A NEW PROOF OF NISHIOKA’S THEOREM IN MAHLER’S METHOD

. In a recent work [3], the authors established new results about general linear Mahler systems in several variables from the per-spective of transcendental number theory, such as a multivariate extension of Nishioka’s theorem. Working with functions of several variables and with diﬀerent Mahler transformations leads to a number of compli-cations, including the need to prove a general vanishing theorem and to use tools from ergodic Ramsey theory and Diophantine approximation ( e.g. , a variant of the p -adic Schmidt subspace theorem). These compli-cations make the proof of the main results proved in [3] rather intricate. In this article, we describe our new approach in the special case of linear Mahler systems in one variable. This leads to a new, elementary, and self-contained proof of Nishioka’s theorem, as well as of the lifting theorem more recently obtained by Philippon [22] and the authors [1]. Though the general strategy remains the same as in [3], the proof turns out to be greatly simpliﬁed. Beyond its own interest, we hope that reading this article will facilitate the understanding of the proof of the main results obtained in [3].


Introduction
Throughout this paper, we let q ≥ 2 denote a fixed integer.An M qfunction is a power series f (z) ∈ Q [[z]] satisfying a linear equation of the form p 0 (z)f (z) + p 1 (z)f (z q ) + • • • + p m (z)f (z q m ) = 0 , where p 0 (z), . . ., p m (z) ∈ Q[z] are not all zero.In the study of M q -functions, it is often more convenient to consider, instead of linear Mahler equations, linear systems of functional equations of the form (1.1) . . .
where A(z) ∈ GL m (Q(z)) and f 1 (z), . . ., f m (z) ∈ Q [[z]].Then, each power series f i (z) is an M q -function.We recall that an M q -function is meromorphic in the open unit disc of C (see, for instance, [9,Théorème 31]).Furthermore, it admits the unit circle as a natural boundary, unless it is a rational function [24,Théorème 4.3].A point α ∈ C is said to be regular with respect to (1.1) if the matrix A(α q k ) is both well-defined and invertible for all integers k ≥ 0.
In this framework, the main aim of Mahler's method is to transfer results about the absence of algebraic (resp.linear) relations between the functions f 1 (z), . . ., f m (z) over Q(z) to the absence of algebraic (resp.linear) relations over Q between their values at non-zero algebraic points lying in the open unit disc (assuming, of course, that these values are well-defined).In 1990, Ku. Nishioka [19] proved the following theorem, which is the analog of the Siegel-Shidlovskii theorem in the theory of Siegel E-functions (see [25]).Given a field K, a field extension L of K, and elements a 1 , . . ., a m in L, we let tr.degK (a 1 , . . ., a m ) denote the transcendence degree over K of the field extension K(a 1 , . . ., a m ).
Theorem 1.1 (Nishioka's theorem).Let f 1 (z), . . ., f m (z) be M q -functions related by a q-Mahler system of the form (1.1) and let α ∈ Q, 0 < |α| < 1, be regular with respect to this system.Then Nishioka's theorem is undoubtedly a landmark result in Mahler's method, but it also suffers from some limitation which prevent it to cover important applications (see the discussion in Sections 1 and 2 of [1] and also the results in [2]).For such applications, the following refinement of Nishioka's theorem, which we called lifting theorem (or théorème de permanence in French), is needed.
Again, Theorem 1.2 has an analog in the theory of E-functions: the lifting theorem proved by Beukers [8] using André's theory of arithmetic Gevrey series [5,6].A slightly weaker version of Theorem 1.2 was first proved by Philippon [22].Theorem 1.2 was then deduced in [1] from Philippon's lifting theorem.In [1,22], the lifting theorem is derived from Nishioka's theorem.Thanks to the work of André [7], pursued by Naguy and Szamuely [17], we now have a general approach based on a suitable Galois theory of linear differential and difference equations that allows one to deduce theorems of the type of Theorem 1.2 from theorems of the type of Theorem 1.1.
The proof of Nishioka's theorem deeply relies on tools from commutative algebra, related to elimination theory, which were introduced and developed by Nesterenko in the framework of transcendental number theory at the end of the 1970s (see, for instance, [18]).Recently, Fernandes [11] observed that Nishioka's theorem can also be derived from a general algebraic independence criterion due to Philippon [23].However, Philippon's criterion is also based on the same tools, so that, in the end, both proofs rely on the same argument.The proof of Nishioka's theorem has the advantage that it can be quantified (see, for instance, [19]), leading to algebraic independence measures.Its main deficiency is that it can hardly be generalised to Mahler systems in several variables.
In this note, we use the approach recently introduced by the authors [3] to provide new and more elementary proofs of both Nishioka's theorem and the lifting theorem.This approach takes its roots in the original one initiated by Mahler [16] and developed much later by Kubota [13], Loxton and van der Poorten [15], and Nishioka [20,21].The main improvement comes from the introduction of the so-called relation matrices whose existence is ensured by Hilbert Nullstellensatz.In contrast with [1,22], we first prove the lifting theorem and then deduce Nishioka's theorem by using a classical argument, as in Shidlovskii's proof of the Siegel-Shidlovskii theorem (see [25] or [10]).Beyond its elementary aspect, this new approach has the great advantage of being generalisable within the framework of Mahler's method in several variables, as has been done in [3].We hope that reading first this article will facilitate the understanding of the proof of the main results in [3].In order to avoid the proofs of Theorems 1.1 and 1.2 being buried in tedious computations, we occasionally just outline the main argument and provide more detail in the appendix at the end of the paper.There, we also prove some auxiliary results that can be used to make our proofs of Theorems 1.1 and 1.2 as elementary and self-contained as possible.

