# Mahler’s method in several variables and finite automata

Boris Adamczewski  Univ Lyon, Université Claude Bernard Lyon 1
CNRS UMR 5208, Institut Camille Jordan
F-69622 Villeurbanne Cedex, France<EMAIL_ADDRESS>and  Colin
Faverjon  Univ Lyon, Université Claude Bernard Lyon 1
CNRS UMR 5208, Institut Camille Jordan
F-69622 Villeurbanne Cedex, France<EMAIL_ADDRESS>
###### Abstract.

We develop a theory of linear Mahler systems in several variables from the
perspective of transcendence and algebraic independence, which also includes
the possibility of dealing with several systems associated with sufficiently
independent matrix transformations. Our main results go far beyond the
existing literature, also surpassing those of two unpublished preprints the
authors made available on the arXiv in 2018. The main new feature is that they
apply now without any restriction on the matrices defining the corresponding
Mahler systems. As a consequence, we settle several problems concerning
expansions of numbers in multiplicatively independent bases. For instance, we
prove that no irrational real number can be automatic in two multiplicatively
independent integer bases, and we give a new proof and a broad algebraic
generalization of Cobham’s theorem in automata theory. We also provide a new
proof and a multivariate generalization of Nishioka’s theorem, a landmark
result in Mahler’s method.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
under the Grant Agreement No 648132.

###### Contents

  1. 1 Introduction
  2. 2 Mahler’s method in several variables
  3. 3 Notation
  4. 4 Admissibility conditions
  5. 5 A new vanishing theorem
  6. 6 Mahler’s method in families
  7. 7 Hilbert’s Nullstellensatz and relation matrices
  8. 8 Proof of Theorem 6.2
  9. 9 Proofs of Theorems 2.3, 2.6, 2.8, and of Corollaries 2.5 and 2.9
  10. 10 Proof of Theorem 1.1
  11. A Representing numbers in independent bases

##  1\. Introduction

It is commonly expected that expansions of numbers in multiplicatively
independent bases, such as $2$ and $10$, should have no common structure.
However, it seems extraordinarily difficult to confirm this naive heuristic
principle in some way or another. In the late 1960s, Furstenberg [26, 27]
suggested a series of conjectures, which became famous, and aim to capture
this heuristic (see Conjecture A.1 in Appendix A). Despite recent remarkable
progress by Shmerkin [58] and Wu [61], Conjecture A.1 remains totally out of
reach of the current methods. As always when mathematicians have to face such
an enormous gap between heuristic and knowledge, it becomes essential to find
out _good problems_. By that, we mean problems which, on the one hand,
formalize and express the general heuristic, and, on the other hand, whose
solution does not seem desperately out of reach. While Furstenberg’s
conjectures take place in a dynamical setting, we use instead the language of
automata theory to formulate some related conjectures that, hopefully, belong
to the above category. These conjectures are introduced and discussed in
Appendix A. Thanks to the work of Cobham [20], various problems involving
numbers generated by finite automata can be translated and extended to
problems concerning transcendence and algebraic independence of values of
$M$-functions. Furthermore, such problems fall into the scope of _Mahler’s
method_. In the end, we are able to settle our conjectures after proving the
apparently unrelated Theorem 1.1.

Let $\overline{\mathbb{Q}}\subset\mathbb{C}$ denote the field of algebraic
numbers and, given a field $\mathbb{K}\subset\mathbb{C}$, let
$\mathbb{K}\\{z\\}$ denote the ring of convergent power series with
coefficients in $\mathbb{K}$. Given an integer $q\geq 2$,
$f(z)\in\overline{\mathbb{Q}}\\{z\\}$ is said to be a $q$-Mahler function if
there exist polynomials $p_{0}(z),\ldots,p_{m}(z)\in\overline{\mathbb{Q}}[z]$,
not all zero, such that

(1.1) $p_{0}(z)f(z)+p_{1}(z)f(z^{q})+\cdots+p_{m}(z)f(z^{q^{m}})\ =\ 0.$

If $f(z)$ is $q$-Mahler for some $q$, we simply say that $f(z)$ is a Mahler
function, or an $M$-function. The coefficients of an $M$-function generate
only a finite field extension of $\mathbb{Q}$. Let us also recall that nonzero
complex numbers $x_{1},\ldots,x_{r}$ are multiplicatively independent if there
is no nonzero tuple of integers $(n_{1},\ldots,n_{r})$ such that
$x_{1}^{n_{1}}\cdots x_{r}^{n_{r}}=1$.

######  Theorem 1.1.

Let $r\geq 1$ be an integer and $\mathbb{K}\subseteq\overline{\mathbb{Q}}$ be
a field. For every integer $i$, $1\leq i\leq r$, we let $q_{i}\geq 2$ be an
integer, $f_{i}(z)\in\mathbb{K}\\{z\\}$ be a $q_{i}$-Mahler function, and
$\alpha_{i}\in\mathbb{K}$, $0<|\alpha_{i}|<1$, be such that $f_{i}(z)$ is
well-defined at $\alpha_{i}$. Let us assume that one of the two following
properties holds.

  * (i)

The numbers $\alpha_{1},\ldots,\alpha_{r}$ are multiplicatively independent.

  * (ii)

The numbers $q_{1},\ldots,q_{r}$ are pairwise multiplicatively independent.

Then $f_{1}(\alpha_{1}),f_{2}(\alpha_{2}),\ldots,f_{r}(\alpha_{r})$ are
algebraically independent over $\overline{\mathbb{Q}}$, unless one of them
belongs to $\mathbb{K}$.

Until now, Theorem 1.1 was only proved when $r=1$. This special case,
conjectured by Cobham [20] in 1968 and settled by the authors [8] in 2017,
implies that the decimal expansion of algebraic irrational numbers cannot be
generated by finite automata111This result was first proved by Bugeaud and the
first author [4] by means of the subspace theorem.. Also, an algorithm to
determine whether the numbers $f_{i}(\alpha_{i})$ belong to $\mathbb{K}$ or
not is described in [9].

Let us point out the four main difficulties we have to face when trying to
prove Theorem 1.1.

  * (i)

We have to consider a bunch of _arbitrary $M$-functions_. In contrast, many
results in the past where restricted to inhomogeneous order one equations (see
Section 2). That is, equations of the form
$p_{-1}(z)+p_{0}(z)f(z)+p_{1}(z)f(z^{q})=0$. Being able to deal with arbitrary
equations becomes essential for applications involving automata.

  * (ii)

Given an $M$-function, we have to consider its values at _arbitrary algebraic
points_ where it is well-defined, while a classical feature of results in this
framework is that they are only available for points which are _regular_
222See Definition 2.2. with respect to the underlying Mahler system.

  * (iii)

We have to consider simultaneously values of $M$-functions at _different
algebraic points_. In the setting of Siegel $E$-functions, the study of
algebraic relations between values of $E$-functions at different algebraic
points can be achieved by considering different $E$-functions at the same
point. Indeed, if $f(z)$ is an $E$-function and $\alpha$ is an algebraic
number, then the function $f(\alpha z)$ is still an $E$-function. However,
this trick no longer works for $M$-functions.

  * (iv)

We have to consider $M$-functions associated with _different transformations_
(i.e., $z\mapsto z^{q}$ with different $q$).

Thanks to the work of Ku. Nishioka [49], the transcendence theory of linear
Mahler systems in one variable is well-developed. It has even reached a rather
definitive stage after the recent works of Philippon [53] and the authors [8].
These new results provide tools to overcome (ii), and also (i) in some
situations. However, Theorem 1.1 does not fall into the scope of Mahler’s
method in one variable. In particular, the problem raised by (iii) requires a
major development of Mahler’s method in several variables. Partial results in
this direction are due to Mahler [42], Kubota [31], Loxton and van der Poorten
[37, 39], and Nishioka [51]. Last but not least, (iv) is a source of well-
known difficulties and only limited results, though of great interest, have
been obtained by Nishioka [50] and Masser [45].

In what follows, we develop a theory of linear Mahler systems in several
variables from the perspective of transcendence and algebraic independence,
which also includes the possibility of dealing with several systems associated
with sufficiently independent matrix transformations. It is condensed in three
main general results, Theorems 2.3, 2.6, and 2.8, which go far beyond the
existing literature. These results also surpass those of two unpublished
preprints [10, 11] the authors made available on the arXiv in 2018. The main
new feature with respect to these two preprints is that our results applies
now without any restriction on the matrices defining the systems under
consideration. In fine, the new approach we follow allows us to overcome all
the aforementioned difficulties. To tell the truth, proving Conjectures A.2
and A.3, and Corollary A.4 stated in Appendix A, was our initial goal. In
order to measure the relevance of the theory eventually developed in Section 2
to reach this goal, the reader is thus encouraged to look at Appendix A.
However, in our opinion, this theory is equally valuable in its own right.

### Organization of the paper

In Section 2, we state our main results concerning the study of linear Mahler
systems in several variables, namely Theorems 2.3, 2.6, and 2.8. We also
discuss the three main new ingredients of our approach in Section 2.4. Some
notation are introduced in Section 3. As made clear in Section 2, the strength
of our results strongly depends on our ability to provide simple and natural
conditions that ensure certain admissibility conditions. This problem is
addressed in Section 4 where concrete and optimal conditions are given. In
Section 5 we prove a new vanishing theorem that is a key ingredient for
proving Theorem 2.8. In Section 6, we state Theorem 6.2, a general result
dealing with families of linear Mahler systems associated with sufficiently
independent transformations. Some preliminary results for proving Theorem 6.2
are gathered in Section 7. Then Theorem 6.2 is proved in Section 8, while
Theorems 2.3, 2.6, 2.8, and Corollaries 2.5 and 2.9 are derived from Theorem
6.2 in Section 9. Finally, we deduce Theorem 1.1 from Theorems 2.6 and 2.8 in
Section 10, and Conjectures A.2, A.3, and Corollary A.4 from Theorem 1.1 in
Appendix A.

##  2\. Mahler’s method in several variables

Let $n\geq 1$ be an integer and ${\boldsymbol{z}}=(z_{1},\ldots,z_{n})$ be a
$n$-tuple of indeterminates. We let
$\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}$ denote the ring of $n$ variables
convergent power series with algebraic coefficients, and we set
$\overline{\mathbb{Q}}^{\star}:=\overline{\mathbb{Q}}\setminus\\{0\\}$. Given
a field extension $\mathbb{L}$ of a field $\mathbb{K}$, and
$a_{1},\ldots,a_{m}$ in $\mathbb{L}$, we let ${\rm
tr.deg}_{\mathbb{K}}(a_{1},\ldots,a_{m})$ denote the transcendence degree of
$\mathbb{K}(a_{1},\ldots,a_{m})$ over $\mathbb{K}$.

Given
$f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}})\in\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}$
related by a system of the form (2.2) (see below) and
${\boldsymbol{\alpha}}\in(\overline{\mathbb{Q}}^{\star})^{n}$ a point at which
these functions are well-defined, Mahler’s method aims at transferring results
about the absence of algebraic (resp., linear) relations between
$f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}})$ over
$\overline{\mathbb{Q}}({\boldsymbol{z}})$ to the absence of algebraic (resp.,
linear) relations over $\overline{\mathbb{Q}}$ between the complex numbers
$f_{1}({\boldsymbol{\alpha}}),\ldots,f_{m}({\boldsymbol{\alpha}})$. In
particular, a reoccurring theme consists in establishing the equality

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

under some reasonable assumptions on $A({\boldsymbol{z}})$, $T$, and
${\boldsymbol{\alpha}}$. This problem goes back to the pioneering work of
Mahler [40, 41, 42] at the end of the 1920s.

###  2.1. Mahler’s transformations and linear Mahler systems

Let $T=(t_{i,j})_{1\leq i,j\leq n}$ be an $n\times n$ matrix with nonnegative
integer coefficients. We set

$T{\boldsymbol{z}}=(z_{1}^{t_{1,1}}z_{2}^{t_{1,2}}\cdots
z_{n}^{t_{1,n}},\ldots,z_{1}^{t_{n,1}}z_{2}^{t_{n,2}}\cdots
z_{n}^{t_{n,n}})\,,$

and we let also $T$ act on $\mathbb{C}^{n}$ in a similar way.

######  Definition 2.1.

A _linear $T$-Mahler system_, or simply a _Mahler system_ , is a system of
functional equations of the form

(2.2) $\left(\begin{array}[]{c}f_{1}(T{\boldsymbol{z}})\\\ \vdots\\\
f_{m}(T{\boldsymbol{z}})\end{array}\right)=A({\boldsymbol{z}})\left(\begin{array}[]{c}f_{1}({\boldsymbol{z}})\\\
\vdots\\\ f_{m}({\boldsymbol{z}})\end{array}\right)\,,$

where $A({\boldsymbol{z}})\in{\rm
GL}_{m}(\overline{\mathbb{Q}}({\boldsymbol{z}}))$. A _Mahler function_
$f({\boldsymbol{z}})\in\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}$ is a
coordinate of a vector representing a solution to a linear Mahler system.

######  Definition 2.2.

A point ${\boldsymbol{\alpha}}\in(\overline{\mathbb{Q}}^{\star})^{n}$ is said
to be _regular_ with respect to the Mahler system (2.2) if the matrix
$A({\boldsymbol{z}})$ is well-defined and invertible at
$T^{k}{\boldsymbol{\alpha}}$ for all nonnegative integers $k$.

### Warning

Independently of the choice of the Mahler system (2.2), there are some
unavoidable restrictions that one has to impose on the matrix transformation
$T$ and on the algebraic point ${\boldsymbol{\alpha}}$. When these conditions
are fulfilled, the pair $(T,{\boldsymbol{\alpha}})$ is said to be
_admissible_. We postpone the definition of an admissible pair to Section 4,
but let us just already say that, in this respect, our results are as general
as possible. With this formalism, all our results are concerned with values at
some algebraic point ${\boldsymbol{\alpha}}$ of some Mahler functions
$f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}})$ related by a system
of the form (2.2) under the assumption that:

  * (a)

the pair $(T,{\boldsymbol{\alpha}})$ is admissible,

  * (b)

${\boldsymbol{\alpha}}$ is regular with respect to (2.2).

As discussed in [1], these conditions are typical in Mahler’s method.

###  2.2. The lifting theorem

As a first contribution, we prove the following result. Let us recall that a
field extension $\mathbb{L}$ of a field $\mathbb{K}$ is said to be _regular_
333The reader will take care that we use two totally different notions of
regularity in this paper. if $\mathbb{K}$ is algebraically closed in
$\mathbb{L}$ and $\mathbb{L}$ is separable over $\mathbb{K}$. If
${\boldsymbol{\alpha}}\in\overline{\mathbb{Q}}^{n}$, we let
$\overline{\mathbb{Q}({\boldsymbol{z}})}_{{\boldsymbol{\alpha}}}$ denote the
algebraic closure of $\overline{\mathbb{Q}}({\boldsymbol{z}})$ in
$\overline{\mathbb{Q}}\\{{\boldsymbol{z}}-{\boldsymbol{\alpha}}\\}$444 The
ring $\overline{\mathbb{Q}}\\{{\boldsymbol{z}}-{\boldsymbol{\alpha}}\\}$ is
the ring of convergent power series in
$({\boldsymbol{z}}-{\boldsymbol{\alpha}})$..

######  Theorem 2.3 (Lifting).

Let
$f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}})\in\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}$
be related by a system of functional equations of the form (2.2). Let us
assume that ${\boldsymbol{\alpha}}\in(\overline{\mathbb{Q}}^{\star})^{n}$ is a
regular point with respect to (2.2) and that the pair
$(T,{\boldsymbol{\alpha}})$ is admissible. Then for every homogeneous
polynomial $P\in\overline{\mathbb{Q}}[X_{1},\ldots,X_{m}]$ such that

$P(f_{1}({\boldsymbol{\alpha}}),\ldots,f_{m}({\boldsymbol{\alpha}}))=0\,,$

there exists a homogeneous polynomial
$Q\in\overline{\mathbb{Q}({\boldsymbol{z}})}_{{\boldsymbol{\alpha}}}[X_{1},\ldots,X_{m}]$
such that

$\displaystyle
Q({\boldsymbol{z}},f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}}))=0$
and $\displaystyle
Q({\boldsymbol{\alpha}},X_{1},\ldots,X_{m})=P(X_{1},\ldots,X_{m}).$

Furthermore, if
$\overline{\mathbb{Q}}({\boldsymbol{z}})(f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}}))$
is a regular extension of $\overline{\mathbb{Q}}({\boldsymbol{z}})$, then
there exists such a polynomial $Q$ in
$\overline{\mathbb{Q}}[{\boldsymbol{z}},X_{1},\ldots,X_{m}]$.

Theorem 2.3 is the first that applies to _all_ linear Mahler systems in
several variables, that is, without any restriction on the matrix
$A({\boldsymbol{z}})$. Furthermore, the quantitative Equality (2.1) is
replaced by a qualitative statement: any algebraic relation over
$\overline{\mathbb{Q}}$ between the values
$f_{1}({\boldsymbol{\alpha}}),\ldots,f_{m}({\boldsymbol{\alpha}})$ can be
lifted to a similar algebraic relation over
$\overline{\mathbb{Q}({\boldsymbol{z}})}$ between the functions
$f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}})$. Such a qualitative
refinement is a key for applications.

######  Remark 2.4.

Theorems 2.3 also applies to nonhomogeneous polynomials, for we can always
turn an inhomogeneous relation into an homogeneous one by adding the constant
function $f_{0}\equiv 1$ to the system and replacing the matrix
$A({\boldsymbol{z}})$ by

$\left(\begin{array}[]{c|ccc}1&&0&\\\ \hline\cr\\\ 0&&A({\boldsymbol{z}})&\\\
&&&\end{array}\right)\,.$

As a corollary of the lifting theorem, we deduce the following result.

######  Corollary 2.5.

Let
$f_{1}({\boldsymbol{z}}),\ldots,f_{m}({\boldsymbol{z}})\in\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}$
be related by a system of the form (2.2). Let us assume furthermore that
${\boldsymbol{\alpha}}\in(\overline{\mathbb{Q}}^{\star})^{n}$ is a regular
point with respect to (2.2) and that the pair $(T,{\boldsymbol{\alpha}})$ is
admissible. Then

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

Now, let us compare Theorem 1.1 with previous results on the subject.

#### The case $n=1$

In that case, the operator $T$ takes the simple form $z\mapsto z^{q}$, where
$q\geq 2$ is an integer, and the pair $(T,\alpha)$ is admissible as soon as
$0<|\alpha|<1$. Furthermore, the field extension
$\overline{\mathbb{Q}}(z)(f_{1}(z),\ldots,f_{m}(z))$ is always regular. After
several partial results due to Mahler, Kubota, and Loxton and van der Poorten,
Ku. Nishioka [49] finally proved in 1990 that

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

for all matrices $A(z)$ and all regular points
$\alpha\in\overline{\mathbb{Q}}$, $0<|\alpha|<1$. This is certainly a landmark
result in Mahler’s method. The proof of Nishioka’s theorem is based on some
technics from commutative algebra first introduced in the framework of
algebraic independence by Nesterenko. More recently, Philippon [53] and then
the authors [8] refine Nishioka’s theorem by proving the case $n=1$ of Theorem
2.3, which we refer to as Philippon’s lifting theorem. Similar lifting
theorems have first been obtained in the framework of linear differential
equations (e.g., Siegel $E$-functions) by Nesterenko and Shidlovskii [47], by
Beukers [17] using some results of André [13, 14] on the theory of
$E$-operators, and then by André [15]. A recent proof of Philippon’s lifting
theorem in the spirit of [15] is also given in [48]. In [53, 8, 48], the
latter is derived from Nishioka’s theorem, while our proof of Theorem 2.3 has
little in common with these papers and [49]. In particular, it provides a new
and more elementary way to prove the theorems of Nishioka and Philippon.

