).
While computable numbers can be generated by general Turing machines,
automatic numbers are those whose expansion in some base can be generated by a
finite automaton. Broadly speaking, a finite automaton is a Turing machine
without any memory tape, all its memory being stored in the finite state
control. This severe restriction justifies that these numbers are considered
as especially simple. For example, the Thue-Morse number $\tau$ is automatic
in base $2$. We refer the reader to [5] and the references therein for a
discussion on these different models of computation. In this new framework,
our general heuristic naturally leads to the following conjecture.

######  Conjecture A.2.

Let $p$ and $q$ be two multiplicatively independent natural numbers. A real
number cannot be automatic in both bases $p$ and $q$, unless it is rational.

This conjecture turns out to be a very special case of Conjecture A.1 for
being automatic in some base implies having zero entropy in that base.
Nevertheless, Conjecture A.2 remains quite challenging since, to date, not a
single real number has been proved to be at once automatic in some base and
not automatic in another one. We also mention that a weaker version of
Conjecture A.2 appears as Open Problems 7 in [12, Chap. 13].

With a more Diophantine flavor, Conjecture A.2 can be strongly strengthened as
follows.

######  Conjecture A.3.

Let $r\geq 1$ be an integer. Let $b_{1},\ldots,b_{r}$ be multiplicatively
independent positive integers, and, for every $i$, $1\leq i\leq r$, let
$\xi_{i}$ be a real number that is automatic in base $b_{i}$. Then the numbers
$\xi_{1},\ldots,\xi_{r}$ are algebraically independent over
$\overline{\mathbb{Q}}$, unless one of them is rational.

Conjecture A.3 is not implied by Furstenberg’s conjecture. The former does not
only imply that the Thue-Morse number $\tau$ cannot be automatic in base $10$,
but also that this is the case for any number obtained from $\tau$ by using
algebraic numbers and algebraic operations (addition, multiplication,
division, taking $n$th roots…). The case $r=1$ was a long-standing conjecture
first proved by Bugeaud and the first author [4] by means of the subspace
theorem. See also [8, 53] for a recent different proof based on Mahler’s
method. So far, Conjecture A.3 has only be settled in that particular case.

####  A.2.1. Connection with Mahler’s method and Theorem 1.1

In 1968, Cobham [20] first noticed the following fundamental connection
between automatic sequences and $M$-functions. If the sequence $(a_{n})_{n\geq
0}$ is $q$-automatic, then the generating function

$f(z):=\sum_{n=0}^{\infty}a_{n}z^{n}$

is a $q$-Mahler function. In turn, problems about transcendence and algebraic
independence of automatic real numbers can be translated and extended to
problems concerning transcendence and algebraic independence of $M$-functions
at algebraic points. In particular, Conjecture A.3, and hence Conjecture A.2,
easily follow from Part (i) of Theorem 1.1.

###### Proof of Conjecture A.3.

Replacing $\xi_{1},\ldots,\xi_{r}$ by their fractional part if necessary, we
can assume without any loss of generality that $0\leq\xi_{i}<1$, for every
$i$, $1\leq i\leq r$. By assumption, the number $\xi_{i}$ is automatic in base
$b_{i}$. This means that, for some integer $q_{i}\geq 2$, there exists a
$q_{i}$-automatic sequence $(a_{i,n})_{n\geq 0}$ with values in
$\\{0,1,\ldots,b_{i}-1\\}$ such that $\xi_{i}=f_{i}(1/b_{i})$, where
$f_{i}(z):=\sum_{n=0}^{\infty}a_{i,n}z^{n}\in\mathbb{Q}\\{z\\}$. A discussed
in [20], the fact that the sequence $(a_{i,n})_{n\geq 0}$ is $q_{i}$-automatic
implies that $f_{i}(z)$ is a $q_{i}$-Mahler function. Now, let us assume that
$\xi_{1},\ldots,\xi_{r}$ are all irrational. By [8, Corollaire 1.8], we obtain
that these numbers are all transcendental. Since by assumption the numbers
$1/b_{1},\ldots,1/b_{r}$ are multiplicatively independent, Part (i) of Theorem
1.1 implies that the numbers
$\xi_{1}=f_{1}(1/b_{1}),\ldots,\xi_{r}=f_{r}(1/b_{r})$ are algebraically
independent, as wanted. ∎