Lifting the linear relations
We first prove Theorem 1.2 in the particular case of linear relations.
The proof of this theorem is dividing in three subsections.We first establish the existence and properties of some special matrices which can be associated with a linear Mahler system.We call them the relation matrices.Then, we construct an auxiliary function and use it to prove a key lemma about the structure of the linear relations between f 1 (z), . . ., f m (z).Finally, we show how this lemma allows us to lift any linear relation over Throughout this section, we keep the notation of Theorem 2.1.

2.1.
Notation.Let d be a positive integer and R be a commutative ring.Given an indeterminate x, we let R[[x]] denote the ring of formal power series with coefficients in R. If R ⊂ C, we let R{x} denote the ring of convergent power series with coefficients in R, that is those elements of R [[x]] that are analytic in some neighborhood of the origin.Given a d-tuple of non-negative Given an m × n matrix M := (m i,j ) with coefficients in R and an m × n matrix µ = (µ i,j ) with nonnegative integer coefficients, we set We use the standard Landau notation O.We also use the notation ≫ as follows.Writing that some property holds true for all integers λ ≫ 1 means that the corresponding property holds true for all λ large enough; writing that some property holds true for all integers λ 1 ≫ λ 2 , λ 3 means that the corresponding property holds true for all λ 1 that is sufficiently large w.r.t.λ 2 and λ 3 ; writing that some property holds true for all integers λ 1 ≫ λ 2 ≫ λ 3 means that the corresponding property holds true for all λ 1 that is sufficiently large w.r.t.λ 2 , assuming that λ 2 is itself sufficiently large w.r.t.λ 3 .

Relation matrices.
To shorten the notation, we set For every integer k ≥ 0, we set Let Y := (y i,j ) 1≤i,j≤m denote a matrix of indeterminates.Given a field K and a non-negative integer δ 1 , we let K[Y ] δ 1 denote the set of polynomials of degree at most δ 1 in each indeterminate y i,j .Given two non-negative integers δ 1 and δ 2 , we let K[Y , z] δ 1 ,δ 2 denote the set of polynomials P ∈ K[Y , z] of degree at most δ 1 in every indeterminate y i,j and of degree at most δ 2 in z.
The identity theorem and the fact that α is a regular point with respect to (1.1) ensure that every polynomial P ∈ Q(z)[Y ] is well-defined at the point (A k (α), α q k ) for all k ≫ 1. Set