#### The case $n\geq 2$

Unfortunately, the method used for proving Nishioka’s theorem hardly
generalizes to higher dimension. In 1982, Loxton and van der Poorten [39]
published a paper claiming that

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

when the matrix $A(\boldsymbol{0})$ is well-defined and nonsingular, the pair
$(T,{\boldsymbol{\alpha}})$ is admissible, and ${\boldsymbol{\alpha}}$ is a
regular algebraic point. It was the main result published in this area, but
unfortunately some argument in their proof is flawed. This is reported, for
instance, by Nishioka in [49]. In the end, Mahler’s method in several
variables has been applied successfully only for the two following much more
restricted classes of matrices. In 1977, Kubota [31] proved that Equality
(2.5) holds true when the matrix $A({\boldsymbol{z}})$ is _almost diagonal_ ,
that is, when the functions $f_{i}({\boldsymbol{z}})$ satisfy a system of
equations of the form

(2.6) $\left(\begin{array}[]{c}1\\\ f_{1}(T{\boldsymbol{z}})\\\ \vdots\\\
f_{m}(T{\boldsymbol{z}})\end{array}\right)=\left(\begin{array}[]{c|ccc}1&0\cdots&&0\\\
\hline\cr b_{1}({\boldsymbol{z}})&a_{1}({\boldsymbol{z}})&&\\\
\vdots&&\ddots&\\\
b_{m}({\boldsymbol{z}})&&&a_{m}({\boldsymbol{z}})\end{array}\right)\left(\begin{array}[]{c}1\\\
f_{1}({\boldsymbol{z}})\\\ \vdots\\\
f_{m}({\boldsymbol{z}})\end{array}\right)\,$

where
$a_{i}({\boldsymbol{z}}),b_{i}({\boldsymbol{z}})\in\overline{\mathbb{Q}}(z)$
have no pole at $\boldsymbol{0}$, and $a_{i}(\boldsymbol{0})\neq 0$. Such
systems are precisely those arising from the study of several inhomogeneous
equations of order one. A variant of this result is due to Nishioka [51], who
proved in 1996 that Equality (2.5) also holds true when the matrix
$A({\boldsymbol{z}})$ is _almost constant_ , that is, for systems of the form

(2.7) $\left(\begin{array}[]{c}1\\\ f_{1}(T{\boldsymbol{z}})\\\ \vdots\\\
f_{m}(T{\boldsymbol{z}})\end{array}\right)=\left(\begin{array}[]{c|ccc}1&0&\cdots&0\\\
\hline\cr b_{1}({\boldsymbol{z}})&&&\\\ \vdots&&B&\\\
b_{m}({\boldsymbol{z}})&&&\end{array}\right)\left(\begin{array}[]{c}1\\\
f_{1}({\boldsymbol{z}})\\\ \vdots\\\
f_{m}({\boldsymbol{z}})\end{array}\right)\,$

where $B\in{\rm GL}_{m}(\overline{\mathbb{Q}})$, and
$b_{i}({\boldsymbol{z}})\in\overline{\mathbb{Q}}({\boldsymbol{z}})$ have no
pole at $\boldsymbol{0}$. The proof of these results (and also of the failed
attempt by Loxton and van der Poorten) follow closely the approach initiated
by Mahler in [42]. We stress that, so far, this remained the only available
strategy to tackle this problem (see Section 2.4).

###  2.3. The two purity theorems

According to the lifting theorem, the study of the algebraic relations between
the values of Mahler functions related by a system of equations of the form
(2.2) can be reduced to the easier study of the algebraic relations between
the functions themselves. However, easier does not necessarily mean _easy_ ,
and, so far, only the linear relations between $M$-functions have been fully
understood [8, 9]. Our second main result is of a different nature. It states
that, when evaluated at sufficiently independent algebraic points, Mahler
functions associated with transformations having the same spectral radius
_always_ behave independently. The main feature of this result is that there
is no need to check any kind of independence between the Mahler functions
under consideration.

To state this result properly, we first need some notation. Let us consider
several tuples of complex numbers

$\mathcal{E}_{1}=(\zeta_{1,1},\ldots,\zeta_{1,s_{1}}),\ldots,\mathcal{E}_{r}=(\zeta_{r,1},\ldots,\zeta_{r,s_{r}})\,.$

With every $i$, $1\leq i\leq r$, we associate a vector of indeterminates
$\boldsymbol{X}_{i}=(X_{i,1},\ldots,X_{i,s_{i}})$, and we let

${\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i}):=\left\\{P(\boldsymbol{X}_{i})\in\overline{\mathbb{Q}}[\boldsymbol{X}_{i}]:P(\zeta_{i,1},\ldots,\zeta_{i,s_{i}})=0\right\\}\,$

denote the ideal of algebraic relations over $\overline{\mathbb{Q}}$ between
the coordinates of $\mathcal{E}_{i}$. We also consider the tuple
$\mathcal{E}=(\zeta_{1,1},\ldots,\zeta_{r,s_{r}})$ obtained by concatenation
of the tuples $\mathcal{E}_{i}$, and we set
$\boldsymbol{X}:=(\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{r})$ and

${\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}):=\left\\{P(\boldsymbol{X})\in\overline{\mathbb{Q}}[\boldsymbol{X}]:P(\zeta_{1,1},\ldots,\zeta_{r,s_{r}})=0\right\\}\,.$

We say that $P\in{\rm Alg}_{\overline{\mathbb{Q}}}(\mathcal{E})$ is a _pure
algebraic relation_ with respect to $\mathcal{E}_{i}$ if it belongs to the
extended ideal

${\rm Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i}\mid\mathcal{E}):={\rm
span}_{\overline{\mathbb{Q}}[\boldsymbol{X}]}\\{P(\boldsymbol{X}_{i}):P\in{\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i})\\}\,.$

Our second main result reads as follows.

######  Theorem 2.6 (Purity–Independent points).

Let $r\geq 2$ be an integer. For every integer $i$, $1\leq i\leq r$, let us
consider a linear Mahler system

(2.8.$i$) $\left(\begin{array}[]{c}f_{i,1}(T_{i}{\boldsymbol{z}}_{i})\\\
\vdots\\\
f_{i,m_{i}}(T_{i}{\boldsymbol{z}}_{i})\end{array}\right)=A_{i}({\boldsymbol{z}}_{i})\left(\begin{array}[]{c}f_{i,1}({\boldsymbol{z}}_{i})\\\
\vdots\\\ f_{i,m_{i}}({\boldsymbol{z}}_{i})\end{array}\right)$

where $A_{i}({\boldsymbol{z}}_{i})$ belongs to ${\rm
GL}_{m_{i}}(\overline{\mathbb{Q}}({\boldsymbol{z}}_{i}))$,
${\boldsymbol{z}}_{i}:=(z_{i,1},\ldots,z_{i,n_{i}})$ is a tuple of
indeterminates, and $T_{i}$ is an $n_{i}\times n_{i}$ matrix with nonnegative
integer coefficients and with spectral radius $\rho(T_{i})$. Let
${\boldsymbol{\alpha}}_{i}=(\alpha_{i,1},\ldots,\alpha_{i,n_{i}})\in(\overline{\mathbb{Q}}^{\star})^{n_{i}}$,
$\mathcal{E}_{i}$ be a subtuple of
$(f_{i,1}({\boldsymbol{\alpha}}_{i}),\ldots,f_{i,m_{i}}({\boldsymbol{\alpha}}_{i}))$,
and $\mathcal{E}=(\mathcal{E}_{1},\ldots,\mathcal{E}_{r})$. Suppose that the
two following conditions hold.

  1. (i)

For every $i$, ${\boldsymbol{\alpha}}_{i}$ is regular w.r.t. (2.8.$i$) and
$(T_{i},{\boldsymbol{\alpha}}_{i})$ is admissible.

  2. (ii)

$\rho(T_{1})=\cdots=\rho(T_{r})$ and there is no nonzero tuple
$({\boldsymbol{\mu}}_{1},\ldots,{\boldsymbol{\mu}}_{r})\in\mathbb{Z}^{N}$,
$N=n_{1}+\cdots+n_{r}$, such that
$(T_{1}^{k}{\boldsymbol{\alpha}}_{1})^{{\boldsymbol{\mu}}_{1}}\cdots(T_{r}^{k}{\boldsymbol{\alpha}}_{r})^{{\boldsymbol{\mu}}_{r}}=1$,
for all $k$ in an arithmetic progression.

Then

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

In other words, the only algebraic relations between the coordinates of
$\mathcal{E}$ are those that can be trivially derived from the pure algebraic
relations with respect to the coordinates of each $\mathcal{E}_{i}$.

The first results dealing with values of Mahler functions at independent
points are due to Mahler [41] and are limited to linear independence over
$\overline{\mathbb{Q}}$. Some generalization are due to Kubota [31] and to
Loxton and van der Poorten [37]. All these results are restricted to the study
of several inhomogeneous equations of order one.

######  Remark 2.7.

Condition (ii) is clearly satisfied when all the algebraic numbers
$\alpha_{1,1},\ldots,\alpha_{r,n_{r}}$ are multiplicatively independent.

Let us turn to our third main result. It states that values at algebraic
points of Mahler functions associated with sufficiently independent
transformations _always_ behave independently. As with Theorem 2.6, the main
advantage is that there is no need to check any kind of functional
independence. Again, this result is expressed in terms of purity.

######  Theorem 2.8 (Purity–Independent transformations).

We continue with the notation of Theorem 2.6. Suppose that the two following
conditions hold.

  1. (i)

For every $i$, ${\boldsymbol{\alpha}}_{i}$ is regular w.r.t. (2.8.$i$) and
$(T_{i},{\boldsymbol{\alpha}}_{i})$ is admissible.

  2. (ii)

The spectral radii $\rho(T_{1}),\ldots,\rho(T_{r})$ are pairwise
multiplicatively independent.

Then

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

In 1976, Kubota [30] and van der Poorten [54], first envisaged the possibility
of extending Mahler’s method in order to consider simultaneously several
Mahler systems associated with independent transformations. In [30], Kubota
gave a sketch of proof in a very specific case and announced a paper on this
problem, but the latter never appeared in print. Then Loxton and van der
Poorten [38] stated some related result, but the corresponding proof is
incomplete (see [50, p. 89]). In 1987, van der Poorten [55] made this guess
more ambitious and precise, pointing out several striking consequences that
would follow from results he expected to prove in his collaboration with
Loxton. However, these authors did not publish any new paper on this problem.
In the end, only examples limited to the study of several inhomogeneous
equations of order one have been worked out by Nishioka [50] and Masser [45].
In contrast, Theorem 2.8 applies to arbitrary linear Mahler systems, and to a
much larger class of transformation matrices and algebraic points.

Of course, Theorems 2.6 and 2.8 are strong statements about algebraic
independence.

######  Corollary 2.9.

We continue with the assumptions of Theorems 2.6 or 2.8. The following
equality holds true:

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

######  Remark 2.10.

In geometric terms, Theorems 2.6 and 2.8 can be rephrased by saying that the
affine $\overline{\mathbb{Q}}$-variety associated with the ideal ${\rm
Alg}_{\overline{\mathbb{Q}}}(\mathcal{E})$ is isomorphic to the cartesian
product of the affine $\overline{\mathbb{Q}}$-varieties associated with the
ideals ${\rm Alg}_{\overline{\mathbb{Q}}}(\mathcal{E}_{i})$, $1\leq i\leq r$.
Indeed, we prove that their coordinate rings are isomorphic. That is,

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

###  2.4. Main new ingredients

As already mentioned, all previous results concerning the transcendence theory
of linear Mahler systems in several variables are very much inspired by the
early work of Mahler [42]. We also start with the same initial strategy, but
we add a number of fundamental new ingredients, including Hilbert’s
Nullstellensatz, tools from ergodic Ramsey theory, and a new vanishing
theorem.

In all previous works, a crucial step consists in expressing the coordinates
of the iterated matrix
$A_{k}({\boldsymbol{z}}):=A({\boldsymbol{z}})A(T{\boldsymbol{z}})\cdots
A(T^{k-1}{\boldsymbol{z}})$ associated with the Mahler system (2.2) in terms
of linear combination some convergent power series of the form
$g_{i}(T^{k}{\boldsymbol{z}})$, possibly twisted by some multivariate
exponential polynomials. This is really of great importance for one can then
apply some vanishing theorems to the power series $g_{i}({\boldsymbol{z}})$.
This step has gradually become more difficult in the aforementioned works, as
the matrices under consideration have taken a more general form. Its
complexity culminated in [10]. Unfortunately, one cannot expect to find this
kind of expression when $A({\boldsymbol{z}})$ is not regular singular in the
sense of [10, Definition 1.1]. Hence this strategy suffers from an intrinsic
limitation, which prevents from dealing with arbitrary Mahler systems. We
overcome this main deficiency by defining the so-called _relation matrices_ in
Section 7. Their existence and main properties are obtained by means of
Hibert’s Nullstellensatz and the notion of piecewise syndetic set. Introducing
these matrices is a cornerstone of the present work and the main novelty with
respect to our two unpublished preprints [10, 11].

In order to apply our results to transformations $T$ and points
${\boldsymbol{\alpha}}$ that are as general as possible, it is of great
importance to prove suitable _vanishing theorems_. That is, results that
guarantee the nonvanishing of arbitrary multivariate analytic functions at
special sets of points (typically, certain subsets of
$\\{T^{k}{\boldsymbol{\alpha}},\,k\geq 0\\}$). In the case of a single
transformation, Masser [44] solved this problem in a rather definitive way. We
note that Masser’s vanishing theorem (in fact a refinement using the notion of
piecewise syndetic set) is already strong enough to prove Theorems 2.3 and
2.6. In fact, we only need the identity theorem for reproving Nishioka’s
theorem and Philippon’s lifting theorem. Unfortunately, Masser’s vanishing
theorem is not suited to deal with Mahler systems associated with independent
transformations. First results towards this goal were proved by Nishioka [50]
and, again, by Masser [45]. Unfortunately, they remain too restricted for
proving Theorem 2.8. In 2005, Corvaja and Zannier [22, Theorem 3] deduced from
the subspace theorem a general result concerning the vanishing at
$\mathcal{S}$-units of analytic multivariate power series with algebraic
coefficients. They already noticed that it could be relevant for Mahler’s
method. Using the flexibility of their result and the notion of piecewise
syndetic set, we cook up in Section 5 our own vanishing theorem, which is
specifically shaped for our purpose.

###  2.5. Relevance of Mahler’s formalism

To end this section, we recall two major advantages that this multivariate
formalism offers.

First, adding variables makes it possible to deal with values of $M$-functions
at _different algebraic points_. Let us give a basic example. With the
function $\mathfrak{f}(z)=\sum_{n=0}^{\infty}z^{2^{n}}$, we can associate the
two variables linear $T$-Mahler system

(2.9) $\begin{pmatrix}1\\\ \mathfrak{f}(z_{1}^{2})\\\
\mathfrak{f}(z_{2}^{2})\end{pmatrix}=\begin{pmatrix}1&0&0\\\ -z_{1}&1&0\\\
-z_{2}&0&1\\\ \end{pmatrix}\begin{pmatrix}1\\\ \mathfrak{f}(z_{1})\\\
\mathfrak{f}(z_{2})\end{pmatrix}\,,\quad\mbox{where }\;T=\begin{pmatrix}2&0\\\
0&2\\\ \end{pmatrix}\,.$

By Theorem 4.6, the point ${\boldsymbol{\alpha}}:=(1/2,1/3)$ is regular w.r.t.
(2.9) and the pair $(T,{\boldsymbol{\alpha}})$ is admissible. The key point is
that the transcendence of $\mathfrak{f}(z)$ gives _for free_ the algebraic
independence over $\overline{\mathbb{Q}}(z_{1},z_{2})$ of the functions
$\mathfrak{f}(z_{1})$ and $\mathfrak{f}(z_{2})$. By Corollary 2.5, it follows
that $\mathfrak{f}(1/2)$ and $\mathfrak{f}(1/3)$ are algebraically independent
over $\overline{\mathbb{Q}}$. This important principle really takes shape, and
acquires great generality, with Theorems 1.1 and 2.6.

The second advantage of Mahler’s multivariate formalism comes from the
possibility of dealing with a much larger class of one-variable functions
obtained by suitable specializations of Mahler functions in several variables.
Mahler’s favorite example is the family of the Hecke-Mahler functions

$\mathfrak{f}_{\omega}(z)=\sum_{n=0}^{\infty}\lfloor n\omega\rfloor z^{n}\,,$

where $\omega$ is a quadratic irrational real number. Though
$\mathfrak{f}_{\omega}(z)$ is not an $M$-function555This fact only very
recently became known. It is an easy consequence of Theorem 2.8, but it can
also be obtained by combining the results in [3] and [23]., we have that
$\mathfrak{f}_{\omega}(z)=F_{\omega}(z,1)$, where

$F_{\omega}(z_{1},z_{2})=\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\lfloor
n_{1}\omega\rfloor}z_{1}^{n_{1}}z_{2}^{n_{2}}$

is a Mahler function in two variables. In another direction, Cobham [20]
proved that generating functions of morphic sequences are specializations of
the form $F(z,\ldots,z)$ for some multivariate Mahler functions
$F(z_{1},\ldots,z_{n})$. Some related applications of our main results can be
found in [11].

##  3\. Notation

We fix here some notation that we will use all along this paper. Let $d$ be a
positive integer and
${\boldsymbol{\alpha}}=(\alpha_{1},\ldots,\alpha_{d})\in(\mathbb{C}\setminus\\{0\\})^{d}$.
If $T=(t_{i,j})_{1\leq i,j\leq d}$ is a $d\times d$ matrix with nonnegative
integer coefficients, we let $\rho(T)$ denote its spectral radius.
Furthermore, we recall that

$T{\boldsymbol{\alpha}}=(\alpha_{1}^{t_{1,1}}\alpha_{2}^{t_{1,2}}\cdots\alpha_{d}^{t_{1,d}},\ldots,\alpha_{1}^{t_{d,1}}\alpha_{2}^{t_{d,2}}\cdots\alpha_{d}^{t_{d,d}})\,$

and that we let $T({\boldsymbol{x}})$ denote the usual matrix product between
$T$ and a column vector ${\boldsymbol{x}}\in\mathbb{C}^{d}$. If
$T_{1},\ldots,T_{d}$ are matrices, we let $T_{1}\oplus\cdots\oplus T_{d}$
denote the direct sum of these matrices. That is,

$T_{1}\oplus\cdots\oplus T_{d}=\left(\begin{array}[]{ccc}T_{1}&&\\\
&\ddots&\\\ &&T_{d}\end{array}\right)\,.$

Let ${\boldsymbol{k}}=(k_{1},\ldots,k_{d})\in\mathbb{Z}^{d}$, then
${\boldsymbol{\alpha}}^{{\boldsymbol{k}}}$ stands for
$\alpha_{1}^{k_{1}}\cdots\alpha_{d}^{k_{d}}$, so that
$(T{\boldsymbol{\alpha}})^{\boldsymbol{k}}={\boldsymbol{\alpha}}^{{\boldsymbol{k}}T}$.
Given a $d$-tuple of natural numbers ${\boldsymbol{k}}=(k_{1},\ldots,k_{d})$,
we set $|{\boldsymbol{k}}|=k_{1}+\cdots+k_{d}$. We use the same symbol
$\|\cdot\|$ to denote both the maximum norm of $\mathbb{C}^{d}$ and the
maximum norm of $\mathcal{M}_{d}(\mathbb{C})$. We let $|\xi|$ denote the
module of the complex number $\xi$.