As with Furstenberg’s conjecture, Theorem 1.1 has also valuable consequences
about expansions of natural numbers. Let us first recall that a set
$\mathcal{E}\subset\mathbb{N}$ is $q$-automatic if its elements, when written
in base $q$, can be recognized by a finite automaton (See, for instance, [12,
Chapter 5]). In this framework, there is a famous theorem by Cobham [21] that
can be stated as follows. If $\mathcal{E}\subset\mathbb{N}$ is both $p$\- and
$q$-automatic, where $p$ and $q$ are multiplicatively independent, then
$\mathcal{E}$ is a periodic set, meaning that $\mathcal{E}$ is the union of a
finite set and finitely many arithmetic progressions.

Cobham’s theorem can be rephrased in terms of power series. Indeed, it is
equivalent to the fact that, given any aperiodic $p$-automatic set
$\mathcal{E}_{p}$ and any aperiodic $q$-automatic set $\mathcal{E}_{q}$, the
corresponding generating functions cannot be equal. That is,

$\sum_{n\in\mathcal{E}_{p}}z^{n}=:f_{p}(z)\not=f_{q}(z):=\sum_{n\in\mathcal{E}_{q}}z^{n}\,.$

In 1987, Loxton and van der Poorten [55] conjectured the following
generalization: a power series in $\overline{\mathbb{Q}}[[z]]$ cannot be both
$p$-Mahler and $q$-Mahler, unless it is rational. This conjecture was first
proved by Bell and the first author in [2], while a different proof was given
by Schäfke and Singer [57]. Very recently, the authors of [7] even proved a
stronger result also conjectured by Loxton and van der Poorten [55]: a
$p$-Mahler function $f_{p}(z)\in\overline{\mathbb{Q}}[[z]]$ and a $q$-Mahler
function $f_{q}(z)\in\overline{\mathbb{Q}}[[z]]$ are algebraically independent
over $\overline{\mathbb{Q}}(z)$, unless one of them is rational. This result
refines Cobham’s theorem by expressing, in algebraic terms, the discrepancy
between aperiodic automatic sets associated with multiplicatively independent
input bases. The proof given in [7] is based on a suitable parametrized Galois
theory associated with linear Mahler equations and follows the strategy
initiated in [6].

Part (ii) of Theorem 1.1 leads to the following significant generalization of
all the aforementioned results, providing in particular a totally new proof of
Cobham’s theorem.

######  Corollary A.4.

Let $r\geq 1$ be an integer. For every integer $i$, $1\leq i\leq r$, let
$q_{i}\geq 2$ be an integer and $f_{i}(z)\in\overline{\mathbb{Q}}\\{z\\}$ be a
$q_{i}$-Mahler function. Assume that $q_{1},\ldots,q_{r}$ are pairwise
multiplicatively independent. Then $f_{1}(z),\ldots,f_{r}(z)$ are
algebraically independent over $\overline{\mathbb{Q}}(z)$, unless one of them
is rational.

The case $r=1$ is a classical result (see, for instance, [52, Theorem 5.1.7]),
while, as previously mentioned, the case $r=2$ is much harder and was only
recently proved in [7].

###### Proof.

Let us assume that the functions $f_{1}(z),\ldots,f_{r}(z)$ are all
irrational. Then, by [52, Theorem 5.1.7], they are all transcendental over
$\overline{\mathbb{Q}}(z)$. Combining Nishioka’s theorem and [16, Lemma 6], we
deduce that there exists $r>0$ such that for all algebraic numbers $\alpha$,
with $0<|\alpha|<r$, the numbers $f_{1}(\alpha),\ldots,f_{r}(\alpha)$ are all
transcendental. Picking such $\alpha$ and applying Part (ii) of Theorem 1.1,
we obtain that the numbers $f_{1}(\alpha),\ldots,f_{r}(\alpha)$ are
algebraically independent over $\overline{\mathbb{Q}}$. Hence the functions
$f_{1}(z),\ldots,f_{r}(z)$ are algebraically independent over
$\overline{\mathbb{Q}}(z)$, as wanted. ∎

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