2.2.1.
Estimates for the dimension of certain vector spaces.Let δ 1 and δ 2 be two non-negative integers.Set I(δ 1 ) : There exists a positive integer c 1 (δ 1 ), that does not depend on δ 2 , such that (2.2) Since these polynomials only depend on δ 1 (and I), there exists δ ′ 1 ≥ 0, which only depends on δ 1 (and I), such that b i,j (z) =: where the numbers p j,λ belong to Q and p j,λ := 0 if λ > δ 2 or λ < 0. By (2.2), P belongs to I(δ 1 , δ 2 ) if and only if The number of linearly independent equations in (2.3) is equal to the dimension of I ⊥ (δ 1 , δ 2 ).As δ 2 tends to infinity, it is equivalent to the number of linearly independent equations in (2.3) where we let e 1 (Y ), . . ., e 2 m 2 (Y ) denote the 2 m 2 distinct monomials of degree at most 1 in each y i,j , and where the polynomials Hence, the decomposition (2.4) defines a linear map that is surjective.The result follows.
2.2.2.Nullstellensatz and relation matrices.In this section, we show how Hilbert's Nullstellensatz allows us to ensure the existence of a matrix φ, whose coordinates are all algebraic over Q(z), and which we call a relation matrix.Such a matrix encodes the linear relations over Q(z) between the functions f 1 (z), . . ., f m (z) and is the cornerstone of the proof of Theorem 2.1.
We first prove the following lemma.
Hence P ∈ I and I is a radical ideal.
Throughout this article, we let A ⊂ d≥1 Q((z 1/d )) denote the algebraic closure of Q(z) in the field of Puiseux series.By the Newton-Puiseux Theorem, A is algebraically closed.Lemma 2.5.There exists a matrix φ(z) ∈ GL m (A) such that P (φ(z), z) = 0 , for all polynomials P ∈ I.
Proof.Let us consider the affine algebraic set V associated with the radical ideal I.That is, According to the weak form of Hilbert's Nullstellensatz (see, for instance, [14, Theorem 1.4, p. 379]), V is non-empty as soon as I is a proper ideal of Q(z) [Y ].But the definition of I implies that non-zero constant polynomials do not belong to I. Hence V is non-empty.Now, let us assume by contradiction that det φ(z) = 0 for all φ(z) in V.By Hilbert's Nullstellensatz (see, for instance, [14, Theorem 1.5, p. 380]), the polynomial det Y belongs to the radical of the ideal I. Hence det Y ∈ I for I is radical.Thus, det A k (α) = 0 for k ≫ 1.This provides a contradiction since A k (α) is invertible for all k ≥ 0. We thus deduce that there exists an invertible matrix φ(z) in V, as wanted.Definition 2.6.A matrix φ(z) ∈ GL m (A) satisfying the property of Lemma 2.5 is called a relation matrix.
The next lemma plays a central role in the proof of Theorem 2.1.
Lemma 2.7.Let φ(z) ∈ GL m (A) be a relation matrix.Then P φ(z)A k (z), z q k = 0 , for all P ∈ I and all k ≥ 0.
Proof.Let P ∈ I, φ(z) ∈ GL m (A) be a relation matrix, and k be a nonnegative integer.Set Q(Y , z) is well-defined and we have as wanted.

Analyticity and relation matrices.
We address now the question of the analyticity of relation matrices.
Lemma 2.8.Let φ(z) ∈ GL m (A) be a relation matrix.Then the three following properties holds for k ≫ 1.
(a) The point α q k belongs to the disc of convergence of each of the functions Proof.Since lim k→∞ α q k = 0 and f 1 (z), . . ., f m (z) are analytic on some neighborhood of 0, Property (a) holds for k ≫ 1. Recall that an algebraic function has only finitely many singularities and finitely many zeros.Hence, for k ≫ 1, α q k is neither a singularity of one of the coordinates of φ(z) nor a zero of det φ(z).We deduce that Properties (b) and (c) hold for k ≫ 1.

The key
where we recall that f (z , where I m is the identity matrix of size m, we obtain that (2.5) The point α being regular with respect to (1.1), we deduce from (2.5) that (2.7) Item (a) in Lemma 2.8 ensures the existence of a positive real number r 1 < 1 such that 0 < |ξ| < r 1 and such that all the power series f 1 (z), . . ., f m (z) have a radius of convergence larger than r 1 .Then, by Item (b) in the same lemma, we can choose r 2 > 0 satisfying 0 < |ξ| + r 2 < r 1 and such that the coefficients of the matrix φ(z) are analytic on the disc D(ξ, r 2 ).For every k ≥ k 0 , we set Remark 2.10.By Lemma 2.8, the coefficients of Θ k 0 (z) are analytic on the disc D(ξ, r 2 ).On the other hand, one has This implies that, for every k ≥ k 0 , the coefficients of Θ k (z) are analytic on some neighborhood of ξ, that is on some disc D(ξ, r k ) ⊂ D(ξ, r 2 ).In what follows, we will consider the expression F (Θ k (z), z q k−k 0 ).Formally, it is a polynomial in f 1 (z q k−k 0 ), . . ., f m (z q k−k 0 ) and the coordinates of Θ k (z).Note that it also defines an analytic function on D(ξ, r k ) ⊂ D(ξ, r 2 ).In addition, F (Θ k 0 (z), z) is analytic on D(ξ, r 2 ).Indeed, the functions f 1 (z), . . ., f m (z) are analytic on D(0, r 1 ) ⊃ D(ξ, r 2 ), while our choice of k 0 ensures that the coordinates of Θ k 0 (z) are analytic on D(ξ, r 2 ).