Let ${\boldsymbol{\lambda}}=(\lambda_{1},\ldots,\lambda_{d})\in\mathbb{N}^{d}$
and ${\boldsymbol{\gamma}}=(\gamma_{1},\ldots,\gamma_{d})\in\mathbb{N}^{d}$.
We define a partial order on $\mathbb{N}^{n}$ by setting
${\boldsymbol{\lambda}}\leq{\boldsymbol{\gamma}}$ if
$\lambda_{i}\leq\gamma_{i}$, $1\leq i\leq d$. We also set

$\binom{{\boldsymbol{\lambda}}}{{\boldsymbol{\gamma}}}=\prod_{i=1}^{d}\binom{\lambda_{i}}{\gamma_{i}}\,,$

the product of binomial coefficients associated with each coordinate of
${\boldsymbol{\lambda}}$ and ${\boldsymbol{\gamma}}$. Given a positive integer
$h$, a matrix $M=(m_{i,j})_{1\leq i,j\leq h}$ with coefficients in some ring,
and a matrix ${\boldsymbol{\mu}}=(\mu_{i,j})_{1\leq i,j\leq h}$ with
nonnegative integer coefficients, we set

$M^{\boldsymbol{\mu}}=\prod_{1\leq i,j,\leq h}m_{i,j}^{\mu_{i,j}}\,.$

We use the standard Landau notation $\mathcal{O}$. We also use the notation
$\gg$ as follows. Writing that some property holds true for all integers
$\lambda_{1}\gg\lambda_{2},\lambda_{3}$ means that the corresponding property
holds true for all $\lambda_{1}$ that is sufficiently large w.r.t.
$\lambda_{2}$ and $\lambda_{3}$, while writing that some property holds true
for all integers $\lambda_{1}\gg\lambda_{2}\gg\lambda_{3}$ means that the
corresponding property holds true for all $\lambda_{1}$ that is sufficiently
large w.r.t. $\lambda_{2}$, assuming that $\lambda_{2}$ is itself sufficiently
large w.r.t. $\lambda_{3}$.

Given a positive real number $R$ and ${\boldsymbol{\alpha}}\in\mathbb{C}^{d}$,
we let

$\mathcal{D}({\boldsymbol{\alpha}},R)=\\{{\boldsymbol{\theta}}\in\mathbb{C}^{d}\
:\ \|{\boldsymbol{\theta}}-{\boldsymbol{\alpha}}\|<R\\}\,$

denote the open polydisc of center ${\boldsymbol{\alpha}}$ and radius $R$. By
definition, an element $g\in\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}$ has a
unique expansion of the form

$g({\boldsymbol{z}})=\sum_{{\boldsymbol{\lambda}}\in\mathbb{N}^{d}}g_{\boldsymbol{\lambda}}{\boldsymbol{z}}^{\boldsymbol{\lambda}}\,,$

which converges in some neighborhood of the origin. The _radius of
convergence_ of $g$ is defined as the supremum of the positive real numbers
$R$ such that the power series defining $g$ is convergent on
$\mathcal{D}(\boldsymbol{0},R)$. By [19, Proposition 2.2], when the radius of
convergence of $g$ is finite and equal to $R_{0}$, the power series defining
$g$ is absolutely convergent on $\mathcal{D}(\boldsymbol{0},R_{0})$. By
specialization, we deduce from the Cauchy-Hadamard theorem that if
$g({\boldsymbol{z}})=\sum_{{\boldsymbol{\lambda}}\in\mathbb{N}^{d}}g_{\boldsymbol{\lambda}}{\boldsymbol{z}}^{\boldsymbol{\lambda}}\in\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}$,
then

(3.1)
$|g_{\boldsymbol{\lambda}}|=\mathcal{O}\left(R^{-|{\boldsymbol{\lambda}}|}\right)\,,$

for all positive real numbers $R$ smaller than the radius of convergence of
$g$.

We let $H(\cdot)$ denote the _absolute Weil height_ over the projective space
$\mathbb{P}^{d}(\overline{\mathbb{Q}})$. Given
$\boldsymbol{\beta}=(\beta_{1},\ldots,\beta_{d})\in\overline{\mathbb{Q}}^{d}$,
we also write $H(\boldsymbol{\beta})$ instead of
$H(\beta_{1}:\cdots:\beta_{d}:1)$. We will only use basic properties of the
Weil height and we refer the interested reader to [60, Chapter 3] for more
details.

##  4\. Admissibility conditions

A well-known feature of Mahler’s method is that, independently of the choice
of the matrix $A({\boldsymbol{z}})$ defining the system (2.2), some
unavoidable restrictions on the transformation $T$ and on the point
${\boldsymbol{\alpha}}$ are required.

######  Definition 4.1.

Let $T$ be an $n\times n$ matrix with nonnegative integer coefficients and
${\boldsymbol{\alpha}}\in(\overline{\mathbb{Q}}^{\star})^{n}$. The pair
$(T,{\boldsymbol{\alpha}})$ is said to be _admissible_ if there exist two real
numbers $\rho>1$ and $c>0$ such that the following three conditions hold.

  1. (a)

The coefficients of the matrix $T^{k}$ belong to $\mathcal{O}(\rho^{k})$.

  2. (b)

Set $T^{k}{\boldsymbol{\alpha}}:=(\alpha_{1}^{(k)},\ldots,\alpha_{n}^{(k)})$.
Then $\log|\alpha_{i}^{(k)}|\leq-c\rho^{k}$, for every integer $i$, $1\leq
i\leq n$, and all sufficiently large integers $k$.

  3. (c)

If $f({{\boldsymbol{z}}})\in\mathbb{C}\\{{{\boldsymbol{z}}}\\}$ is nonzero,
then there are infinitely many integers $k$ such that
$f(T^{k}\boldsymbol{\alpha})\not=0$.

The strength of our results strongly depends on our ability to provide simple
and natural conditions that imply Conditions (a), (b), and (c). The latter are
in fact necessary to apply Mahler’s method (see [40]). Though they appear
naturally in proofs, it is not that easy, at first glance, to see how to check
them. We provide here a simple characterization of matrices and algebraic
points satisfying these conditions, gathering results of Kubota [31], Loxton
and van der Poorten [36, 37], and mainly Masser [44].

######  Definition 4.2.

Let $T$ be an $n\times n$ matrix with nonnegative integer coefficients and
with spectral radius $\rho(T)$. We say that $T$ belongs to the class
$\mathcal{M}$ if it satisfies the following three conditions.

  * (i)

It is nonsingular.

  * (ii)

None of its eigenvalues are roots of unity.

  * (iii)

There exists an eigenvector with positive coordinates associated with the
eigenvalue $\rho(T)$.

In particular, if $T$ belongs to the class $\mathcal{M}$, then $\rho(T)>1$.

######  Remark 4.3.

Let us consider $r$ Mahler systems associated with transformations
$T_{1},\ldots,T_{r}$ in $\mathcal{M}$, all having the same spectral radius
$\rho$. Then the direct sum $T_{1}\oplus\cdots\oplus T_{r}$ also belongs to
the class $\mathcal{M}$. More generally, given $r$ Mahler systems, associated
with transformation matrices $T_{1},\ldots,T_{r}\in\mathcal{M}$ with pairwise
multiplicatively dependent spectral radius, it is possible to gather them into
a larger Mahler system whose transformation matrix also belongs to the class
$\mathcal{M}$, and then to apply Theorem 2.3.

Given a one-variable Mahler system associated with a matrix $A(z)$, we can
consider the same system twice but with different variables. That is, the
system associated with the matrix

$\left(\begin{array}[]{cc}A(z_{1})&0\\\ 0&A(z_{2})\end{array}\right)\,.$

This shows that some kind of minimal independence between the coordinates of
the point ${\boldsymbol{\alpha}}=(\alpha_{1},\alpha_{2})$ is required in order
to apply Mahler’s method. Typically, we cannot consider a point of the form
$(\alpha,\alpha)$ in that case.

######  Definition 4.4.

A point ${\boldsymbol{\alpha}}\in(\overline{\mathbb{Q}}^{\star})^{n}$ is said
to be $T$-independent if there is no nonzero $n$-tuple of integers
${\boldsymbol{\mu}}$ for which
$(T^{k}{\boldsymbol{\alpha}})^{\boldsymbol{\mu}}=1$ for all $k$ in an
arithmetic progression.

######  Remark 4.5.

According to Definition 4.4, Condition (ii) of Theorem 2.6 is equivalent to
the fact that the point
${\boldsymbol{\alpha}}:=({\boldsymbol{\alpha}}_{1},\ldots,{\boldsymbol{\alpha}}_{r})$
is $T$-independent with respect to the direct sum $T=T_{1}\oplus\cdots\oplus
T_{r}$.

With these definitions, we have the following characterization of
admissibility, which makes our main results very convenient to apply.

######  Theorem 4.6.

Let $T$ be an $n\times n$ matrix with nonnegative integer coefficients and
${\boldsymbol{\alpha}}\in(\overline{\mathbb{Q}}^{\star})^{n}$. Then the pair
$(T,{\boldsymbol{\alpha}})$ is admissible if and only if $T$ belongs to the
class $\mathcal{M}$,
$\lim_{k\to\infty}T^{k}{\boldsymbol{\alpha}}=\boldsymbol{0}$, and
${\boldsymbol{\alpha}}$ is $T$-independent.

######  Remark 4.7.

There is no difficulty in checking whether or not a matrix belongs to the
class $\mathcal{M}$. Furthermore, if
$\alpha_{1},\ldots,\alpha_{n}\in\mathbb{C}\setminus\\{0\\}$ are
multiplicatively independent complex numbers, then
${\boldsymbol{\alpha}}=(\alpha_{1},\ldots,\alpha_{n})$ is _a fortiori_
$T$-independent.

###### Proof of Theorem 4.6.

We first prove the reverse direction. Let us assume that $T$ belongs to
$\mathcal{M}$, ${\boldsymbol{\alpha}}\in(\overline{\mathbb{Q}}^{\star})^{n}$
is $T$-independent, and that $\lim_{k\to\infty}T^{k}{\boldsymbol{\alpha}}=0$.
Then there exists $k_{0}$ such that $\|T^{k_{0}}{\boldsymbol{\alpha}}\|<1$. We
observe that if the pair $(T,T^{k_{0}}{\boldsymbol{\alpha}})$ is admissible,
then so is the pair $(T,{\boldsymbol{\alpha}})$. Thus, we can assume without
any loss of generality that $\|{\boldsymbol{\alpha}}\|<1.$ We let
${\boldsymbol{x}}$ denote the transpose of the vector

$\left(-\log|\alpha_{1}|,\ldots,-\log|\alpha_{n}|\right)\,,$

whose coordinates are all positive. By assumption, $T$ has a positive
eigenvector associated with the eigenvalue $\rho(T)$. Let us choose such an
eigenvector ${\boldsymbol{\mu}}$ whose coordinates are all smaller than those
of ${\boldsymbol{x}}$. We have

$-\log\|T^{k}{\boldsymbol{\alpha}}\|=\|T^{k}({\boldsymbol{x}})\|=\|T^{k}({\boldsymbol{\mu}})+T^{k}({\boldsymbol{x}}-{\boldsymbol{\mu}})\|>\|T^{k}({\boldsymbol{\mu}})\|=\rho(T)^{k}\|{\boldsymbol{\mu}}\|\,,$

for all $k\in\mathbb{N}$, because $T^{k}({\boldsymbol{x}}-{\boldsymbol{\mu}})$
has positive coordinates. Condition (b) is thus satisfied with $\rho=\rho(T)$.
Following Loxton and van der Poorten [36], Condition (a) is satisfied for the
matrix $T$ belongs to $\mathcal{M}$. Finally, Masser vanishing theorem [44]
implies that Condition (c) holds since ${\boldsymbol{\alpha}}$ is
$T$-independent. Hence, the pair $(T,{\boldsymbol{\alpha}})$ is admissible.

Now, we prove the forward direction. Let $(T,{\boldsymbol{\alpha}})$ be an
admissible pair. We first note that Condition (b) implies that
$\lim_{k\to\infty}T^{k}{\boldsymbol{\alpha}}=0$.

Replacing ${\boldsymbol{\alpha}}$ by $T^{k}{\boldsymbol{\alpha}}$ for some $k$
if necessary, we can thus assume without any loss of generality that
$\|{\boldsymbol{\alpha}}\|<1$. By [31], Condition (c) implies that the matrix
$T$ is nonsingular and that none of its eigenvalues is a root of unity. Hence
its spectral radius $\rho(T)$ is larger than $1$. Let us prove that there
exists an eigenvector with positive coordinates associated with the eigenvalue
$\rho(T)$. Let $\rho$ be as in Definition 4.1. We first infer from Conditions
(a) and (b) that $\rho=\rho(T)$. Since the coefficients of $T$ are nonnegative
integers, for every eigenvalue $\rho^{\prime}$ with $|\rho^{\prime}|=\rho$,
there is a root of unity $\mu$ such that $\rho^{\prime}=\mu\rho$ (see, for
instance, [28, Theorem 2, p. 65 & Chapter III §4]). Replacing $T$ by some
power if necessary, we can assume that $\rho$ is larger than every other
eigenvalue of $T$. Let $E_{\rho}$ denote the eigenspace associated with
$\rho$. Condition (a) implies that the characteristic space associated with
$\rho$ is equal to $E_{\rho}$, since otherwise the sequence $T^{k}/\rho^{k}$
would not be bounded. Hence $E_{\rho}$ has a $T$-invariant complement, say
$E_{\rho}^{\perp}$. From Condition (b), we infer the existence of a positive
real number $\gamma$ such that every coordinate of $T^{k}({\boldsymbol{x}})$
is larger than $\gamma\rho^{k}$. Set

${\boldsymbol{x}}=\boldsymbol{e}+\boldsymbol{e}^{\perp},\,\text{ where
}\boldsymbol{e}\in E_{\rho}\text{ and }\boldsymbol{e}^{\perp}\in
E_{\rho}^{\perp}\,.$

Suppose that there is no vector in $E_{\rho}$ with positive coordinates. Then,
for some $j$, the $j$th coordinate of $\boldsymbol{e}$ is nonpositive. Since
$T^{k}({\boldsymbol{x}})=\rho^{k}\boldsymbol{e}+T^{k}(\boldsymbol{e}^{\perp})$,
we deduce that the $j$th coordinate of $T^{k}(\boldsymbol{e}^{\perp})$ is
larger than $\gamma\rho^{k}$. Since all eigenvalues of $T$ on
$E_{\rho}^{\perp}$ are smaller than $\rho$, we obtain a contradiction. This
shows that $T$ belongs to the class $\mathcal{M}$.

Finally, ${\boldsymbol{\alpha}}$ is $T$-independent. Indeed, otherwise there
would exist two tuples of nonnegative integers $(s_{1},\ldots,s_{n})$ and
$(t_{1},\ldots,t_{n})$, not both zero, such that
$P(T^{k}{\boldsymbol{\alpha}})=0$ for infinitely many $k$, where
$P({\boldsymbol{z}})=z_{1}^{s_{1}}\cdots z_{n}^{s_{n}}$ \-
$z_{1}^{t_{1}}\cdots z_{n}^{t_{n}}$, providing a contradiction with Condition
(c). ∎

##  5\. A new vanishing theorem

As already mentioned, it is of great importance to find natural conditions
that ensure nonvanishing properties similar to Condition (c) in Definition
4.1. Of course, our goal is to obtain a _vanishing theorem_ that can be
applied to transformation matrices and points which are as general as
possible. Our contribution to this problem is Theorem 5.4.

In the framework of Mahler’s method, several vanishing theorems have been
formulated by saying that a nonzero multivariate power series cannot vanish at
all points in some well-structured large sets. The latter are obtained by
iteration of the transformation matrix and usually involve arithmetic
progressions. In order to prove our main theorems, we need to replace these
_well-structured sets_ by sets which remain large but offer much more
flexibility. We use the notion of piecewise syndetic set, which is classical
in Ramsey theory, especially in its ergodic counterpart. As we just said, it
can be though of as a notion of largeness for subsets of $\mathbb{N}$.
Furthermore, Brown’s lemma (see (ii) in Lemma 5.2) shows that such sets are
partition regular, and thus much more robust in terms of partitions than
arithmetic progressions.

######  Definition 5.1.

A set ${\mathcal{Z}}\subset\mathbb{N}$ is said to be _piecewise syndetic_ if
there exists a natural number $B\geq 1$ such that for any given integer $C\geq
2$ there exist $l_{1}<\cdots<l_{C}$ in ${\mathcal{Z}}$ such that

$l_{i+1}-l_{i}\leq B,\qquad 1\leq i<C\,.$

In this case, we say that $B$ is a bound for ${\mathcal{Z}}$. A set
${\mathcal{Z}}\subset\mathbb{N}$ is said to be _negligible_ if it is not
piecewise syndetic, while it is said to be _full_ if its complement is
negligible.

Let us recall that a subset of $\mathbb{N}$ is said to be syndetic, or
sometimes _relatively dense_ , if it has bounded gaps. A subset of
$\mathbb{N}$ is said to be _thick_ if it contains arbitrarily long intervals.
Thus piecewise syndetic sets are those that can be obtained as the
intersection of a syndetic set and a thick set. In the rest of this section,
as well as all along Section 7, we will use heavily the following results.

######  Lemma 5.2.

Let ${\mathcal{Z}}\subset\mathbb{N}$ be a piecewise syndetic set with bound
$B$. Then the following properties hold.

  1. (i)

If ${\mathcal{Z}}\subset{\mathcal{Z}}^{\prime}\subset\mathbb{N}$, then
${\mathcal{Z}}^{\prime}$ is also piecewise syndetic.

  2. (ii)

If ${\mathcal{Z}}\subset\cup_{i=1}^{s}{\mathcal{Z}}_{i}$, then at least one of
the sets ${\mathcal{Z}}_{i}$ is piecewise syndetic.

  3. (iii)

Let $l_{0}$ be a natural number. The set

${\mathcal{Z}}_{0}:=\left\\{l\in\mathcal{Z}:(l+\\{l_{0},\ldots,l_{0}+B\\})\cap{\mathcal{Z}}\neq\emptyset\right\\}$

is piecewise syndetic.

  4. (iv)

The set ${\mathcal{Z}}$ contains arbitrarily long arithmetic progressions.

###### Proof.

The point (i) immediately follows from the definition, while points (ii) and
(iv) correspond to classical results respectively known as Brown’s lemma (see
[18]) and Szemerédi’s theorem [59]. Let us prove (iii). Let $l_{0}$ and $C$ be
two natural numbers and let $a$ be the smallest integer such that $aB\geq
l_{0}$. Since ${\mathcal{Z}}$ is piecewise syndetic, there exists a sequence
$l_{1}<l_{2}<\cdots<l_{C+aB}$ of integers in ${\mathcal{Z}}$ such that
$l_{i+1}-l_{i}<B$. Let $i\in\\{1,\ldots,C\\}$. Then, $l_{i}+l_{0}\leq
l_{i}+aB\leq l_{i+aB}$. There thus exists an integer $j\leq aB$ such that
$l_{i}+l_{0}\leq l_{i+j}\leq l_{i}+l_{0}+B$. Hence
$l_{i}\in{\mathcal{Z}}_{0}$. Thus, $l_{1},\ldots,l_{C}$ all belong to the set
${\mathcal{Z}}_{0}$, which proves that this set is piecewise syndetic. ∎

Part of Lemma 5.2 can be naturally rephrased as follows.

######  Lemma 5.3.

The following properties hold.

  1. (i)

A finite union of negligible sets is negligible.

  2. (ii)

A finite intersection of full sets is full.

  3. (iii)

If ${\mathcal{Z}}_{1}$ is full and ${\mathcal{Z}}_{2}$ is negligible, then
${\mathcal{Z}}_{1}\setminus{\mathcal{Z}}_{2}$ is full.