2.3.3.
The key lemma.The end of the section is devoted to proof of the following result.
Lemma 2.11.One has In what follows, we argue by contradiction, assuming that (2.10) We divide the proof of Lemma 2.11 into the four steps (AF), (UB), (NV), and (LB), following the classical proof scheme in transcendental number theory.In Step (AF) we build an auxiliary function by considering some sort of Padé approximant of type I for the first powers of F (Y , z).In Step (UB) we compute some upper bound for the absolute value of the evaluation of our auxiliary function at (A k (α), α q k ), for large k, by means of analytic estimates.In Step (NV) we prove that our auxiliary function is non-vanishing at (A k (α), α q k ) for infinitely many k.In Step (LB), we provide a lower bound for the absolute value of the evaluation of our auxiliary function at (A k (α), α q k ), for infinitely many k, by using Liouville's inequality.Finally, we show that the steps (UB) and (LB) lead to a contradiction.
Step (AF).Given a formal power series and an integer p > 0, we let denote the truncation of E at order p with respect to z.We recall that Lemma 2.12.Let δ 1 ≥ 0 and δ 2 ≫ δ 1 be two integers.Let p := δ 1 δ 2 2 m 2 +2 .Then there exist polynomials P i ∈ I ⊥ (δ 1 , δ 2 ), 0 ≤ i ≤ δ 1 , not all zero, such that the formal power series The Q-vector spaces J (δ 1 , δ 2 ) and I(δ 1 , δ 2 ) have same dimension.This follows directly from the fact that the map is an isomorphism, the matrix A k 0 (α)φ(α q k 0 ) −1 being invertible.Furthermore, we have , and Lemma 2.7 implies that For k ≥ k 0 , replacing k by k − k 0 in the previous equality, we obtain that By (2.9), we thus have P (Θ k (z), z q k−k 0 ) = 0.
Let p be as in the lemma and let us consider the three Q-linear maps: Note that these maps are well-defined.By Lemma 2.2, the dimension of the Q-vector space I ⊥ (δ 1 , δ 2 ) is at least equal to c 1 (δ 1 ) 2 δ 2 , assuming that δ 2 is large enough.It follows that (2.12) For every pair of non-negative integers (u, v), set Since J (δ 1 , δ 2 ) and I(δ 1 , δ 2 ) have same dimension, Lemma 2.3 implies that On the other hand, the choice of p ensures that and (2.12) implies that Hence the Q-linear map defined by has a non-trivial kernel.We deduce the existence of polynomials P 0 , . . ., P δ 1 in I ⊥ (δ 1 , δ 2 ), not all zero, such that E p ∈ J (2δ 1 , p − 1).By (2.11), we obtain that E p (Θ k (z), z q k−k 0 ) = 0 for all k ≥ k 0 .This ends the proof.
] be a formal power series satisfying the properties of Lemma 2.12 and let v 0 be the smallest index such that the polynomial P v 0 is non-zero.Then the formal power series is the auxiliary function that we were looking for.Note that we have Warning.The function E(Θ k (z), z q k−k 0 ) can be though of as a simultaneous Padé approximant of type I for the first δ 1 th powers of F (Θ k (z), z q k−k 0 ).However, we have to be careful: F (Θ k (z), z q k−k 0 ) it is not necessarily a power series in z.It is a linear combination of f 1 (z q k−k 0 ), . . ., f m (z q k−k 0 ) whose coefficients are only known to be algebraic over Q(z).We only know that F (Θ k (z), z q k−k 0 ) is analytic in some neighborhood of the point ξ.
Step (UB).The aim of this step is to prove that there exists a real number According to Remark 2.10, the functions E(Θ k (z), z q k−k 0 ), F (Θ k 0 (z), z) v 0 , and E(Θ k (z), z q k−k 0 ) are all analytic on the disc D(ξ, r k ).Hence they respectively have power series expansions of the form We need the following result whose proof is postponed after the end of the argument for proving our main upper bound (2.14).
Lemma 2.13.Let p be defined as in Lemma 2.12.There exists a real number γ > 0 that does not depend on the integers δ 1 , δ 2 , λ, and k, and such that Using (2.6), we get that By (2.13), we thus have for all k ≥ k 0 and all z ∈ D(ξ, r k ).We use now our assumption that

10))
. There thus exists at least one non-zero coefficient a λ in (2.16).Let us consider the least integer λ 0 such that a λ 0 = 0.
Identifying the coefficients of (z − ξ) λ 0 in the power series expansion of both sides of (2.18) with the help of (2.15), (2.16), and (2.17), we obtain that )) and a λ 0 depends only on δ 1 but not on k, we infer from Lemma 2.13, Equality (2. 19), and the definition of p (see Lemma 2.12), the existence of a real number c 2 > 0 that does not depend on δ 1 , δ 2 , and k, such that This proves the upper bound (2.14), as wanted.Now, it remains to prove Lemma 2.13.
Proof of Lemma 2.13.Set where p is defined as in Lemma 2.12.By Lemma 2.12, we have Let ν 1 , . . ., ν s denote an enumeration of all the m × m matrices with coefficients in the set {0, 1, . . ., 2δ 1 }.There exists a unique decomposition of the form where g λ,i ∈ Q.For every i, 1 ≤ i ≤ s, we define the formal power series By definition of F (Y , z), these series belong to Q[z, f (z)].In particular, they are analytic on some disc D(0, r) with r > r 1 (where r 1 is defined at the beginning of Section 2.3.2).From the Cauchy-Hadamard inequality, there exists a positive real number γ 1 (δ 1 , δ 2 ) such that For every k ≥ k 0 , G i (z q k−k 0 ) can thus be written as with g λ,i,k ∈ Q.Furthermore, this power series is absolutely convergent on the disc D(0, r 1 ).Since r 1 ≤ 1, we deduce from (2.21) that , for all λ ≥ 0, i ∈ {1, . . ., s}, and k ≥ k 0 .On the other hand, every function G i (z q k−k 0 ), 1 ≤ i ≤ s, k ≥ k 0 , is analytic on the disc D(ξ, r k ).Thus, we can write where h λ,i,k ∈ C. Since by assumption D(ξ, r k ) ⊂ D(0, r 1 ), the two power series expansions (2.22) and (2.24) match on D(ξ, r k ).Using the equality (2.25) and identifying, for every λ ≥ 0, the coefficients of (z − ξ) λ in (2.22) and (2.24), we deduce that For γ ≥ λ, one has Given λ ≥ 0, we have that λ < q k−k 0 p as soon as k is large enough, and since |ξ| < r 1 , we infer from (2.23) and (2.26) the existence of a real number γ 2 > 0 that does not depend on δ 1 , δ 2 , λ, and k, such that

Now, we proceed to bound the absolute value of the coefficients of the power series expansion in
For all k large enough, ξ q k−k 0 belongs to the domain of analyticity of Q(z).Using again (2.25) and (2.27) we obtain that, for every λ ≥ 0, |q λ,k | = O(1) as k tends to infinity, where the underlying constant in the O notation depends both on Q(z) and λ.Fix some λ ≥ 0. Let v ≥ 0 be an integer such that the coordinates of z v A(z) have no poles at 0. The coordinates of z v A(z) are convergent power series at 0, and the points ξ q k−k 0 belong to their domain of analyticity for k large enough.Then, the coefficients of (z − ξ) λ in the power series expansion in z − ξ of each of the coordinates of z vq k−k 0 A(z q k−k 0 ) belong to O(1) as k tends to infinity.Using (2.25), we write Using (2.27), we deduce the existence of a real number γ 3 > 0 which does not depend on k and such that |r λ,k | = O(e γ 3 q k ) as k tends to infinity.It follows that the absolute value of the coefficient of (z − ξ) λ , in the power series expansion in z − ξ of each of the coordinates of A(z q k−k 0 ), belongs to O(e γ 3 q k ) as k tends to infinity, where the underlying constant in the O notation depends on λ but not on δ 1 , δ 2 , and k.By Remark 2.10, the monomial Θ k (z) ν i is analytic on D(ξ, r k ) for every i, 1 ≤ i ≤ s, and every k ≥ k 0 .Thus, we can write where θ λ,i,k ∈ C. Using the recurrence relation we obtain the existence of a real number γ 4 (λ) > 0 that does not depend on i, δ 1 , δ 2 , and k, such that the absolute value of the coefficient of (z − ξ) λ in each of the coordinates of Θ k (z) is at most e γ 4 (λ)q k .Since |ν i | ≤ 2m 2 δ 1 for each i, there exists a real number γ 5 (λ) > 0 that does not depend on i, δ 1 , δ 2 , and k, such that From (2.17), (2.20), (2.24), and (2.29), we deduce that (2.31) Finally, identifying the coefficents of (z − ξ) λ in both sides of (2.31), we have Note that p ≫ δ 1 when δ 2 ≫ δ 1 .Inequalities (2.28) and (2.30) imply the existence of a real number γ 6 > 0 that does not depend on δ 1 , δ 2 , λ, and k, and such that Setting γ := γ 6 , this ends the proof.
Step (NV).Let us first recall that by (2.7) we have By construction of our auxiliary function, we deduce that Furthermore, since this construction ensures that P v 0 / ∈ I, there exists an infinite set of positive integers E such that Without any loss of generality, we assume that k ≥ k 0 for all k ∈ E.
Step (LB).Given an algebraic number β, we let h(β) denote the absolute logarithmic Weil height of β (see [26,Chapter 3] or Section A.2 for a definition).In order to prove our lower bound, we only need the following basic properties of the Weil height.The use of the Weil height simplifies some computations but any other standard notion of height would also do the job.Given two algebraic numbers β and γ, one has (see [26,Property 3.3]): and β 1 , . . ., β n ∈ Q, we deduce from [26,Lemma 3.7] that (2.33) Given a number field k, we have the fundamental Liouville inequality (see [26, p. 82]): We are going to use (2.34) to find a lower bound for |E(A k (α), α q k )|.A simple computation by induction on k shows that the height of each coordinate of A k (α) is at most γq k for some γ > 0 that does not depend on k (see Section A.2 for more detail).The polynomial P v 0 (Y , z) has degree at most δ 1 in each indeterminate y i,j and degree at most δ 2 in z.Furthermore, its coefficients are algebraic numbers which only depend on the parameters δ 1 and δ 2 .Using (2.32) and (2.33), we obtain that the height of the algebraic number P v 0 (A k (α), α q k ) is at most cq k δ 2 for some constant c that does not depend on δ 1 , δ 2 , and k, assuming that k ≫ δ 2 ≥ δ 1 .Since these algebraic numbers belong to a fixed number field, Liouville's inequality ensures the existence of c 3 > 0 that does not depend on δ 1 , δ 2 , and k, and such that (2.35) We are now ready to end the proof of our key lemma.
Proof of Lemma 2.11.By Inequalities (2.14) and (2.35), we obtain that We deduce that Since c 2 and c 3 are positive numbers which do not depend on δ 1 , this provides a contradiction, as soon as δ 1 is large enough.