###### Proof.

Point (i) follows directly from Point (ii) of Lemma 5.2. Let
${\mathcal{Z}}_{1},\ldots,{\mathcal{Z}}_{r}$ be full sets. By (i), we obtain
that $(\cap{\mathcal{Z}}_{i})^{c}=\cup({\mathcal{Z}}_{i}^{c})$ is negligible,
which proves (ii). Let us prove (iii). By assumption, ${\mathcal{Z}}_{1}^{c}$
is negligible. By (i), the set ${\mathcal{Z}}_{2}\cup{\mathcal{Z}}_{1}^{c}$ is
also negligible. Since
${\mathcal{Z}}_{2}\cup{\mathcal{Z}}_{1}^{c}=({\mathcal{Z}}_{1}\setminus{\mathcal{Z}}_{2})^{c}$,
we obtain that ${\mathcal{Z}}_{1}\setminus{\mathcal{Z}}_{2}$ is full, as
wanted. ∎

We are now ready to state our vanishing theorem.

######  Theorem 5.4.

Let $T_{1},\ldots,T_{r}$ be matrices in the class $\mathcal{M}$ whose spectral
radii $\rho(T_{1}),\ldots,\rho(T_{r})$ are pairwise multiplicatively
independent. Let $n_{i}$ denote the size of the matrix $T_{i}$ and set
$N:=\sum_{i=1}^{r}n_{i}$. Set

(5.1)
$\Theta:=\left(\frac{1}{\log\rho(T_{1})},\ldots,\frac{1}{\log\rho(T_{r})}\right)\,.$

For every $l\in\mathbb{N}$, we let
${\boldsymbol{k}}_{l}:=(k_{l,1},\ldots,k_{l,r})$ denote a $r$-tuple of
positive integers. Let us assume that

(5.2) $\|{\boldsymbol{k}}_{l}-l\Theta\|=\mathcal{O}(1)\,.$

Let
${\boldsymbol{\alpha}}:=({\boldsymbol{\alpha}}_{1},\ldots,{\boldsymbol{\alpha}}_{r})\in(\overline{\mathbb{Q}}^{\star})^{N}$
be such that the pair $(T_{i},{\boldsymbol{\alpha}}_{i})$ is admissible for
every $i$, and let
$g({\boldsymbol{z}})\in\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}$ be
nonzero. Then the set

$\left\\{l\in\mathbb{N}:g(T_{1}^{k_{l,1}}{\boldsymbol{\alpha}}_{1},\ldots,T_{r}^{k_{l,r}}{\boldsymbol{\alpha}}_{r})=0\right\\}$

is negligible.

Applying Mahler’s method to several Mahler systems requires some uniform speed
of convergence to the origin for the orbits of each algebraic point
${\boldsymbol{\alpha}}_{i}$ under the matrix transformations $T_{i}$. As
noticed by van der Poorten [55], one way to overcome this difficulty is to
iterate each transformation $T_{i}$ $k_{i}$-times, and to choose the iteration
vector ${\boldsymbol{k}}=(k_{1},\ldots,k_{r})$ so that asymptotically the
matrices $T_{i}^{k_{i}}$ have essentially the same radius of convergence. As
explained by Lemma 5.5, this forces us to consider only iteration vectors
${\boldsymbol{k}}$ that remain at bounded distance from the real line
$\mathbb{R}\Theta$, where $\Theta$ is defined by (5.1). This explains why the
assumption (5.2) is natural in this framework. In the rest of this section, we
set

$T_{{\boldsymbol{k}}}:=T_{1}^{k_{1}}\oplus\cdots\oplus T_{r}^{k_{r}}\;\mbox{
and
}\;T_{{\boldsymbol{k}}}{\boldsymbol{\alpha}}:=(T_{1}^{k_{1}}{\boldsymbol{\alpha}}_{1},\ldots,T_{r}^{k_{r}}{\boldsymbol{\alpha}}_{r}).$

######  Lemma 5.5.

Let $T_{1},\ldots,T_{r}$,
${\boldsymbol{\alpha}}_{1},\ldots,{\boldsymbol{\alpha}}_{r}$ be as in Theorem
5.4 and let $({\boldsymbol{k}}_{l})_{l\in\mathbb{N}}$ be an arbitrary sequence
with values in $\mathbb{N}^{r}$. Then the following properties are equivalent.

  * •

There exist a real numbers $\rho>1$ and $c>0$ such that

$\left\|T_{{\boldsymbol{k}}_{l}}\right\|=\mathcal{O}(\rho^{l})\,\mbox{ and
}\log\left\|T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}\right\|\leq-c\rho^{l}\,\mbox{
for all large enough $l$.}$

  * •

There exists a real number $\lambda$ such that $\|{\boldsymbol{k}}_{l}-\lambda
l\Theta\|=\mathcal{O}(1)$, where $\Theta$ is defined as in (5.1). In that
case, one has $\rho=e^{\lambda}$. In particular, if $\lambda=1$, then
$\rho=e$.

###### Proof.

Let us assume that there exists a real number $\lambda$ such that
$\|{\boldsymbol{k}}_{l}-\lambda l\Theta\|=\mathcal{O}(1)$. There thus exists a
positive real number $B$, such that, for every $l\in\mathbb{N}$ and $i$,
$1\leq i\leq r$, we have $k_{i,l}=\lambda l/\log(\rho(T_{i}))+\epsilon(i,l)$,
for some real number $\epsilon(i,l)$ with $|\epsilon(i,l)|\leq B$. By
Conditions (a) and (b) in Definition 4.1, we have, on the one hand, that

$\left\|T_{i}^{k_{i,l}}\right\|=\mathcal{O}\left(\rho(T_{i})^{k_{i,l}}\right)=\mathcal{O}\left(\rho(T_{i})^{\lambda
l/\log(\rho(T_{i}))}\right)=\mathcal{O}\left(e^{\lambda l}\right)\,,$

while, on the other hand,

$\log\left\|T_{i}^{k_{i,l}}{\boldsymbol{\alpha}}_{i}\right\|\leq-
c_{i}\rho(T_{i})^{k_{i,l}}\leq-c^{\prime}_{i}e^{\lambda l}\,,$

for all large enough $l$, where $c_{i}$ and $c^{\prime}_{i}$ are positive real
numbers. Setting $c:=\min_{i}\\{c^{\prime}_{1},\ldots,c^{\prime}_{r}\\}$, we
obtain the desired estimate.

Now, let us assume that

$\left\|T_{{\boldsymbol{k}}_{l}}\right\|=\mathcal{O}(\rho^{l})\,\mbox{ and
}\log\left\|T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}\right\|\leq-c\rho^{l}\,\mbox{
for all large enough $l$.}$

Set $\lambda:=\log(\rho)$ and let $i$ be an integer with $1\leq i\leq r$.
Since the pair $(T_{i},{\boldsymbol{\alpha}}_{i})$ satisfies Conditions (a)
and (b) of Definition 4.1, we have

$\left\|T_{i}^{k_{i,l}}\right\|=\mathcal{O}\left(\rho(T_{i})^{k_{i,l}}\right)\,\mbox{
and }\log\left\|T_{i}^{k_{i,l}}{\boldsymbol{\alpha}}_{i}\right\|\leq-
c_{i}\rho(T_{i})^{k_{i,l}}\,,$

for some positive real number $c_{i}$ and all large enough $l$. There thus
exist two positive real number $\kappa_{i}$ and $\gamma_{i}$ such that

$\kappa_{i}e^{\lambda l}\leq\rho(T_{i})^{k_{i,l}}\leq\gamma_{i}e^{\lambda
l}\,,$

for all $l\in\mathbb{N}$. Taking the logarithm, it follows that

$\log(\kappa_{i})+\lambda
l\leq\log(\rho(T_{i}))k_{i,l}\leq\log(\gamma_{i})+\lambda l\,.$

Dividing by $\log(\rho(T_{i}))$, we see that $k_{i,l}$ remains at bounded
distance from $\lambda l/\log(\rho(T_{i}))$. This ends the proof. ∎

Before proving Theorem 5.4, we need the two following auxiliary results. The
proof of Theorem 5.4 is based on a vanishing theorem due to Corvaja and
Zannier [22, Theorem 3]. The latter states that if the set of zeros of a
multivariate analytic function with algebraic coefficients contains an
infinite sequence of $\mathcal{S}$-unit points whose height does not grow too
fast, then these points all belong to a finite number of translates of tori.
The goal of the following two lemmas is to show that most points of the form
$T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}$ , $l\in\mathbb{N}$, avoid
these tori.

######  Lemma 5.6.

We continue with the assumptions of Theorem 5.4. Then, for every nonzero
integer $N$-tuple ${\boldsymbol{\mu}}$, the set

${\mathcal{Z}}:=\left\\{l\in\mathbb{N}:(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})^{{\boldsymbol{\mu}}}=1\right\\}\,$

is negligible.

###### Proof.

We argue by contradiction, assuming that ${\mathcal{Z}}$ is piecewise
syndetic. For every pair of nonnegative integers $(l,m)$, with $m>0$, we
define the $r$-tuple of natural numbers
${\boldsymbol{e}}:={\boldsymbol{e}}(l,m)$ by

(5.3) ${\boldsymbol{e}}={\boldsymbol{k}}_{l+m}-{\boldsymbol{k}}_{l}\,$

and we set $\mathcal{E}:=\\{{\boldsymbol{e}}(l,m):\ l,\,m\in\mathbb{N}\\}$.
Since by (5.2) we have ${\boldsymbol{k}}_{l}=l\Theta+\mathcal{O}(1)$, we
obtain that

(5.4) ${\boldsymbol{e}}(l,m)=m\Theta+\mathcal{O}(1)\,,$

which shows that $\mathcal{E}$ is infinite. However, given any positive
integer $m_{0}$, the set $\\{{\boldsymbol{e}}(l,m_{0}):\ l\in\mathbb{N}\\}$ is
finite.

Let us remark that given any pair $(\beta_{1},\beta_{2})$ of nonzero complex
numbers that are not roots of unity, and any pair of natural numbers $(i,j)$,
$1\leq i<j\leq r$, the set

$\mathcal{E}_{1}:=\left\\{{\boldsymbol{e}}=(e_{1},\ldots,e_{r})\in\mathcal{E}:\beta_{1}^{e_{i}}=\beta_{2}^{e_{j}}\right\\}$

is finite. Indeed, the set of natural numbers $u$ such that there exists a
natural number $v$ for which $\beta_{1}^{u}=\beta_{2}^{v}$ is an ideal of
$\mathbb{Z}$. Let $u_{0}\geq 0$ be a generator of this ideal and let
$v_{0}\in\mathbb{N}$ be such that $\beta_{1}^{u_{0}}=\beta_{2}^{v_{0}}$. For
every $(e_{1},\ldots,e_{r})\in\mathcal{E}_{1}$, there exists an integer $a$
such that $e_{i}=au_{0}$. We obtain that

$\beta_{2}^{e_{j}}=\beta_{1}^{e_{i}}=\beta_{1}^{au_{0}}=\beta_{2}^{av_{0}}\,.$

Since $\beta_{2}$ is nonzero and is not a root of unity, we have
$e_{j}=av_{0}$. Hence $e_{i}/e_{j}=u_{0}/v_{0}\in\mathbb{Q}$. Since
$e_{i}=k_{i,l+m}-k_{i,l}$ and $e_{j}=k_{j,l+m}-k_{j,l}$, we get that

$\frac{k_{j,l+m}-k_{j,l}}{k_{i,l+m}-k_{i,l}}=\frac{u_{0}}{v_{0}}\in\mathbb{Q}\,.$

Now let us assume by contradiction that $\mathcal{E}_{1}$ is infinite. Then,
there exist arbitrarily large integers $m$ with this property. Letting $m$
tends to infinity, we deduce from (5.4) that the ratio
$\log\rho(T_{i})/\log\rho(T_{j})$ is rational. This provides a contradiction
since by assumption $\rho(T_{i})$ and $\rho(T_{j})$ are multiplicatively
independent. Hence $\mathcal{E}_{1}$ is finite.

Let us recall that, by assumption, none of the eigenvalues of the matrices
$T_{i}$ is equal to zero or to a root of unity. The previous reasoning shows
that there exists a positive integer $m_{0}$ such that, for every $m\geq
m_{0}$, every $l\in\mathbb{N}$, every eigenvalue $\lambda_{i}$ of $T_{i}$, and
every eigenvalue $\lambda_{j}$ of $T_{j}$, $i\neq j$, we have

(5.5) $\lambda_{i}^{e_{i}}\neq\lambda_{j}^{e_{j}}\,,$

where ${\boldsymbol{e}}={\boldsymbol{e}}(l,m)=(e_{1},\ldots,e_{r})$. For such
a vector ${\boldsymbol{e}}$, set

$T_{\boldsymbol{e}}:=T_{1}^{e_{1}}\oplus\cdots\oplus T_{r}^{e_{r}}\,.$

Given a vector space $V\subset\mathbb{C}^{N}$, we have

$V\subset\bigoplus_{i=1}^{r}\iota_{i}\circ\pi_{i}(V)\,,$

where we let
$\pi_{i}:\mathbb{C}^{N}=\mathbb{C}^{n_{1}+\cdots+n_{r}}\rightarrow\mathbb{C}^{n_{i}}$
denote the projection defined by
$\pi_{i}({\boldsymbol{x}}_{1},\ldots,{\boldsymbol{x}}_{r})={\boldsymbol{x}}_{i}$
and $\iota_{i}:\mathbb{C}^{n_{i}}\rightarrow\mathbb{C}^{N}$ be such that
$\pi_{i}\circ\iota_{i}$ is the identity on $\mathbb{C}^{n_{i}}$. By (5.5), if
$V$ is invariant under $T_{\boldsymbol{e}}$, then

(5.6) $V=\bigoplus_{i=1}^{r}\iota_{i}\circ\pi_{i}(V)\,.$

We are now ready to proceed with the proof of the lemma. Let us consider the
column vector ${\boldsymbol{x}}$ whose transpose is the vector

$(\log\alpha_{1,1},\log\alpha_{1,2},\ldots,\log\alpha_{1,n_{1}},\log\alpha_{2,1},\ldots,\log\alpha_{r,n_{r}})\
,$

where $\log$ stands for a suitable determination of the logarithm (that is,
the corresponding branch cut avoids all the coordinates of the points
$T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}$, $l\in\mathbb{N}$). By
assumption, we have

$\langle{\boldsymbol{\mu}}\,,\,T_{{\boldsymbol{k}}_{l}}({\boldsymbol{x}})\rangle=0\,$

for all $l\in{\mathcal{Z}}$, where we let $\langle\,,\rangle$ denote the usual
scalar product. Let $U$ denote the orthogonal complement to the vector
${\boldsymbol{\mu}}$ in $\mathbb{C}^{N}$. This is a proper subspace of
$\mathbb{C}^{N}$ defined over $\mathbb{Q}$, which contains all vectors
$T_{{\boldsymbol{k}}_{l}}({\boldsymbol{x}})$, $l\in{\mathcal{Z}}$. Given
${\mathcal{Z}}^{\prime}\subset{\mathcal{Z}}$, we let
$U({\mathcal{Z}}^{\prime})$ denote the smallest vector subspace of
$\mathbb{C}^{N}$ defined over $\mathbb{Q}$ and containing all
$T_{{\boldsymbol{k}}_{l}}({\boldsymbol{x}})$, $l\in{\mathcal{Z}}^{\prime}$. It
follows that $U({\mathcal{Z}})\subset U$. Furthermore, if
${\mathcal{Z}}^{\prime\prime}\subset{\mathcal{Z}}^{\prime}$, then
$U({\mathcal{Z}}^{\prime\prime})\subset U({\mathcal{Z}}^{\prime})$. The
subspace $U({\mathcal{Z}})$ having finite dimension, there exists a subset
${\mathcal{Z}}_{1}\subset{\mathcal{Z}}$ that is piecewise syndetic, and such
that for all piecewise syndetic set
${\mathcal{Z}}^{\prime}\subset{\mathcal{Z}}_{1}$, one has
$U({\mathcal{Z}}^{\prime})=U({\mathcal{Z}}_{1})$. Let $B$ denote a bound for
${\mathcal{Z}}_{1}$ and set

$\mathcal{E}_{0}:=\\{{\boldsymbol{e}}(l,m):m\in[m_{0},m_{0}+B],l\in{\mathcal{Z}}_{1},l+m\in{\mathcal{Z}}_{1}\\}\,,$

where $m_{0}$ is defined as in the first part of the proof (just before
(5.5)). This is a finite set. Let

${\mathcal{Z}}_{2}:=\\{l\in{\mathcal{Z}}_{1}:\exists m\in[m_{0},m_{0}+B]\mbox{
such that }l+m\in{\mathcal{Z}}_{1}\\}\,.$

By Lemma 5.2, the set ${\mathcal{Z}}_{2}$ is piecewise syndetic. Now, given
${\boldsymbol{e}}={\boldsymbol{e}}(l,m)\in\mathcal{E}_{0}$, we set

${\mathcal{Z}}_{\boldsymbol{e}}:=\\{l\in{\mathcal{Z}}_{2}:T_{{\boldsymbol{e}}}(T_{{\boldsymbol{k}}_{l}}({\boldsymbol{x}}))=T_{{\boldsymbol{k}}_{l+m}}({\boldsymbol{x}})\in
U({\mathcal{Z}}_{1})\\}\,.$

If $l\in{\mathcal{Z}}_{2}$, then there exists $m\in[m_{0},m_{0}+b]$ such that
$l\in{\mathcal{Z}}_{{\boldsymbol{e}}}$ for some
${\boldsymbol{e}}={\boldsymbol{e}}(l,m)$. Hence,
${\mathcal{Z}}_{2}\subset\cup_{{\boldsymbol{e}}\in\mathcal{E}_{0}}{\mathcal{Z}}_{\boldsymbol{e}}$.
Since ${\mathcal{Z}}_{2}$ is piecewise syndetic, Lemma 5.2 ensures the
existence of ${\boldsymbol{e}}\in\mathcal{E}_{0}$ such that
${\mathcal{Z}}_{{\boldsymbol{e}}}$ is piecewise syndetic. Furthermore,
${\mathcal{Z}}_{{\boldsymbol{e}}}\subset{\mathcal{Z}}_{1}$. Thus we obtain
that

$U({\mathcal{Z}}_{{\boldsymbol{e}}})=U({\mathcal{Z}}_{1})\,.$

Hence, the vector space $U({\mathcal{Z}}_{1})$ is invariant under
$T_{{\boldsymbol{e}}}$. Indeed, if
$T_{{\boldsymbol{k}}_{l}}({\boldsymbol{x}})\in
U({\mathcal{Z}}_{1})=U({\mathcal{Z}}_{{\boldsymbol{e}}})$, then
$T_{{\boldsymbol{e}}}(T_{{\boldsymbol{k}}_{l}}({\boldsymbol{x}}))\in
U({\mathcal{Z}}_{1})$. By (5.6), there is a decomposition of the form

$U({\mathcal{Z}}_{1})=\bigoplus_{i=1}^{r}\iota_{i}(U_{i})\,,$

where, for every $i$,
$U_{i}=\pi_{i}(U({\mathcal{Z}}_{1}))\subset\mathbb{C}^{n_{i}}$ is a
$T_{i}^{e_{i}}$-invariant vector space defined over $\mathbb{Q}$. Since
$U({\mathcal{Z}}_{1})$ is a proper subspace of $\mathbb{C}^{N}$, there exists
$i$, $1\leq i\leq r$, such that $U_{i}$ is a proper subspace of
$\mathbb{C}^{n_{i}}$. This vector space being defined over $\mathbb{Q}$, it
has a nonzero vector ${\boldsymbol{\nu}}_{0}\in\mathbb{Z}^{n_{i}}$ in its
orthogonal complement. We thus have

(5.7)
$\langle{\boldsymbol{\nu}}_{0}\,,\,T_{i}^{e_{i}k_{i,l}}({\boldsymbol{x}}_{i})\rangle=0\,,$

for all $l\in{\mathcal{Z}}_{1}$, where
${\boldsymbol{x}}_{i}:=\pi_{i}({\boldsymbol{x}})$. The set ${\mathcal{Z}}_{1}$
being piecewise syndetic, we infer from the definition of the sequence
${\boldsymbol{k}}_{l}$ that the set
${\mathcal{Z}}_{2}:=\\{k_{i,l}:l\in{\mathcal{Z}}_{1}\\}$ is also piecewise
syndetic. By property (iv) of Lemma 5.2, it contains arbitrarily long
arithmetic progressions. Let us consider an arithmetic progression of length
$n_{i}$ in ${\mathcal{Z}}_{2}$, say

$a,a+b,a+2b,\ldots,a+(n_{i}-1)b\,,$

where $a,b\in\mathbb{N}$. Let us also consider the sequence of vector spaces

$V_{0}\subset\cdots\subset V_{n_{i}-1}\subset{\boldsymbol{\nu}}_{0}^{\perp}$

defined by

$V_{j}={\rm
Vect}_{\mathbb{Q}}\left\\{T_{i}^{e_{i}a}({\boldsymbol{x}}_{i}),\ldots,T_{i}^{e_{i}(a+jb)}({\boldsymbol{x}}_{i})\right\\}\,.$

Since $\dim V_{n_{i}-1}<n_{i}$, there exists $j_{0}$ such that
$V_{j_{0}}=V_{j_{0}+1}$. The vector space $V_{j_{0}}$ is then invariant under
$T^{e_{i}b}$ and we get that
$\langle{\boldsymbol{\nu}}_{0}\,,\,T^{e_{i}(a+kb)}({\boldsymbol{x}}_{i})\rangle=0$,
or equivalently that

$\left(T_{i}^{e_{i}(a+kb)}{\boldsymbol{\alpha}}_{i}\right)^{{\boldsymbol{\nu}}_{0}}=1\,,\;\mbox{for
all }k\in\mathbb{N}\,.$

Hence ${\boldsymbol{\alpha}}_{i}$ is not $T_{i}$-independent. By Theorem 4.6,
this provides a contradiction with the assumption that the pair
$(T_{i},{\boldsymbol{\alpha}}_{i})$ is admissible. ∎

######  Lemma 5.7.

We continue with the notation of Theorem 5.4. Let
$\gamma\in\overline{\mathbb{Q}}^{\star}$ and ${\boldsymbol{\mu}}$ be a nonzero
integer $N$-tuple. Then the set

${\mathcal{Z}}:=\left\\{l\in\mathbb{N}:\left(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}\right)^{\boldsymbol{\mu}}=\gamma\right\\}$

is negligible.