2.4.
End of the proof of Theorem 2.1.The coordinates of φ(z) being algebraic over Q(z), they generate a finite extension of Q(z).Let k ⊂ A denote this extension and let γ ≥ 1 be the degree of k.We recall that A is the algebraic closure of Q(z) in the field of Puiseux series.Choosing a primitive element ϕ(z) in k, we obtain a decomposition of the form where the matrices φ j (z), 0 denote a common denominator of the coordinates of the matrices φ j (z).Without any loss of generality, we can assume that in (2.8) the integer k 0 has been chosen large enough so that ϕ(z) is analytic at ξ = α q k 0 and d(α q k 0 ) = 0. Let q(z) denote the least common multiple of the denominators of the coordinates of the matrix A −1 k 0 (z).Since α is assumed to be regular with respect to the Mahler system (1.1), we have that q(α) = 0.
By Lemma 2.11, we know that F (Θ k 0 (z), z) = 0, and substituting z q k 0 for z, we obtain that F (Θ k 0 (z q k 0 ), z q k 0 ) = 0.The function F (Y , z) being linear in Y , we deduce that Now, let us consider the linear form in X 1 , . . ., X n defined by: Thus, the coefficient of each Finally, it follows from (2.37) that There is only one point left to address: we have lifted the linear relation between and A are linearly disjoint over Q(z) (see [14,Chapter VIII]).Let δ denote the degree of ϕ(z q k 0 ) over Q(z), so that the functions ϕ(z q k 0 ) j , 0 ≤ j ≤ δ − 1, are linearly independent over Q(z).Since Q(z)(f (z)) and A are linearly disjoint over Q(z), these functions remain linearly independent over Q(z)(f (z)).Thus, splitting the linear form Q as where Q j (z, X) ∈ Q[z, X] are linear forms, we deduce that Finally, setting we obtain that L(z, f (z)) = 0 and L(α, X) = L(X), as wanted.This ends the proof of Theorem 2.1.

From linear to algebraic relations
In this section, we end the proof of Theorem 1.2.In order to deduce Theorem 1.2 from Theorem 2.1, the key observation is that, given M q -functions f 1 (z), . . ., f m (z) related by a q-Mahler system, the M q -functions obtained by considering all monomials of a given degree in f 1 (z), . . ., f m (z) are also related by a q-Mahler system with no additional singularity.
Let us first recall some notation.Let A = (a i,j ) and B be matrices with entries in a given commutative ring, with dimension, respectively, (m, n) and (p, q).The Kronecker product of A and B is the matrix A ⊗ B, of size (mp, nq) with block decomposition If d ≥ 1 is an integer, we also let , denote the dth Kronecker power of the matrix A.
Proof of Theorem 1.2.Let d denote the total degree of P and λ 1 , . . ., λ t be an enumeration of the set {λ ∈ (Z ≥0 ) m : |λ| = d}.Then, we have where p j ∈ Q and X := (X 1 , . . ., X m ).Set f (z) := (f 1 (z), . . ., f m (z)) ⊤ .The coordinates of the vector f (z) ⊗d are precisely the monomials of degree d in f 1 (z), . . ., f m (z), with some of them appearing several times (for example, the product f 1 (z)f 2 (z) appears twice in f (z) ⊗2 ).Using [12, Lemma 4.2.10] or (i) of Lemma A.2 in Section A.3 and a straightforward induction on d, we obtain that Since α is a regular point with respect to the system (1.1) the matrix A(z) is well-defined and invertible at α q k for all integers k ≥ 0. The entries of the matrix A(z) ⊗d being products of the entries of A(z), the matrix A(z) ⊗d is well-defined at α q k for all integers k ≥ 0. Furthermore, since det A(α q k ) = 0 we have det A(α q k ) ⊗d = 0 (see [12,Corollary 4.2.11] or (ii) of Lemma A.2 in Section A.3), for all integers k ≥ 0. Hence α is a regular point with respect to the system (3.1).For each j, 1 ≤ j ≤ t, let I j ⊂ {1, . . ., m d } denote the set of integers i for which the ith entry of X ⊗d is X λ j .For each j, we pick an integer i j in I j .Let Y 1 , . . ., Y m d be a family of indeterminates and let us consider the linear form L defined by We also let g 1 , . . ., g m d denote the coordinates of f (z) ⊗d .By construction g i (z) = f (z) λ j when i ∈ I j .Thus, On the one hand, we have while, on the other hand, we have p j X λ j = P (X) .
This ends the proof.