###### Proof.

Let us assume by contradiction that ${\mathcal{Z}}$ is piecewise syndetic and
let $B$ be a bound for ${\mathcal{Z}}$. Set

$\mathcal{E}:=\\{{\boldsymbol{e}}(l,m):l\in{\mathcal{Z}},l+m\in{\mathcal{Z}},m\leq
B\\}\,.$

This is a finite set. For every ${\boldsymbol{e}}\in\mathcal{E}$, set

${\mathcal{Z}}_{\boldsymbol{e}}:=\left\\{l\in\mathbb{N}:\left(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}\right)^{{\boldsymbol{\mu}}-{\boldsymbol{\mu}}T_{\boldsymbol{e}}}=1\right\\}\,$

and

${\mathcal{Z}}^{\prime}:=\\{l\in{\mathcal{Z}}:\exists m\leq B\mbox{ such that
}l+m\in{\mathcal{Z}}\\}\,.$

Lemma 5.2 implies that ${\mathcal{Z}}^{\prime}$ is piecewise syndetic. For
$l\in{\mathcal{Z}}^{\prime}$, there exists
${\boldsymbol{e}}=e(l,m)\in\mathcal{E}$ such that $m\leq B$ and
$l+m\in{\mathcal{Z}}$. Thus,

$\left(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}\right)^{{\boldsymbol{\mu}}-{\boldsymbol{\mu}}T_{\boldsymbol{e}}}=\gamma/\gamma=1\,$

and we get that
${\mathcal{Z}}^{\prime}\subset\cup_{{\boldsymbol{e}}\in\mathcal{E}}{\mathcal{Z}}_{\boldsymbol{e}}$.
Lemma 5.2 ensures the existence of ${\boldsymbol{e}}\in\mathcal{E}$ such that
${\mathcal{Z}}_{\boldsymbol{e}}$ is piecewise syndetic. By Lemma 5.6, it
follows that ${\boldsymbol{\mu}}-T_{\boldsymbol{e}}{\boldsymbol{\mu}}=0$ for
such a vector ${\boldsymbol{e}}$, which contradicts the assumption that none
of the eigenvalues of $T_{i}$ is a root of unity. ∎

We are now ready to prove Theorem 5.4.

###### Proof of Theorem 5.4.

Set

${\mathcal{Z}}:=\left\\{l\in\mathbb{N}:g(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=0\right\\}\,.$

The assumption that the pairs $(T_{i},{\boldsymbol{\alpha}}_{i})$ are
admissible allows us to apply [22, Theorem 3 ] to the sequence of points
$(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})_{l\in\mathbb{N}}$. In order
to apply the result of Corvaja and Zannier, we need to prove that the
following three conditions are satisfied.

  * (i)

There exists a finite set of places $\mathcal{S}$ such that the algebraic
points $T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}$ are
$\mathcal{S}$-units.

  * (ii)

The sequence
$(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})_{l\in\mathbb{N}}$ tends to
$0$.

  * (iii)

One has $\log
H(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=\mathcal{O}(-\log\|T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}\|)$,
where we let $H$ denote the absolute Weil height (see Section 3).

Condition (i) is easy to check. Indeed, any finite number of nonzero algebraic
numbers are $\mathcal{S}$-units for some $\mathcal{S}$. The coordinates of the
vector ${\boldsymbol{\alpha}}$ are thus $\mathcal{S}$-units for some fixed
$\mathcal{S}$, and it follows directly that all
$T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}$ are $\mathcal{S}$-units too.
Since by assumption the pairs $(T_{i},{\boldsymbol{\alpha}}_{i})$ are
admissible, Theorem 4.6 implies that the sequence
$(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})_{l\in\mathbb{N}}$ tends to
$0$, and thus (ii) is satisfied. Next we check that (iii) holds. We infer from
Lemma 5.5 that

$\|T_{{\boldsymbol{k}}_{l}}\|=\mathcal{O}(e^{l})\ \mbox{ and }\
\log\|T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}\|\leq-ce^{l}\,,$

for some positive real number $c$ and all sufficiently large integers $l$. On
the other hand, we have $\log
H(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=\mathcal{O}(\|T_{{\boldsymbol{k}}_{l}}\|)$.
It thus follows that

$\log
H(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=\mathcal{O}(-\log\|T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}\|)\,,$

which shows that Condition (iii) is satisfied.

Applying [22, Theorem 3] to the sequence of algebraic points
$(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})_{l\in\mathbb{N}}$ and to the
function $g({\boldsymbol{z}})$, we obtain the existence of a finite number of
$N$-tuples ${\boldsymbol{\mu}}_{1},\ldots,{\boldsymbol{\mu}}_{s}$ and of
algebraic numbers $\gamma_{1},\ldots,\gamma_{s}$, such that

${\mathcal{Z}}\subset\bigcup_{i=1}^{s}{\mathcal{Z}}_{i}$

where

${\mathcal{Z}}_{i}:=\left\\{l\in\mathbb{N}:\left(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}\right)^{{\boldsymbol{\mu}}_{i}}=\gamma_{i}\right\\}\,.$

By Lemma 5.7, the sets ${\mathcal{Z}}_{i}$ are all negligible. It thus follows
from Lemma 5.3 that ${\mathcal{Z}}$ is also negligible, which proves the
theorem. ∎

##  6\. Mahler’s method in families

In this section, we state Theorem 6.2, a general lifting theorem dealing with
families of Mahler systems associated with sufficiently independent
transformations. In Section 9, Theorems 2.3, 2.6, and 2.8, as well as
Corollary 2.9, will be deduced from this result.

###  6.1. Statement of Theorem 6.2

Let $r$ be a positive integer. For every $i$, $1\leq i\leq r$, let us consider
a Mahler system

(6.1.$i$) $\left(\begin{array}[]{c}f_{i,1}({\boldsymbol{z}}_{i})\\\ \vdots\\\
f_{i,m_{i}}({\boldsymbol{z}}_{i})\end{array}\right)=A_{i}({\boldsymbol{z}}_{i})\left(\begin{array}[]{c}f_{i,1}(T_{i}{\boldsymbol{z}}_{i})\\\
\vdots\\\ f_{i,m_{i}}(T_{i}{\boldsymbol{z}}_{i})\end{array}\right)$

where $n_{i}$ and $m_{i}$ are positive integers,
${\boldsymbol{z}}_{i}:=(z_{i,1},\ldots,z_{i,n_{i}})$ is a vector of
indeterminates, $T_{i}$ is an $n_{i}\times n_{i}$ matrix with nonnegative
integer coefficients and with spectral radius $\rho(T_{i})$,
$A_{i}({\boldsymbol{z}}_{i})$ belongs to $\in{\rm
GL}_{m_{i}}(\overline{\mathbb{Q}}({\boldsymbol{z}}_{i}))$, and
$f_{i,1}({\boldsymbol{z}}_{i}),\ldots,f_{i,m_{i}}({\boldsymbol{z}}_{i})$
belong to $\overline{\mathbb{Q}}\\{{\boldsymbol{z}}_{i}\\}$. We also let
${\boldsymbol{\alpha}}_{i}\in(\overline{\mathbb{Q}}^{\star})^{n_{i}}$ and
$\boldsymbol{X}_{i}:=(X_{i,1},\ldots,X_{i,m_{i}})$ denote a vector of
indeterminates. Set
${\boldsymbol{z}}:=({\boldsymbol{z}}_{1},\ldots,{\boldsymbol{z}}_{r})$ and
${\boldsymbol{\alpha}}:=({\boldsymbol{\alpha}}_{1},\ldots,{\boldsymbol{\alpha}}_{r})$.

######  Remark 6.1.

Note that one has to replace $A_{i}({\boldsymbol{z}}_{i})$ by
$A_{i}({\boldsymbol{z}}_{i})^{-1}$ to obtain a system as in (2.2). However, it
is more natural in our proof to work with systems written in the form
(6.1.$i$). We recall that
$\overline{\mathbb{Q}({\boldsymbol{z}})}_{{\boldsymbol{\alpha}}}$ denote the
algebraic closure of $\overline{\mathbb{Q}}({\boldsymbol{z}})$ in
$\overline{\mathbb{Q}}\\{{\boldsymbol{z}}-{\boldsymbol{\alpha}}\\}$.

######  Theorem 6.2.

We continue with the above assumptions. Let us assume that the two following
conditions hold.

  1. (i)

For every $i$, ${\boldsymbol{\alpha}}_{i}$ is regular w.r.t. (6.1.$i$) and
$(T_{i},{\boldsymbol{\alpha}}_{i})$ is admissible.

  2. (ii)

$\rho(T_{1}),\ldots,\rho(T_{r})$ are pairwise multiplicatively independent.

Then for every polynomial
$P\in\overline{\mathbb{Q}}[\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{r}]$ that
is homogeneous with respect to each family of indeterminates
$\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{r}$, and such that

$P(f_{1,1}({\boldsymbol{\alpha}}_{1}),\ldots,f_{r,m_{r}}({\boldsymbol{\alpha}}_{r}))=0\,,$

there exists a polynomial
$Q\in\overline{\mathbb{Q}({\boldsymbol{z}})}_{{\boldsymbol{\alpha}}}[\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{r}]$,
homogeneous with respect to each family of indeterminates
$\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{r}$, and such that

$Q({\boldsymbol{z}},f_{1,1}({\boldsymbol{z}}_{1}),\ldots,f_{r,m_{r}}({\boldsymbol{z}}_{r}))=0\mbox{
and }\
Q({\boldsymbol{\alpha}},\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{r})=P(\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{r})\,.$

Furthermore, if
$\overline{\mathbb{Q}}({\boldsymbol{z}})(f_{1,1}({\boldsymbol{z}}_{1}),\ldots,f_{r,m_{r}}({\boldsymbol{z}}_{r}))$
is a regular extension of $\overline{\mathbb{Q}}({\boldsymbol{z}})$, then
there exists such a polynomial $Q$ in
$\overline{\mathbb{Q}}[{\boldsymbol{z}},\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{r}]$.

###  6.2. Notation

In order to lighten the notation, we let
${\boldsymbol{f}}_{i}({\boldsymbol{z}}_{i})$ denote the column vector formed
by the functions
$f_{i,1}({\boldsymbol{z}}_{i}),\ldots,f_{i,m_{i}}({\boldsymbol{z}}_{i})$. We
also set

(6.2) $M:=\sum_{i=1}^{r}m_{i}\qquad\text{ and }\qquad N:=\sum_{i=1}^{r}n_{i}.$

Iterating $k$ times the system (6.1.$i$), one obtains the new system

(6.3.$i$)
${\boldsymbol{f}}_{i}({\boldsymbol{z}}_{i})=A_{i,k}({\boldsymbol{z}}_{i}){\boldsymbol{f}}_{i}(T_{i}^{k}{\boldsymbol{z}}_{i})\,,$

where

$A_{i,k}({\boldsymbol{z}}_{i}):=A_{i}({\boldsymbol{z}}_{i})A_{i}(T_{i}{\boldsymbol{z}}_{i})\cdots
A_{i}(T_{i}^{k-1}{\boldsymbol{z}}_{i})\,.$

Set ${\boldsymbol{z}}:=({\boldsymbol{z}}_{1},\ldots,{\boldsymbol{z}}_{r})$ and
by abuse of notation
${\boldsymbol{f}}_{i}({\boldsymbol{z}}):={\boldsymbol{f}}_{i}({\boldsymbol{z}}_{i})$.
For every $r$-tuple of positive integers
${\boldsymbol{k}}=(k_{1},\ldots,k_{r})$, one can gather the systems (6.3.$i$)
into a single one as follows:

(6.4) $\left(\begin{array}[]{c}{\boldsymbol{f}}_{1}({\boldsymbol{z}})\\\
\vdots\\\
{\boldsymbol{f}}_{r}({\boldsymbol{z}})\end{array}\right)=\left(\begin{array}[]{ccc}A_{1,k_{1}}({\boldsymbol{z}}_{1})&&\\\
&\ddots&\\\
&&A_{r,k_{r}}({\boldsymbol{z}}_{r})\end{array}\right)\left(\begin{array}[]{c}{\boldsymbol{f}}_{1}(T_{\boldsymbol{k}}{\boldsymbol{z}})\\\
\vdots\\\
{\boldsymbol{f}}_{r}(T_{\boldsymbol{k}}{\boldsymbol{z}})\end{array}\right)\,,$

where $T_{\boldsymbol{k}}:=T_{1}^{k_{1}}\oplus\cdots\oplus T_{r}^{k_{r}}$.
Finally, we let ${\boldsymbol{f}}({\boldsymbol{z}})$ denote the column vector
formed by all functions $f_{i,j}({\boldsymbol{z}}_{i})$, and
$A_{\boldsymbol{k}}({\boldsymbol{z}})$ denote the block diagonal matrix
defined so that (6.4) can be shortened to

(6.5)
${\boldsymbol{f}}({\boldsymbol{z}})=A_{\boldsymbol{k}}({\boldsymbol{z}}){\boldsymbol{f}}(T_{\boldsymbol{k}}{\boldsymbol{z}})\,.$

We keep this notation for the rest of the paper.

###  6.3. Choice of the sequence ${\boldsymbol{k}}_{l}$

In order to prove Theorem 6.2, one needs to choose a sequence
$({\boldsymbol{k}}_{l})_{l\in\mathbb{N}}\subset\mathbb{N}^{r}$ satisfying the
asymptotic

(6.6) ${\boldsymbol{k}}_{l}=\Theta l+\mathcal{O}(1),\qquad
l\rightarrow\infty\,,$

where $\Theta$ is defined as in (5.1), as well as some additional properties.
Let us define a partial order over $\mathbb{Z}^{r}$ by setting
${\boldsymbol{k}}_{1}\preceq{\boldsymbol{k}}_{2}$ when the vector
${\boldsymbol{k}}_{2}-{\boldsymbol{k}}_{1}$ has nonnegative coordinates. Let
$V$ denote the orthogonal complement to the vector $\Theta$ in
$\mathbb{Z}^{r}$. That is,

$V:=\\{{\boldsymbol{\mu}}\in\mathbb{Z}^{r}\,:\,\langle{\boldsymbol{\mu}},\Theta\rangle=0\\}\,.$

Let $V^{\perp}$ denote the orthogonal complement to $V$ in $\mathbb{Z}^{r}$.
That is,

$V^{\perp}:=\\{{\boldsymbol{k}}\in\mathbb{Z}^{r}\,:\,\langle{\boldsymbol{k}},{\boldsymbol{\mu}}\rangle=0\mbox{
for all }{\boldsymbol{\mu}}\in V\\}\,.$

We also set $V_{+}^{\perp}:=V^{\perp}\cap\mathbb{N}^{r}$.

######  Lemma 6.3.

There exists a sequence of nonnegative $r$-tuples
$({\boldsymbol{k}}_{l})_{l\in\mathbb{N}}\subset V_{+}^{\perp}$, satisfying the
asymptotic (6.6), and such that
${\boldsymbol{k}}_{l}\preceq{\boldsymbol{k}}_{l+1}$ for all $l\in\mathbb{N}$.

###### Proof.

We first note that the set $V$ could possibly be reduced to $\\{0\\}$; this is
the case when the numbers

$1/\log\rho(T_{1}),\ldots,1/\log\rho(T_{r})$

are linearly independent over the rational numbers666When $r>2$, It is not
known whether the pairwise multiplicative independence of the numbers
$\rho(T_{i})$ implies that their reciprocals are linearly independent over
$\mathbb{Q}$.. In that case, one can simply choose

${\boldsymbol{k}}_{l}:=\left(\left\lfloor\frac{l}{\rho(T_{1})}\right\rfloor,\ldots,\left\lfloor\frac{l}{\rho(T_{r})}\right\rfloor\right)\,.$

In contrast, the set $V^{\perp}$ is a nonempty $\mathbb{Z}$-module. Indeed,
the $\mathbb{R}$-vector space generated by $V^{\perp}$ in $\mathbb{R}^{r}$
contains the vector $\Theta$. Let
$\boldsymbol{e}_{1},\ldots,\boldsymbol{e}_{s}$ be a $\mathbb{Z}$-basis of
$V^{\perp}$. For all $l\in\mathbb{N}$, there exist real numbers
$\lambda_{1}(l),\ldots,\lambda_{s}(l)$ such that

$l\Theta=\lambda_{1}(l)\boldsymbol{e}_{1}+\cdots+\lambda_{s}(l)\boldsymbol{e}_{s}\,.$

We deduce that

(6.7)
$\|l\Theta-\lfloor\lambda_{1}(l)\rfloor\boldsymbol{e}_{1}+\cdots+\lfloor\lambda_{s}(l)\rfloor\boldsymbol{e}_{s}\|\leq\sum_{i=1}^{s}\|\boldsymbol{e}_{i}\|\,.$

Since all coordinates of $\Theta$ are positive, there exists a nonnegative
integer $l_{0}$ such that, for all $l\geq l_{0}$, the vector
$\lfloor\lambda_{1}(l)\rfloor\boldsymbol{e}_{1}+\cdots+\lfloor\lambda_{s}(l)\rfloor\boldsymbol{e}_{s}$
has positive coordinates. For every $l\geq l_{0}$, set

${\boldsymbol{a}}_{l}:=\lfloor\lambda_{1}(l)\rfloor\boldsymbol{e}_{1}+\cdots+\lfloor\lambda_{s}(l)\rfloor\boldsymbol{e}_{s}\in
V_{+}^{\perp}\,.$

The sequence $({\boldsymbol{a}}_{l})_{l\geq l_{0}}$ agrees with the asymptotic
(6.6), but not necessarily with the order $\preceq$. Let
$e:=\sum_{i=1}^{s}\|\boldsymbol{e}_{i}\|$ and let $\theta>0$ denote the
minimum of the modules of the coordinates of the vector $\Theta$. If
$l_{1}\geq l_{0}$ and $l_{2}\geq l_{1}+2e/\theta$, then (6.7) implies that
${\boldsymbol{a}}_{l_{1}}\preceq{\boldsymbol{a}}_{l_{2}}$. Set $b:=\lceil
2e/\theta\rceil$. Let us define the sequence
$({\boldsymbol{k}}_{l})_{l\in\mathbb{N}}\subset\mathbb{N}^{r}$ by setting

${\boldsymbol{k}}_{l_{0}+lb+j}:={\boldsymbol{a}}_{l_{0}+lb}$

for $l\in\mathbb{N}$ and $0\leq j<b$, and
${\boldsymbol{k}}_{l}=\boldsymbol{0}$ for $l<l_{0}$. Then the sequence
$({\boldsymbol{k}}_{l})_{l\in\mathbb{N}}$ has all the required properties. ∎

##  7\. Hilbert’s Nullstellensatz and relation matrices

In this section, we gather some preliminary results needed for proving Theorem
6.2. In particular, we introduce the so-called _relation matrices_ and study
some of their properties.

Let us fix some notation. Let $r$, and $m_{1},\ldots,m_{r}$, be some positive
integers, and let $d_{1},\ldots,d_{r}$ be some nonnegative integers. Set
$M=m_{1}+\cdots+m_{r}$ and let
${\boldsymbol{\mu}}_{1},\ldots,{\boldsymbol{\mu}}_{t}$ denote an enumeration
of all distinct $M$-tuples
${\boldsymbol{\mu}}:=(\mu_{1,1},\ldots,\mu_{1,m_{1}},\mu_{2,1},\ldots,\mu_{r,m_{r}})$
such that

$\mu_{i,1}+\cdots+\mu_{i,m_{i}}=d_{i}\,,$

for every $i$, $1\leq i\leq r$. For every $i$, $1\leq i\leq r$, we let $B_{i}$
denote an $m_{i}\times m_{i}$ matrix with coefficients in some commutative
ring $\mathcal{R}$, and we let $B=B_{1}\oplus\cdots\oplus B_{r}$. Given a
vector of indeterminates
$\boldsymbol{X}=(\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{r})$, where
$\boldsymbol{X}_{i}=(X_{i,1},\ldots,X_{i,m_{i}})$, we note that
$(B(\boldsymbol{X}))^{{\boldsymbol{\mu}}_{j}}\in\mathcal{R}[\boldsymbol{X}]$
is a homogeneous polynomial of degree $d_{i}$ in each set of variables
$\boldsymbol{X}_{i}$. We let $R_{j,l}(B)$ denote the elements of $\mathcal{R}$
defined by

(7.1)
$\left(B(\boldsymbol{X})\right)^{{\boldsymbol{\mu}}_{j}}=\sum_{l=1}^{t}R_{j,l}(B)\boldsymbol{X}^{{\boldsymbol{\mu}}_{l}}\,.$

For every $j$ and $l$, $R_{j,l}(B)$ is a polynomial of degree
$d:=\max\\{d_{1},\ldots,d_{r}\\}$ in the coefficients of the matrix $B$. Set
$R(B):=(R_{j,l}(B))_{1\leq j,l\leq t}$. Let $B_{1}$ and $B_{2}$ be two
$M\times M$ block diagonal matrix. Then

(7.2) $R(B_{1}B_{2})=R(B_{1})R(B_{2})$

and

(7.3) $R(B_{1})^{-1}=R(B_{1}^{-1})$

when $B_{1}$ is invertible.

All along this section, we continue with the notation of Section 6.2. We let
$A_{\boldsymbol{k}}({\boldsymbol{z}})\in{\rm
GL}_{m}(\overline{\mathbb{Q}}({\boldsymbol{z}}))$ and
$f_{1,1}({\boldsymbol{z}}),\ldots,f_{r,m_{r}}({\boldsymbol{z}})\in\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}$
be as in (6.5). We fix some
${\boldsymbol{\alpha}}:=({\boldsymbol{\alpha}}_{1},\ldots,{\boldsymbol{\alpha}}_{r})\in(\overline{\mathbb{Q}}^{\star})^{N}$
such that
${\boldsymbol{\alpha}}_{i}\in(\overline{\mathbb{Q}}^{\star})^{n_{i}}$ is
regular with respect to the matrix $A_{i}$. We also fix a sequence of integer
vectors $({\boldsymbol{k}}_{l})_{l\in\mathbb{N}}$ satisfying the assumptions
of Lemma 6.3. Then, for every ${\boldsymbol{k}}\in\mathbb{N}^{r}$, we set

(7.4)
${\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{z}}):=R(A_{\boldsymbol{k}}({\boldsymbol{z}}))\,.$

This is a $t\times t$-matrix with coefficients in
$\overline{\mathbb{Q}}({\boldsymbol{z}})$. One has
${\boldsymbol{R}}_{\boldsymbol{0}}({\boldsymbol{z}})={\rm I}_{t}$ and, given
${\boldsymbol{k}}_{1},{\boldsymbol{k}}_{2}\in\mathbb{N}^{r}$, it follows from
(7.2) that

(7.5)
${\boldsymbol{R}}_{{\boldsymbol{k}}_{1}+{\boldsymbol{k}}_{2}}({\boldsymbol{z}})={\boldsymbol{R}}_{{\boldsymbol{k}}_{1}}({\boldsymbol{z}}){\boldsymbol{R}}_{{\boldsymbol{k}}_{2}}(T_{{\boldsymbol{k}}_{1}}{\boldsymbol{z}})\,.$

Furthermore, since the points ${\boldsymbol{\alpha}}_{i}$ are assumed to be
regular, we infer from (7.3) that the matrix
${\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{\alpha}})$ is well-defined and
invertible for all ${\boldsymbol{k}}\in\mathbb{N}^{r}$, with inverse equal to
$R(A_{\boldsymbol{k}}({\boldsymbol{\alpha}})^{-1})$.

Let $\boldsymbol{Y}:=(y_{i,j})_{1\leq i,j\leq t}$ denote a matrix of
indeterminates. Given a field $\mathbb{K}$ and a nonnegative integer
$\delta_{1}$, we let $\mathbb{K}[\boldsymbol{Y}]_{\delta_{1}}$ denote the set
of polynomials of degree at most $\delta_{1}$ in every indeterminate
$y_{i,j}$. Given two nonnegative integers $\delta_{1}$ and $\delta_{2}$, we
let $\mathbb{K}[\boldsymbol{Y},{\boldsymbol{z}}]_{\delta_{1},\delta_{2}}$
denote the set of polynomials
$P\in\mathbb{K}[\boldsymbol{Y},{\boldsymbol{z}}]$ of degree at most
$\delta_{1}$ in every indeterminate $y_{i,j}$ and of total degree at most
$\delta_{2}$ in the indeterminates $z_{i,j}$. By Theorem 5.4, a polynomial
$P\in\overline{\mathbb{Q}}({\boldsymbol{z}})[\boldsymbol{Y}]$ is well-defined
at the point
$({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$
for all $l$ in a full subset of $\mathbb{N}$. Set

$\mathcal{I}:=\\{P\in\overline{\mathbb{Q}}({\boldsymbol{z}})[\boldsymbol{Y}]\
:P({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=0\,,\;\forall
l\mbox{ in a full subset of $\mathbb{N}$}\\}\,.$

Piecewise syndetic, negligible, and full sets are introduced in Definition
5.1.

###  7.1. Estimates for the dimension of certain vector spaces

Let $\delta_{1}$ and $\delta_{2}$ be two nonnegative integers. Set
$\mathcal{I}(\delta_{1}):=\mathcal{I}\cap\overline{\mathbb{Q}}({\boldsymbol{z}})[\boldsymbol{Y}]_{\delta_{1}}$
and
$\mathcal{I}(\delta_{1},\delta_{2}):=\mathcal{I}\cap\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{\delta_{1},\delta_{2}}$.
Note that $\mathcal{I}(\delta_{1},\delta_{2})$ is a vector subspace of
$\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{\delta_{1},\delta_{2}}$,
and let $\mathcal{I}^{\perp}(\delta_{1},\delta_{2})$ denote a complement to
$\mathcal{I}(\delta_{1},\delta_{2})$ in
$\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{\delta_{1},\delta_{2}}$.

######  Lemma 7.1.

Let $d(\delta_{1},\delta_{2})$ denote the dimension of
$\mathcal{I}^{\perp}(\delta_{1},\delta_{2})$ over $\overline{\mathbb{Q}}$.
There exists a positive real number $c_{1}(\delta_{1})$, that does not depend
on $\delta_{2}$, such that

$d(\delta_{1},\delta_{2})\sim c_{1}(\delta_{1})\delta_{2}^{N}\,,\mbox{ as
$\delta_{2}$ tends to infinity. }$

###### Proof.

Let
$\boldsymbol{Y}^{{\boldsymbol{\nu}}_{1}},\ldots,\boldsymbol{Y}^{{\boldsymbol{\nu}}_{h}}$,
$h:=(\delta_{1}+1)^{t^{2}}$, denote an enumeration of the monomials of degree
at most $\delta_{1}$ in every indeterminates $y_{i,j}$. Let
$P_{1},\ldots,P_{h}$ be polynomials in $\mathcal{I}(\delta_{1})$. Every
$P_{i}$ has a unique decomposition of the form

$P_{i}(\boldsymbol{Y},{\boldsymbol{z}}):=\sum_{j=1}^{h}p_{i,j}({\boldsymbol{z}})\boldsymbol{Y}^{{\boldsymbol{\nu}}_{j}}\,,$

where $p_{i,j}({\boldsymbol{z}})\in\overline{\mathbb{Q}}({\boldsymbol{z}})$,
$1\leq i,j\leq h$. Let $C({\boldsymbol{z}})$ denote the square matrix defined
by $C({\boldsymbol{z}}):=(p_{i,j}({\boldsymbol{z}}))_{1\leq i,j\leq h}$. By
Theorem 5.4, $C({\boldsymbol{z}})$ is well-defined at
$T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}$ for all $l$ in a full set
${\mathcal{Z}}_{0}\subset\mathbb{N}$. For every $i$, $1\leq i\leq h$, let
${\mathcal{Z}}_{i}$ denote the set of nonnegative integers $l$ such that
$P_{i}({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=0$.
Since $P_{1},\ldots,P_{h}\in\mathcal{I}(\delta_{1})$, the sets
${\mathcal{Z}}_{1},\ldots,{\mathcal{Z}}_{h}$ are full. By Lemma 5.3, the set
${\mathcal{Z}}:=\bigcap_{i=0}^{h}{\mathcal{Z}}_{i}$ is full. For every
$l\in{\mathcal{Z}}$, one has

(7.6)
$C(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})\left(\begin{array}[]{c}{\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}})^{{\boldsymbol{\nu}}_{1}}\\\
\vdots\\\
{\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}})^{{\boldsymbol{\nu}}_{h}}\end{array}\right)=\left(\begin{array}[]{c}P_{1}({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})\\\
\vdots\\\
P_{h}({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})\end{array}\right)=0\,.$

Since the matrix
${\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}})$ is
invertible, it is nonzero. Hence the vector
$({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}})^{{\boldsymbol{\nu}}_{1}},\ldots,{\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}})^{{\boldsymbol{\nu}}_{h}})$
is also nonzero, and we deduce from (7.6) that $\det
C(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=0$ for all
$l\in{\mathcal{Z}}$. Since ${\mathcal{Z}}$ is not negligible, Theorem 5.4
implies that $\det C({\boldsymbol{z}})=0$. Hence $\mathcal{I}(\delta_{1})$ is
a strict subspace of
$\overline{\mathbb{Q}}({\boldsymbol{z}})[\boldsymbol{Y}]_{\delta_{1}}$, say of
dimension $d<h$. There thus exist polynomials
$b_{i,j}(z)\in\overline{\mathbb{Q}}[{\boldsymbol{z}}]$, $1\leq i\leq h-d$,
$1\leq j\leq h$, such that for all
$p_{1}({\boldsymbol{z}}),\ldots,p_{h}({\boldsymbol{z}})\in\overline{\mathbb{Q}}({\boldsymbol{z}})$:

(7.7)
$\sum_{j=1}^{h}p_{j}({\boldsymbol{z}})\boldsymbol{Y}^{{\boldsymbol{\nu}}_{j}}\in\mathcal{I}(\delta_{1})\Leftrightarrow\sum_{j=1}^{h}b_{i,j}({\boldsymbol{z}})p_{j}({\boldsymbol{z}})=0\;\;\;\;\forall
i,\;1\leq i\leq h-d\,.$

Since $d+(h-d)=h$, these equations are linearly independent. Now, let us
consider a polynomial
$P:=\sum_{j=1}^{h}p_{j}({\boldsymbol{z}})\boldsymbol{Y}^{{\boldsymbol{\nu}}_{j}}\in\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{\delta_{1},\delta_{2}}$
and set

$p_{j}({\boldsymbol{z}})=\sum_{{\boldsymbol{\lambda}}\,:\,|{\boldsymbol{\lambda}}|\leq\delta_{2}}p_{j,{\boldsymbol{\lambda}}}{\boldsymbol{z}}^{\boldsymbol{\lambda}}\
\text{ and }\
b_{i,j}({\boldsymbol{z}})=\sum_{{\boldsymbol{\kappa}}\,:\,|{\boldsymbol{\kappa}}|\leq\delta_{1}^{\prime}}b_{i,j,{\boldsymbol{\kappa}}}{\boldsymbol{z}}^{\boldsymbol{\kappa}}\,,$

where $\delta_{1}^{\prime}$ is a nonnegative integer depending only on
$\delta_{1}$, and where the numbers $p_{j,{\boldsymbol{\lambda}}}$ and
$b_{i,j,{\boldsymbol{\kappa}}}$ are algebraic for all quadruples
$(i,j,{\boldsymbol{\lambda}},{\boldsymbol{\kappa}})$. By (7.7), $P$ belongs to
$\mathcal{I}(\delta_{1},\delta_{2})$ if and only if

$\sum_{j=1}^{h}\sum_{\begin{subarray}{c}|{\boldsymbol{\lambda}}|\leq\delta_{2},|{\boldsymbol{\kappa}}|\leq\delta_{1}^{\prime}\\\
{\boldsymbol{\lambda}}+{\boldsymbol{\kappa}}={\boldsymbol{\gamma}}\end{subarray}}b_{i,j,{\boldsymbol{\kappa}}}p_{j,{\boldsymbol{\lambda}}}=0\,,\;\;\;\;\forall({\boldsymbol{\gamma}},i)\,.$

If we fix the vector ${\boldsymbol{\gamma}}$, then, as soon as every
coordinate of ${\boldsymbol{\gamma}}$ is larger than $\delta_{1}^{\prime}$ and
that $|{\boldsymbol{\gamma}}|\leq\delta_{2}$, the number of linearly
independent equations does not depend on our choice for
${\boldsymbol{\gamma}}$. Thus, the total number of linearly independent
equations defining the vector space $\mathcal{I}(\delta_{1},\delta_{2})$ is
equivalent to

$c_{1}(\delta_{1})\delta_{2}^{N}\,,$

when $\delta_{2}$ tends to infinity, where $c_{1}(\delta_{1})$ is a positive
real number depending only on $\delta_{1}$. This number is precisely the
dimension of the vector space $\mathcal{I}^{\perp}(\delta_{1},\delta_{2})$,
which ends the proof. ∎

######  Lemma 7.2.

For every pair of nonnegative integers $(\delta_{1},\delta_{2})$, one has

$\dim\mathcal{I}^{\perp}(2\delta_{1},\delta_{2})\leq
2^{t^{2}}\dim\mathcal{I}^{\perp}(\delta_{1},\delta_{2})\,.$

###### Proof.

Let
$P(\boldsymbol{Y},{\boldsymbol{z}})\in\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{2\delta_{1},\delta_{2}}$.
Such a polynomial has a decomposition of the form

(7.8)
$P(\boldsymbol{Y},{\boldsymbol{z}})=\sum_{\ell=1}^{t^{2}}e_{\ell}(\boldsymbol{Y})^{\delta_{1}}P_{\ell}(\boldsymbol{Y},{\boldsymbol{z}})\,,$

where we let $e_{\ell}(\boldsymbol{Y})$ denote the $2^{t^{2}}$ monomials of
degree at most one in the indeterminates $y_{i,j}$, and where each
$P_{\ell}(\boldsymbol{Y},{\boldsymbol{z}})$ belongs to
$\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{\delta_{1},\delta_{2}}$.
If, in such a decomposition, every polynomial $P_{\ell}$ belongs to
$\mathcal{I}(\delta_{1},\delta_{2})$, then $P$ belongs to
$\mathcal{I}(2\delta_{1},\delta_{2})$. The decomposition (7.8) naturally
defines a surjective linear map from
$\left(\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{\delta_{1},\delta_{2}}/\mathcal{I}(\delta_{1},\delta_{2})\right)^{2^{t^{2}}}$
to
$\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{z}}]_{2\delta_{1},\delta_{2}}/\mathcal{I}(2\delta_{1},\delta_{2})$.
It follows that

$\dim_{\overline{\mathbb{Q}}}\mathcal{I}^{\perp}(2\delta_{1},\delta_{2})\leq
2^{t^{2}}\dim_{\overline{\mathbb{Q}}}\mathcal{I}^{\perp}(\delta_{1},\delta_{2})\,,$

as wanted. ∎

###  7.2. Nullstellensatz and relation matrices

In this section, we show how Hilbert’s Nullstellensatz allows us to exhibit a
matrix ${\boldsymbol{\phi}}$, called a _relation matrix_ , whose coordinates
are all algebraic over $\overline{\mathbb{Q}}({\boldsymbol{z}})$, and which
encodes the algebraic relations over $\overline{\mathbb{Q}}({\boldsymbol{z}})$
of degree at most $d_{i}$ in each variables between the functions
$f_{1,1}({\boldsymbol{z}}),\ldots,f_{r,m_{r}}({\boldsymbol{z}})$. These
relation matrices are the cornerstone of the proof of Theorem 6.2.

We first prove the following lemma.

######  Lemma 7.3.

The set $\mathcal{I}$ is a radical ideal of
$\overline{\mathbb{Q}}({\boldsymbol{z}})[\boldsymbol{Y}]$.

###### Proof.

Checking that $\mathcal{I}$ is an ideal of
$\overline{\mathbb{Q}}({\boldsymbol{z}})[\boldsymbol{Y}]$ is not difficult. If
$P_{1},P_{2}\in\mathcal{I}$, and if ${\mathcal{Z}}_{1}$ (resp.
${\mathcal{Z}}_{2}$) are full sets of nonnegative integers $l$ for which
$P_{1}({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=0$
(resp.
$P_{2}({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=0$),
then $P_{1}+P_{2}$ vanishes at the points
$({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$
for all $l\in{\mathcal{Z}}_{1}\cap{\mathcal{Z}}_{2}$. By Lemma 5.3, this set
is full. Hence $P_{1}+P_{2}\in\mathcal{I}$.

Now let $P_{1}\in\mathcal{I}$ and
$P_{2}\in\overline{\mathbb{Q}}({\boldsymbol{z}})[\boldsymbol{Y}]$. On the one
hand,
$P_{1}({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=0$
for all $l$ in a full set ${\mathcal{Z}}_{1}$, while, on the other hand,
Theorem 5.4 ensures that $P_{2}(\boldsymbol{Y},{\boldsymbol{z}})$ is well-
defined at
$({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$
for all nonnegative integers $l$ outside a negligible set ${\mathcal{Z}}_{2}$.
We deduce that

$P_{1}({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})P_{2}({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=0$

for all $l\in{\mathcal{Z}}_{1}\setminus{\mathcal{Z}}_{2}$. By Lemma 5.3, this
is a full set. Hence $P_{1}P_{2}\in\mathcal{I}$.