Deducing Nishioka's theorem from the lifting theorem
In this section, we show how to deduce Nishioka's theorem from the lifting theorem.
Proof of Theorem 1.1.We first note that the inequality always holds.Hence we only have to prove that (4.1) Let d ≥ 0 be an integer.We let ϕ α (d) denote the dimension of the Q-vector space spanned by the monomials of degree at most d in f 1 (α), . . ., f m (α).We also let ϕ z (d) denote the dimension of the Q(z)-vector space spanned by the monomials of degree at most d in f 1 (z), . . ., f m (z).Note that the functions 1, f 1 (z), . . ., f m (z) are related by the q-Mahler system of size m + 1: . . .
Furthermore, the point α remains regular with respect to this new system.Applying Theorem 1.2 to (4.2), we obtain that By a result of Hilbert, ϕ α (d) and ϕ z (d) are polynomials in d of degree respectively equal to t α and t z when d ≫ 1 (see, for instance, the discussion around the Hilbert-Serre theorem in [27, p. 232]).Thus, there exist two positive real numbers β and γ such that ϕ α (d) ≤ βd tα and ϕ z (d) ≥ γd tz , ∀d ≫ 1 .
Remark 4.1.In the proof of Theorem 1.1, we do not need the full strength of Hilbert's result.Suitable estimates for ϕ α (d) and ϕ z (d) can be easily achieved by elementary means (see Section A.4).
Remark 4.2.At the end of our proof of Theorem 1.2, we used the fact that the field extension Q(z, f 1 (z), . . ., f m (z)) is a regular extension of Q(z).We stress that this argument is not needed to deduce Nishioka's theorem.Indeed, without using it, we still obtain that every Q-linear relation between f 1 (α), . . ., f m (α) can be lifted into a linear relation over the algebraic closure A of Q(z) between f 1 (z), . . ., f m (z).Then we could reproduce the previous argument, just replacing Q(z) by A. We would derive the main result since A being by definition algebraic over Q(z).
Appendix A. Addendum to the proofs of Theorems 1.1 and 1.2 In Sections A.1 and A.2, we provide more details about some auxiliary results used in the proof of Theorem 2.1.We also give in Sections A.3 and A.4 the proof of two elementary auxiliary results that can be used to make the proof of Theorems 1.1 and 1.2 as elementary and self-contained as possible.
A.1.Computation of the dimension of I ⊥ (δ 1 , δ 2 ).We provide here a more detailed argument for the proof of Lemma 2.2.
Proof of Lemma 2.2.Set h := (δ 1 + 1) m 2 and let ν 1 , . . ., ν h denote an enumeration of the set of all matrices in M m (Z ≥0 ) whose entries are at most δ 1 .Any polynomial P ∈ Q(z)[Y ] δ 1 has a unique decomposition of the form which are linearly independent over Q(z) and such that for all p 1 (z), . . ., p h (z) ∈ Q(z): Since none of the vectors (b i,1 (z), . . ., b i,h (z)) is zero, we can choose the polynomials b i,j (z) so that for every i, Since these polynomials only depend on δ 1 (and I), there exists a nonnegative integer δ ′ 1 , which only depends on δ 1 (and I), such that they can be written as b i,j (z) =: where the numbers b i,j,κ belong to Q.For every integer κ such that κ > δ ′ 1 or κ < 0, we also set b i,j,κ := 0. Now, let us consider a polynomial P (Y , z) where the numbers p j,λ belong to Q.We also set p j,λ := 0 if λ > δ 2 or λ < 0. By (A.1), P belongs to I(δ 1 , δ 2 ) if and only if For each δ 2 ≥ δ ′ 1 , we let V (δ 1 , δ 2 ) denote the Q-vector subspace of the dual of Q[Y , z] spanned by the linear forms (A.4) Since in (A.3) and (A.4) the parameter γ runs over the same interval up to a finite set that does not depend on δ 2 , it follows that as δ 2 tends to infinity.For every Proof of the first claim.Let us observe that, for every (γ, λ, i, j), we have Hence L γ,i (z δ 2 +1 Y ν j ) = 0 for all γ ≤ δ 2 and all (i, j), while for every i, Second claim.The sequence (c(δ 1 , δ 2 )) δ 2 ≥δ ′ 1 is non-increasing.Proof of the second claim.Note that, by definition, c(δ 1 , δ 2 ) is equal to the number of linear forms in {L δ 2 +1,i : 1 ≤ i ≤ d} which are linearly independent over V (δ 1 , δ 2 ).Let us assume that some of these linear forms, say where θ 1 , . . ., θ t are algebraic numbers, not all zero.Then we are going to show that we also have The second claim follows directly from (A.7).