Let $P\in\overline{\mathbb{Q}}({\boldsymbol{z}})[\boldsymbol{Y}]$ be such that
$P^{r}\in\mathcal{I}$ for some $r$. If $l$ is a nonnegative integer such that
$P({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})^{r}=0$,
then
$P({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=0$.
Hence $P\in\mathcal{I}$ and $\mathcal{I}$ is a radical ideal. ∎

Let $\mathbb{K}$ denote an algebraic closure of
$\overline{\mathbb{Q}}({\boldsymbol{z}})$.

######  Lemma 7.4.

There exists a matrix ${\boldsymbol{\phi}}\in{\rm GL}_{t}(\mathbb{K})$ such
that

$P({\boldsymbol{\phi}},{\boldsymbol{z}})=0\,,$

for all polynomials $P\in\mathcal{I}$.

###### Proof.

Let us consider the affine algebraic set $\mathcal{V}$ associated with the
radical ideal $\mathcal{I}$. That is,

${\mathcal{V}}:=\\{{\boldsymbol{\phi}}\in\mathcal{M}_{t}(\mathbb{K})\ :\
P({\boldsymbol{\phi}},{\boldsymbol{z}})=0\,,\;\forall P\in\mathcal{I}\\}\,.$

According to the weak form of Hilbert’s Nullstellensatz (see, for instance,
[33, Theorem 1.4, p. 379]), $\mathcal{V}$ is nonempty as soon as $\mathcal{I}$
is a proper ideal of
$\overline{\mathbb{Q}}({\boldsymbol{z}})[\boldsymbol{Y}]$. But the definition
of $\mathcal{I}$ clearly implies that nonzero constant polynomials do not
belong to $\mathcal{I}$. Hence $\mathcal{V}$ is nonempty.

Now, let us assume by contradiction that $\det{\boldsymbol{\phi}}=0$ for all
${\boldsymbol{\phi}}$ in $\mathcal{V}$. By Hilbert’s Nullstellensatz (see, for
instance, [33, Theorem 1.5, p. 380]), the polynomial $\det\boldsymbol{Y}$
belongs to the radical of the ideal $\mathcal{I}$. Hence
$\det\boldsymbol{Y}\in\mathcal{I}$ for $\mathcal{I}$ is radical. Thus,
$\det{\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}})=0$ for all
$l$ in a full set. This provides a contradiction since
${\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}})$ is invertible
for all $l$ in $\mathbb{N}$. We thus deduce that there exists an invertible
matrix ${\boldsymbol{\phi}}$ in $\mathcal{V}$, as wanted. ∎

######  Definition 7.5.

A matrix ${\boldsymbol{\phi}}\in{\rm GL}_{t}(\mathbb{K})$ satisfying the
property of Lemma 7.4 is called a _relation matrix_.

The next lemma plays a central role in the proof of Theorem 6.2.

######  Lemma 7.6.

Let ${\boldsymbol{\phi}}\in{\rm GL}_{t}(\mathbb{K})$ be a relation matrix.
Then

$P\left({\boldsymbol{\phi}}{\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{z}}),T_{\boldsymbol{k}}{\boldsymbol{z}}\right)=0\,,$

for all $P\in\mathcal{I}$ and all ${\boldsymbol{k}}\in V_{+}^{\perp}$.

###### Proof.

Let $\mathcal{A}$ denote the subring of
$\overline{\mathbb{Q}}({\boldsymbol{z}})$ formed by all rational functions
with no pole at the points $T_{{\boldsymbol{k}}}{\boldsymbol{\alpha}}$,
${\boldsymbol{k}}\in\mathbb{N}^{r}$. The set $\mathcal{S}$ of polynomials
$P\in\overline{\mathbb{Q}}[{\boldsymbol{z}}]$ that does not vanish at any of
the points $T_{\boldsymbol{k}}{\boldsymbol{\alpha}}$,
${\boldsymbol{k}}\in\mathbb{N}^{r}$, being multiplicatively closed, the ring
$\mathcal{A}$ is the localization of $\overline{\mathbb{Q}}[{\boldsymbol{z}}]$
at $\mathcal{S}$, i.e.,
$\mathcal{A}=\mathcal{S}^{-1}\overline{\mathbb{Q}}[{\boldsymbol{z}}]$. It
follows that $\mathcal{A}$ is a Noetherian ring (see, for instance, [33,
Proposition 1.6, p. 415]). Since by assumption each of the points
${\boldsymbol{\alpha}}_{1},\ldots,{\boldsymbol{\alpha}}_{r}$ is regular with
respect to the corresponding Mahler system (6.1.$i$), the coefficients of the
matrices ${\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{z}})$,
${\boldsymbol{k}}\in\mathbb{N}^{r}$, belong to the ring $\mathcal{A}$. With
any piecewise syndetic set ${\mathcal{Z}}\subset\mathbb{N}$, we associate the
set

$\mathcal{I}_{\mathcal{Z}}:=\left\\{P\in\mathcal{A}[\boldsymbol{Y}]\,:\,P({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=0\,,\;\forall
l\in{\mathcal{Z}}\right\\}\,.$

The proof is divided into the following seven simple results, namely Facts 1
to 7.

######  Fact 1.

The set $\mathcal{I}_{\mathcal{Z}}$ is an ideal of
$\mathcal{A}[\boldsymbol{Y}]$.

Let $P_{1},P_{2}\in\mathcal{I}_{\mathcal{Z}}$. Then $P_{1}+P_{2}$ vanishes at
$({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$
for all $l\in{\mathcal{Z}}$. Hence $P_{1}+P_{2}\in\mathcal{I}_{\mathcal{Z}}$.
Let $P_{1}\in\mathcal{I}_{\mathcal{Z}}$ and
$P_{2}\in\mathcal{A}[\boldsymbol{Y}]$. Then
$P_{1}({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=0$
for all $l\in{\mathcal{Z}}$. On the other hand, by definition of
$\mathcal{A}$, $P_{2}(\boldsymbol{Y},{\boldsymbol{z}})$ has no pole at
$({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$,
$l\in\mathbb{N}$. It follows that $P_{1}P_{2}$ vanishes at
$({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$
for all $l\in{\mathcal{Z}}$. Hence $P_{1}P_{2}\in\mathcal{I}_{\mathcal{Z}}$,
which proves Fact 1.

If ${\mathcal{Z}}^{\prime}$ is a piecewise syndetic set such that
${\mathcal{Z}}^{\prime}\subset{\mathcal{Z}}\subset\mathbb{N}$, one has
$\mathcal{I}_{\mathcal{Z}}\subset\mathcal{I}_{{\mathcal{Z}}^{\prime}}$. Since
$\mathcal{A}$ is Noetherian, $\mathcal{A}[\boldsymbol{Y}]$ is Noetherian too
and any increasing sequence of ideals is stationary. Thus, for every piecewise
syndetic set ${\mathcal{Z}}\subset\mathbb{N}$, there exists a piecewise
syndetic set ${\mathcal{Z}}_{0}\subset{\mathcal{Z}}$ such that
$\mathcal{I}_{{\mathcal{Z}}_{1}}=\mathcal{I}_{{\mathcal{Z}}_{0}}$ for all
piecewise syndetic sets ${\mathcal{Z}}_{1}\subset{\mathcal{Z}}_{0}$. For Facts
2 to 7, we fix such a pair of sets $({\mathcal{Z}},{\mathcal{Z}}_{0})$.

######  Fact 2.

There exist infinitely many $r$-tuples ${\boldsymbol{k}}\in V_{+}^{\perp}$
such that

(7.9) ${\mathcal{Z}}_{0}({\boldsymbol{k}}):=\\{l\in{\mathcal{Z}}_{0}\ :\
\exists
l^{\prime}\in{\mathcal{Z}}_{0},\;{\boldsymbol{k}}_{l^{\prime}}={\boldsymbol{k}}_{l}+{\boldsymbol{k}}\\}$

is a piecewise syndetic set.

Let $B$ be a bound for ${\mathcal{Z}}_{0}$ and $e\geq 0$ be an integer. By
Lemma 5.2, the set

${\mathcal{Z}}_{e}:=\\{l\in{\mathcal{Z}}_{0}\ :\ \exists
l^{\prime}\in{\mathcal{Z}}_{0},e\leq l^{\prime}-l\leq e+B\\}$

is piecewise syndetic. By (6.6), there are only finitely many differences
${\boldsymbol{k}}_{l^{\prime}}-{\boldsymbol{k}}_{l}$ for which $e\leq
l^{\prime}-l\leq e+B$. Furthermore, by Lemma 6.3, such differences all belong
to $V_{+}^{\perp}$. Let $\mathcal{K}_{e}\subset V_{+}^{\perp}$ denote the
finite set formed by these differences. Then

${\mathcal{Z}}_{e}\subset\cup_{{\boldsymbol{k}}\in\mathcal{K}_{e}}{\mathcal{Z}}_{0}({\boldsymbol{k}})\,.$

By Lemma 5.2, at least one of the sets ${\mathcal{Z}}_{0}({\boldsymbol{k}})$,
${\boldsymbol{k}}\in\mathcal{K}_{e}$, is piecewise syndetic. Letting $e$ tend
to infinity, this proves Fact 2.

Now, let

$\mathcal{K}:=\\{{\boldsymbol{k}}\in V_{+}^{\perp}:\
{\mathcal{Z}}_{0}({\boldsymbol{k}})\,\mbox{ is piecewise syndetic}\\}\,$

denote this infinite set.

######  Fact 3.

The $\mathbb{Z}$-module generated by $\mathcal{K}$ in $\mathbb{Z}^{r}$ is
equal to $V^{\perp}$.

Let $W$ denote the $\mathbb{Z}$-module generated by $\mathcal{K}$ in
$\mathbb{Z}^{r}$. It is enough to show that $W^{\perp}$, its orthogonal
complement in $\mathbb{Z}^{r}$, is equal to $V$. Since $W\subset V^{\perp}$,
we have $V\subset W^{\perp}$. Let us prove the reverse inclusion. Let
${\boldsymbol{\lambda}}\in W^{\perp}$. Then ${\boldsymbol{\lambda}}$ is
orthogonal to all ${\boldsymbol{k}}\in\mathcal{K}$. By construction,
$\mathcal{K}$ remains at bounded distance from $\mathbb{R}\Theta$.
Renormalizing and taking the limit for large vectors
${\boldsymbol{k}}\in\mathcal{K}$, we get that ${\boldsymbol{\lambda}}$ is
orthogonal to $\Theta$. Hence, ${\boldsymbol{\lambda}}\in V$. This proves Fact
3.

Given ${\boldsymbol{k}}\in V_{+}^{\perp}$, we define an action from the monoid
$V_{+}^{\perp}$ to $\mathcal{A}[\boldsymbol{Y}]$ by:

$\sigma_{\boldsymbol{k}}:\left\\{\begin{array}[]{rcl}\mathcal{A}[\boldsymbol{Y}]&\rightarrow&\mathcal{A}[\boldsymbol{Y}]\\\
P(\boldsymbol{Y},{\boldsymbol{z}})&\mapsto&P(\boldsymbol{Y}{\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{z}}),T_{\boldsymbol{k}}{\boldsymbol{z}})\,.\end{array}\right.$

Note that the map $\sigma_{\boldsymbol{k}}$ is well-defined. Indeed, we
already observed that the coordinates of
${\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{z}})$ belong to the ring
$\mathcal{A}$ for all ${\boldsymbol{k}}\in\mathbb{N}^{r}$.

######  Fact 4.

For all ${\boldsymbol{k}}\in\mathcal{K}$,
$\sigma_{\boldsymbol{k}}(\mathcal{I}_{{\mathcal{Z}}_{0}})\subset\mathcal{I}_{{\mathcal{Z}}_{0}}$.

Let $P\in\mathcal{I}_{{\mathcal{Z}}_{0}}$, ${\boldsymbol{k}}\in\mathcal{K}$,
and $l\in{\mathcal{Z}}_{0}({\boldsymbol{k}})$. Let
$l^{\prime}\in{\mathcal{Z}}_{0}$ be such that
${\boldsymbol{k}}_{l^{\prime}}={\boldsymbol{k}}+{\boldsymbol{k}}_{l}$. Then,
we have

$\displaystyle\sigma_{\boldsymbol{k}}(P)({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}))$
$\displaystyle=$ $\displaystyle
P({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}){\boldsymbol{R}}_{{\boldsymbol{k}}}(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}),T_{\boldsymbol{k}}T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$
$\displaystyle=$ $\displaystyle
P({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}+{\boldsymbol{k}}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}+{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$
$\displaystyle=$ $\displaystyle
P({\boldsymbol{R}}_{{\boldsymbol{k}}_{l^{\prime}}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l^{\prime}}}{\boldsymbol{\alpha}})$
$\displaystyle=$ $\displaystyle 0\,.$

Thus, $\sigma_{\boldsymbol{k}}(P)$ belongs to the ideal
$\mathcal{I}_{{\mathcal{Z}}_{0}({\boldsymbol{k}})}$, which is equal to
$\mathcal{I}_{{\mathcal{Z}}_{0}}$ by minimality. This proves Fact 4.

######  Fact 5.

Let $P\in\mathcal{I}_{{\mathcal{Z}}_{0}}$ be such that
$P=\sigma_{\boldsymbol{k}}(Q)$ for some $Q\in\mathcal{A}[\boldsymbol{Y}]$ and
${\boldsymbol{k}}\in\mathcal{K}$. Then $Q\in\mathcal{I}_{{\mathcal{Z}}_{0}}$.

Since ${\mathcal{Z}}_{0}({\boldsymbol{k}})$ is piecewise syndetic, we infer
from (6.6) that the set

${\mathcal{Z}}_{0}({\boldsymbol{k}})^{-1}:=\\{l^{\prime}\in{\mathcal{Z}}_{0}\
:\ \exists
l\in{\mathcal{Z}}_{0},\,{\boldsymbol{k}}_{l^{\prime}}={\boldsymbol{k}}_{l}+{\boldsymbol{k}}\\}$

is also piecewise syndetic. By minimality, we obtain
$\mathcal{I}_{{\mathcal{Z}}_{0}({\boldsymbol{k}})^{-1}}=\mathcal{I}_{{\mathcal{Z}}_{0}}$.
Let $l^{\prime}\in{\mathcal{Z}}_{0}({\boldsymbol{k}})^{-1}$ and let
$l\in{\mathcal{Z}}_{0}$ be such that
${\boldsymbol{k}}_{l^{\prime}}:={\boldsymbol{k}}_{l}+{\boldsymbol{k}}$. Then

$\displaystyle
Q({\boldsymbol{R}}_{{\boldsymbol{k}}_{l^{\prime}}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l^{\prime}}}{\boldsymbol{\alpha}})$
$\displaystyle=$ $\displaystyle
Q({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}+{\boldsymbol{k}}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}+{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$
$\displaystyle=$ $\displaystyle
Q({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}){\boldsymbol{R}}_{{\boldsymbol{k}}}(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}),T_{\boldsymbol{k}}T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$
$\displaystyle=$
$\displaystyle\sigma_{\boldsymbol{k}}(Q)({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}))$
$\displaystyle=$ $\displaystyle
P({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}))$
$\displaystyle=$ $\displaystyle 0\,.$

Thus,
$Q\in\mathcal{I}_{{\mathcal{Z}}_{0}({\boldsymbol{k}})^{-1}}=\mathcal{I}_{{\mathcal{Z}}_{0}}$.
This proves Fact 5.

######  Fact 6.

For all ${\boldsymbol{k}}\in V_{+}^{\perp}$,
$\sigma_{\boldsymbol{k}}(\mathcal{I}_{{\mathcal{Z}}_{0}})\subset\mathcal{I}_{{\mathcal{Z}}_{0}}$.

Let ${\boldsymbol{k}}\in V_{+}^{\perp}$ and
$P\in\mathcal{I}_{{\mathcal{Z}}_{0}}$. By Fact 3, there is a decomposition of
the form

${\boldsymbol{k}}:={\boldsymbol{a}}_{1}+\cdots+{\boldsymbol{a}}_{u}-{\boldsymbol{a}}_{u+1}-\cdots-{\boldsymbol{a}}_{v}$

with ${\boldsymbol{a}}_{1},\ldots,{\boldsymbol{a}}_{v}\in\mathcal{K}$. Using
recursively Fact 4 with
$\sigma_{{\boldsymbol{a}}_{1}},\ldots,\sigma_{{\boldsymbol{a}}_{u}}$, we
deduce that
$\sigma_{{\boldsymbol{a}}_{1}+\cdots+{\boldsymbol{a}}_{u}}(P)\in\mathcal{I}_{{\mathcal{Z}}_{0}}$.
On the other hand, we have
$\sigma_{{\boldsymbol{a}}_{u+1}+\cdots+{\boldsymbol{a}}_{v}}(\sigma_{{\boldsymbol{k}}}(P))=\sigma_{{\boldsymbol{a}}_{1}+\cdots+{\boldsymbol{a}}_{u}}(P)\in\mathcal{I}_{{\mathcal{Z}}_{0}}\,.$
Using recursively Fact 5 with
$\sigma_{{\boldsymbol{a}}_{u+1}},\ldots,\sigma_{{\boldsymbol{a}}_{v}}$, we
obtain that $\sigma_{{\boldsymbol{k}}}(P)\in\mathcal{I}_{{\mathcal{Z}}_{0}}$.
This proves Fact 6.

######  Fact 7.

One has $\mathcal{I}_{{\mathcal{Z}}_{0}}\subset\mathcal{I}$.

Let $l_{0}\in\mathbb{N}$ denote the smallest element in ${\mathcal{Z}}_{0}$.
Let $P\in\mathcal{I}_{{\mathcal{Z}}_{0}}$ and let $l\geq l_{0}$ be an integer.
By Lemma 6.3, ${\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}\in
V_{+}^{\perp}$. Set
$Q:=\sigma_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}(P)$. By Fact 6, we
obtain that $Q\in\mathcal{I}_{{\mathcal{Z}}_{0}}$. Since
$l_{0}\in{\mathcal{Z}}_{0}$, we have

$P({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=Q({\boldsymbol{R}}_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})=0\,.$

Hence $P$ vanishes at
$({\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})$,
for all $l\geq l_{0}$. This proves Fact 7.

We are now ready to conclude the proof of Lemma 7.6. Let $P\in\mathcal{I}$,
${\boldsymbol{\phi}}\in{\rm GL}_{t}(\mathbb{K})$ be a relation matrix, and
${\boldsymbol{k}}\in V_{+}^{\perp}$. Let
$b({\boldsymbol{z}})\in\overline{\mathbb{Q}}[{\boldsymbol{z}}]$ be a nonzero
polynomial such that
$b({\boldsymbol{z}})P(\boldsymbol{Y},{\boldsymbol{z}})\in\overline{\mathbb{Q}}[{\boldsymbol{z}},\boldsymbol{Y}]$.
In particular,
$b({\boldsymbol{z}})P(\boldsymbol{Y},{\boldsymbol{z}})\in\mathcal{I}\cap\mathcal{A}[\boldsymbol{Y}]$.
Let ${\mathcal{Z}}$ be the set of integers $l\geq 0$ for which
$b(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})P({\boldsymbol{\phi}}{\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})=0$.
Since $bP\in\mathcal{I}$, ${\mathcal{Z}}$ is full and hence piecewise
syndetic. There thus exists a piecewise syndetic set
${\mathcal{Z}}_{0}\subset{\mathcal{Z}}$ satisfying Facts 2 to 7. By
definition,
$b({\boldsymbol{z}})P(\boldsymbol{Y},{\boldsymbol{z}})\in\mathcal{I}_{{\mathcal{Z}}_{0}}$.
By Fact 6, we also have
$\sigma_{\boldsymbol{k}}(bP)\in\mathcal{I}_{{\mathcal{Z}}_{0}}$, for all
${\boldsymbol{k}}\in V_{+}^{\perp}$. By Fact 7, we deduce that
$\sigma_{\boldsymbol{k}}(bP)\in\mathcal{I}$. Then, we infer from Lemma 7.4
that

$b(T_{\boldsymbol{k}}{\boldsymbol{z}})P({\boldsymbol{\phi}}{\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{z}}),T_{\boldsymbol{k}}{\boldsymbol{z}})=\sigma_{\boldsymbol{k}}(bP)({\boldsymbol{\phi}},{\boldsymbol{z}})=0\,.$

Since $T_{\boldsymbol{k}}$ is nonsingular and $b({\boldsymbol{z}})\not=0$, we
obtain
$P({\boldsymbol{\phi}}{\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{z}}),T_{\boldsymbol{k}}{\boldsymbol{z}})=0$.
∎

###  7.3. Analyticity of relation matrices

Let ${\boldsymbol{\phi}}$ be a relation matrix. All coordinates of
${\boldsymbol{\phi}}$ being algebraic over
$\overline{\mathbb{Q}}({\boldsymbol{z}})$, they generate a finite extension of
$\overline{\mathbb{Q}}({\boldsymbol{z}})$. Let $\mathbf{k}\subset\mathbb{K}$
denote this extension and let $\gamma\geq 1$ be the degree of $\mathbf{k}$.
Choosing a primitive element $\phi$ in $\mathbf{k}$, we obtain a decomposition
of the form

(7.10)
${\boldsymbol{\phi}}=\sum_{j=0}^{\gamma-1}{\boldsymbol{\phi}}_{j}({\boldsymbol{z}})\phi^{j}\,,$

where the matrices ${\boldsymbol{\phi}}_{j}({\boldsymbol{z}})$, $0\leq
j\leq\gamma-1$, have coefficients in
$\overline{\mathbb{Q}}({\boldsymbol{z}})$. The field $\mathbb{K}$ is a priori
an abstract algebraic closure of $\overline{\mathbb{Q}}({\boldsymbol{z}})$,
but we can easily reduce the situation to the case where the coordinates of
${\boldsymbol{\phi}}$ are analytic at some suitable point
$T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}}$.

######  Lemma 7.7.

We continue with the previous notation. There exist an integer $l_{0}\geq 0$,
a neighborhood $\mathcal{V}$ of
$T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}}$, and a function
$\varphi({\boldsymbol{z}})$ that is analytic on $\mathcal{V}$ and algebraic
over $\overline{\mathbb{Q}}({\boldsymbol{z}})$ such that the following
properties holds.

  * (a)

$\|T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}}\|<1$.

  * (b)

$T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}}$ belongs to the disc of
convergence of
$f_{1,1}({\boldsymbol{z}}),\ldots,f_{r,m_{r}}({\boldsymbol{z}})$.

  * (c)

The matrix

${\boldsymbol{\phi}}({\boldsymbol{z}}):=\sum_{j=0}^{\gamma-1}{\boldsymbol{\phi}}_{j}({\boldsymbol{z}})\varphi({\boldsymbol{z}})^{j}\in{\rm
GL}_{t}(\mathcal{M}er(\mathcal{V}))\,$

is a relation matrix. That is, it satisfies Lemmas 7.4 and 7.6.

  * (d)

For every $j$, $1\leq j\leq\gamma-1$, the coordinates of the matrix
${\boldsymbol{\phi}}_{j}({\boldsymbol{z}})$ are analytic on $\mathcal{V}$ and
the matrix
${\boldsymbol{\phi}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})$ is
invertible.

###### Proof.

By Theorem 4.6,
$\lim_{l\to\infty}T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}=0$. This
ensures that (a) and (b) are satisfied for all sufficiently large integers
$l$.

Let $P({\boldsymbol{z}},y)\in\overline{\mathbb{Q}}[{\boldsymbol{z}},y]$ denote
the minimal polynomial of $\phi$ and let
$D({\boldsymbol{z}})\in\overline{\mathbb{Q}}[{\boldsymbol{z}}]$ denote the
discriminant of $P$, seen as a polynomial in the variable $y$. Since
${\boldsymbol{\phi}}$ is a relation matrix, $\det{\boldsymbol{\phi}}$ is
nonzero and algebraic over $\overline{\mathbb{Q}}({\boldsymbol{z}})$. There
thus exist polynomials
$q_{0}({\boldsymbol{z}}),\ldots,q_{\nu}({\boldsymbol{z}})\in\overline{\mathbb{Q}}[{\boldsymbol{z}}]$,
$q_{0}({\boldsymbol{z}})\not=0$, such that

(7.11)
$q_{0}({\boldsymbol{z}})=q_{1}({\boldsymbol{z}})\det{\boldsymbol{\phi}}+q_{2}({\boldsymbol{z}})\det{\boldsymbol{\phi}}^{2}+\cdots+q_{\nu}({\boldsymbol{z}})\det{\boldsymbol{\phi}}^{\nu}\,.$

Let $d({\boldsymbol{z}})$ be the least common multiple of the denominators of
the coefficients of the matrices
${\boldsymbol{\phi}}_{0}({\boldsymbol{z}}),\ldots,{\boldsymbol{\phi}}_{\gamma-1}({\boldsymbol{z}})$.
Since the polynomial
$D({\boldsymbol{z}})q_{0}({\boldsymbol{z}})d({\boldsymbol{z}})$ is nonzero,
Theorem 5.4 ensures the existence of a full set $\mathcal{E}\subset\mathbb{N}$
such that

$D(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})q_{0}(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})d(T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}})\not=0\,,\;\forall
l\in\mathcal{E}\,.$

Let $l_{0}\in\mathcal{E}$ be chosen large enough to guarantee that (a) and (b)
hold. Since $D(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})\not=0$, the
implicit function theorem (see, for instance, [19, Proposition 6.1, p. 138])
implies that there exists a function $\varphi({\boldsymbol{z}})$ that is
analytic on a neighborhood of
$T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}}$, say $\mathcal{V}_{0}$,
and such that $P({\boldsymbol{z}},\varphi({\boldsymbol{z}}))=0$. Note that
there is a $\overline{\mathbb{Q}}({\boldsymbol{z}})$-isomorphism between the
field $\overline{\mathbb{Q}}({\boldsymbol{z}},\phi)$ and
$\overline{\mathbb{Q}}({\boldsymbol{z}},\varphi({\boldsymbol{z}}))$. We thus
deduce that the matrix
${\boldsymbol{\phi}}({\boldsymbol{z}}):=\sum_{j=0}^{\gamma-1}{\boldsymbol{\phi}}_{j}({\boldsymbol{z}})\varphi({\boldsymbol{z}})^{j}$
satisfies the properties of Lemmas 7.4 and 7.6. Furthermore, as
$\det{\boldsymbol{\phi}}\not=0$, we also deduce that
$\det{\boldsymbol{\phi}}({\boldsymbol{z}})\not=0$. Hence
${\boldsymbol{\phi}}({\boldsymbol{z}})\in{\rm
GL}_{t}(\mathcal{M}er(\mathcal{V}_{0}))$. Finally, we deduce that
$\det{\boldsymbol{\phi}}({\boldsymbol{z}})$ also satisfies Equation (7.11).
Since $q_{0}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})\not=0$, we get
that
$\det{\boldsymbol{\phi}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})\not=0$.
Hence the matrix
${\boldsymbol{\phi}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})$ is
invertible. Furthermore, since
$d(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})\not=0$, the coordinates
of ${\boldsymbol{\phi}}_{j}({\boldsymbol{z}})$ are analytic on some
neighborhood of $T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}}$, say
$\mathcal{V}_{1}$. Finally, setting
$\mathcal{V}=\mathcal{V}_{0}\cap\mathcal{V}_{1}$, we obtain that Properties
(a)–(d) are satisfied. ∎

##  8\. Proof of Theorem 6.2

Let $P\in\overline{\mathbb{Q}}[\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{r}]$
be defined as in Theorem 6.2. Let $d_{i}$ denote the total degree of $P$ in
the indeterminates $\boldsymbol{X}_{i}$. Set
$\boldsymbol{X}:=(\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{r})$. We keep on
with the notation of the previous section. The monomials
$\boldsymbol{X}^{{\boldsymbol{\mu}}_{1}},\ldots,\boldsymbol{X}^{{\boldsymbol{\mu}}_{t}}$
are precisely those which are of total degree $d_{i}$ in the indeterminates
$\boldsymbol{X}_{i}$, for every $i$. Hence we have a decomposition of the form

$P(\boldsymbol{X}):=\sum_{j=1}^{t}\tau_{j}\boldsymbol{X}^{{\boldsymbol{\mu}}_{j}}\,,$

where $\tau_{1},\ldots,\tau_{t}\in\overline{\mathbb{Q}}$. Set
${\boldsymbol{\tau}}:=(\tau_{1},\ldots,\tau_{t})\in\overline{\mathbb{Q}}^{t}$
and let
$\boldsymbol{X}^{\boldsymbol{\mu}}\in\overline{\mathbb{Q}}[\boldsymbol{X}]^{t}$
denote the column vector of indeterminates whose coordinates are the monomials
$\boldsymbol{X}^{{\boldsymbol{\mu}}_{1}},\ldots,\boldsymbol{X}^{{\boldsymbol{\mu}}_{t}}$.
We write
$P(\boldsymbol{X})={\boldsymbol{\tau}}\boldsymbol{X}^{\boldsymbol{\mu}}$.
Given a matrix of indeterminates $\boldsymbol{Y}:=(y_{i,j})_{1\leq i,j\leq
t}$, we set

(8.1)
$F(\boldsymbol{Y},{\boldsymbol{z}}):=\sum_{i,j}\tau_{i}y_{i,j}{\boldsymbol{f}}({\boldsymbol{z}})^{{\boldsymbol{\mu}}_{j}}={\boldsymbol{\tau}}\boldsymbol{Y}{\boldsymbol{f}}({\boldsymbol{z}})^{\boldsymbol{\mu}}\in\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}[\boldsymbol{Y}]\,,$

where we let
${\boldsymbol{f}}(z)\in\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}^{M}$ denote
the column vector formed by the functions
$f_{1,1}({\boldsymbol{z}}_{1}),\ldots,f_{r,m_{r}}({\boldsymbol{z}}_{r})$ and
${\boldsymbol{f}}({\boldsymbol{z}})^{{\boldsymbol{\mu}}}\in\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}^{t}$
denote the column vector formed by the functions
${\boldsymbol{f}}({\boldsymbol{z}})^{{\boldsymbol{\mu}}_{1}},\ldots,{\boldsymbol{f}}({\boldsymbol{z}})^{{\boldsymbol{\mu}}_{t}}$.
Note that $F$ is a linear form in $\boldsymbol{Y}$. Evaluating at $({\rm
I}_{t},{\boldsymbol{\alpha}})$, we obtain

(8.2) $F({\rm
I}_{t},{\boldsymbol{\alpha}})=\sum_{j=1}^{t}\tau_{j}{\boldsymbol{f}}({\boldsymbol{\alpha}})^{{\boldsymbol{\mu}}_{j}}=P({\boldsymbol{f}}({\boldsymbol{\alpha}}))=0\,.$

######  Remark 8.1.

We have
$F(\boldsymbol{Y},{\boldsymbol{z}})\in\overline{\mathbb{Q}}[\boldsymbol{Y},{\boldsymbol{f}}({\boldsymbol{z}})]\subset\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}[\boldsymbol{Y}]$.
Also, $F(\boldsymbol{Y},{\boldsymbol{z}})$ can be seen as an element of
$\overline{\mathbb{Q}}[\boldsymbol{Y}][[{\boldsymbol{z}}]]$, as we will
sometimes do in what follows.

###  8.1. Iterated relations

Using Equality (6.5) and (7.1), we obtain

(8.3)
${\boldsymbol{f}}({\boldsymbol{z}})^{{\boldsymbol{\mu}}_{j}}=\left(A_{{\boldsymbol{k}}}({\boldsymbol{z}}){\boldsymbol{f}}(T_{\boldsymbol{k}}{\boldsymbol{z}})\right)^{{\boldsymbol{\mu}}_{j}}=\sum_{l=1}^{t}R_{j,l}(A_{\boldsymbol{k}}({\boldsymbol{z}})){\boldsymbol{f}}(T_{\boldsymbol{k}}{\boldsymbol{z}})^{{\boldsymbol{\mu}}_{l}}\,,$

for every $j$, $1\leq j\leq t$. We deduce from (8.3) that

(8.4)
${\boldsymbol{f}}({\boldsymbol{z}})^{{\boldsymbol{\mu}}}={\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{z}}){\boldsymbol{f}}(T_{\boldsymbol{k}}{\boldsymbol{z}})^{{\boldsymbol{\mu}}}\,,$

for all ${\boldsymbol{k}}\in\mathbb{N}^{r}$, where
${\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{z}})$ is defined as in (7.4).

For every $i$, $1\leq i\leq r$, let
$b_{i}({\boldsymbol{z}}_{i})\in\overline{\mathbb{Q}}[{\boldsymbol{z}}_{i}]$
denote the least common multiple of the denominators of the coordinates of
$A_{i}({\boldsymbol{z}}_{i})$. Hence the matrix
$b_{i}({\boldsymbol{z}}_{i})A({\boldsymbol{z}}_{i})$ has coefficients in
$\overline{\mathbb{Q}}[{\boldsymbol{z}}_{i}]$. For every
${\boldsymbol{k}}=(k_{1},\ldots,k_{r})\in\mathbb{N}^{r}$, we set

$b_{\boldsymbol{k}}({\boldsymbol{z}}):=\prod_{i=1}^{r}\prod_{j=0}^{k_{i}-1}b_{i}(T_{i}^{j}{\boldsymbol{z}}_{i})^{d_{i}}\,,$

so that the matrix
$b_{\boldsymbol{k}}({\boldsymbol{z}}){\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{z}})$
has coefficients in $\overline{\mathbb{Q}}[{\boldsymbol{z}}]$. For all
${\boldsymbol{k}}\in\mathbb{N}^{r}$, Equality (8.4) implies the following
equality in $\overline{\mathbb{Q}}\\{{\boldsymbol{z}}\\}[\boldsymbol{Y}]$:

$\displaystyle
F(\boldsymbol{Y}b_{\boldsymbol{k}}({\boldsymbol{z}}),{\boldsymbol{z}})$
$\displaystyle=$
$\displaystyle{\boldsymbol{\tau}}\boldsymbol{Y}b_{\boldsymbol{k}}({\boldsymbol{z}}){\boldsymbol{f}}({\boldsymbol{z}})^{\boldsymbol{\mu}}$
$\displaystyle=$
$\displaystyle{\boldsymbol{\tau}}\boldsymbol{Y}b_{\boldsymbol{k}}({\boldsymbol{z}}){\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{z}}){\boldsymbol{f}}(T_{\boldsymbol{k}}{\boldsymbol{z}})^{\boldsymbol{\mu}}$
$\displaystyle=$ $\displaystyle
F(\boldsymbol{Y}b_{\boldsymbol{k}}({\boldsymbol{z}}){\boldsymbol{R}}_{\boldsymbol{k}}({\boldsymbol{z}}),T_{\boldsymbol{k}}{\boldsymbol{z}})\,.$

Every point ${\boldsymbol{\alpha}}_{i}$ being regular with respect to the
system (6.1.$i$), the number $b_{\boldsymbol{k}}({\boldsymbol{\alpha}})$ is
nonzero for all ${\boldsymbol{k}}\in\mathbb{N}^{r}$. From (8.2) and the fact
that $F$ is linear in $\boldsymbol{Y}$, we deduce that

(8.6)
$F({\boldsymbol{R}}_{{\boldsymbol{k}}}({\boldsymbol{\alpha}}),T_{{\boldsymbol{k}}}{\boldsymbol{\alpha}})=0\,,\;\forall{\boldsymbol{k}}\in\mathbb{N}^{r}\,.$

###  8.2. The matrix ${\boldsymbol{\Theta}}_{l}({\boldsymbol{z}})$

From now on, we fix a nonnegative integer $l_{0}$ and a relation matrix
${\boldsymbol{\phi}}({\boldsymbol{z}})$ satisfying the properties of Lemma
7.7. Set

${\boldsymbol{\xi}}:=T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}}\,.$

Properties (a) and (b) in Lemma 7.7 ensure the existence of a real number
$r_{1}$ such that $0<\|{\boldsymbol{\xi}}\|<r_{1}<1$ and such that all the
power series $f_{1,1}({\boldsymbol{z}}),\ldots,f_{r,m_{r}}({\boldsymbol{z}})$
have a radius of convergence larger than $r_{1}$. Then, by Properties (c) and
(d) in Lemma 7.7, we can choose a real number $r_{2}$ satisfying
$0<\|{\boldsymbol{\xi}}\|+r_{2}<r_{1}$ and such that the coefficients of the
matrix ${\boldsymbol{\phi}}({\boldsymbol{z}})$ are analytic on the polydisc
$\mathcal{D}({\boldsymbol{\xi}},r_{2})$. Note that Property (a) in Lemma 7.7
and Lemma 6.3 imply that
$\|T_{{\boldsymbol{k}}_{l}}{\boldsymbol{\alpha}}\|\leq\|T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}}\|$
for $l\geq l_{0}$. For every $l\geq l_{0}$, we set

(8.7)
${\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}):={\boldsymbol{R}}_{{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{\alpha}}){\boldsymbol{\phi}}(T_{{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{\alpha}})^{-1}{\boldsymbol{\phi}}({\boldsymbol{z}}){\boldsymbol{R}}_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}})\,.$

By (7.5), we have
${\boldsymbol{\Theta}}_{l}({\boldsymbol{\xi}})={\boldsymbol{R}}_{{\boldsymbol{k}}_{l}}({\boldsymbol{\alpha}})$,
for all $l\geq l_{0}$.

######  Remark 8.2.

By Lemma 7.7, the coefficients of
${\boldsymbol{\Theta}}_{l_{0}}({\boldsymbol{z}})$ are analytic on the polydisc
$\mathcal{D}({\boldsymbol{\xi}},r_{2})$. On the other hand, one has

${\boldsymbol{\Theta}}_{l}({\boldsymbol{z}})={\boldsymbol{\Theta}}_{l-1}({\boldsymbol{z}}){\boldsymbol{R}}_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l-1}}(T_{{\boldsymbol{k}}_{l-1}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})\,,\;\forall
l\geq l_{0}\,.$

This implies that, for every $l\geq l_{0}$, the coefficients of
${\boldsymbol{\Theta}}_{l}({\boldsymbol{z}})$ are analytic on some
neighborhood of ${\boldsymbol{\xi}}$, that is on some polydisc
$\mathcal{D}({\boldsymbol{\xi}},r_{l})\subset\mathcal{D}({\boldsymbol{\xi}},r_{2})$.
Also, the coefficients of
$b_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}({\boldsymbol{z}}){\boldsymbol{\Theta}}_{l}({\boldsymbol{z}})$
are analytic on the polydisc $\mathcal{D}({\boldsymbol{\xi}},r_{2})$. In what
follows, we will consider the expression
$F({\boldsymbol{\Theta}}_{l}({\boldsymbol{z}}),T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$.
Formally, it is a polynomial in
$f_{1,1}(T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}}),\ldots,f_{r,m_{r}}(T_{{\boldsymbol{k}}_{l}-{\boldsymbol{k}}_{l_{0}}}{\boldsymbol{z}})$
and the coordinates of ${\boldsymbol{\Theta}}_{l}({\boldsymbol{z}})$. Note
that it also defines an analytic functions on