By assumption, we can write where the η γ,i are algebraic numbers.In order to prove (A.7), we show more precisely that or, equivalently, that ℓ,j (α q k )| ν ≤ me (γ 1 +γ 2 )q k ≤ e γ 2 q k+1 .This proves (A.10).We deduce from (A.8) that h(a It follows from the first part of the proof that the height of the coordinates of the matrix b k (α)A k (α) is at most γ 4 q k , for some positive real number γ 4 which does not depend on k.Using (2.32) and (2.33), we obtain that h(b(α q k )) = O(q k ) as k tends to infinity.By (2.32), we deduce that there exists a real number γ 5 > 0 such that h(b k (α)) ≤ γ 5 q k , ∀k ≥ 0 .
A.3.Kronecker product.For the sake of completeness, we provide a proof of the basic properties that we used about Kronecker product in Section 3.
Lemma A.2.The following properties hold.In particular, if A is a square matrix of size m and d ≥ 1 is an integer, then det A ⊗d = (det A) md .
Proof.Let us first prove (i).Set A := (a i,j ) i,j and B := (b j,k ) j,k .Then we have the block matrix decompositions A ⊗ C = (a i,j C) i,j , and B ⊗ D = (b j,k D) j,k .
According to this block decomposition, the (i, k)th block of (A ⊗ C)(B ⊗ D) is j a i,j b j,k CD .
In the mean time, AB = ( j a i,j b j,k ) i,k and thus, the (i, k)th block of (AB)⊗ (CD) is precisely j a i,j b j,k CD.Now, let us prove (ii).Consider the matrix A ⊗ I n .After a suitable permutation of the rows and the columns, one obtains the matrix I n ⊗ A, which is a block diagonal matrix.Hence A.4. Hilbert's theorem about transcendence degree.In Section 3, we use a result of Hilbert to deduce Theorem 1.1 from Theorem 1.2.Let K be a field, L be a field extension of K, ξ 1 , . . ., ξ m ∈ L, and ϕ(d) denote the dimension of the K-vector space formed by the polynomials in ξ 1 , . . ., ξ m of total degree at most d.Hilbert's theorem states that ϕ(d) is a polynomial with degree t := tr.degK (ξ 1 , . . ., ξ m ) for all d large enough.As mentionned in Remark 4.1, we do not need the full strength of Hilbert's theorem.The following elementary lemma is indeed sufficient to deduce Theorem 1.1.
Lemma A.3.We continue with the notation above.There exist two positive real numbers γ 1 and γ 2 that does not depend on d and such that Proof.Suppose first that t = 0.Then, K(ξ 1 , . . ., ξ m ) is a finitely generated extension of K which is algebraic over K.It follows that it has finite degree, say δ, over K.Then, for any d, Taking γ 1 = 1 and γ 2 = δ, this proves the lemma when t = 0. Suppose now that t ≥ 1.Without any loss of generality, we can assume that ξ 1 , . . ., ξ t are algebraically independent over K.
We first prove the lower bound.By assumption, all the monomials in ξ 1 , . . ., ξ t are linearly independent over K. Since there are d+t t distinct monomials in ξ 1 , . . ., ξ t of degree at most d, we obtain that Taking γ 1 = (t!) −1 , we obtain the expected lower bound.

( i )
If A,B, C, and D are matrices such that the products AB and CD are well-defined, then (AB) ⊗ (CD) = (A ⊗ C)(B ⊗ D).(ii)If A and B are two square matrices respectively of size m and n, then det(A ⊗ B) = (det A) n (det B) m .
det(A ⊗ I n ) = det(I n ⊗ A) = (det A) n .Similarly, det(I m ⊗ B) = (det B) m .We infer from (i) thatA ⊗ B = (AI m ) ⊗ (I n B) = (A ⊗ I n )(I m ⊗ B) .Hence det(A ⊗ B) = det(I n ⊗ A) det(I m ⊗ B) = (det A) n (det B) mas wanted.The last property follows now by induction on d.
as δ 2 tends to infinity.
3.2.The matrices Θ k (z).From now on, we fix a relation matrix φ(z) and a non-negative integer k 0 satisfying the properties of Lemma 2.8.Set