# Rational approximations to values of $E$-functions

S. Fischler and T. Rivoal

(August 28, 2024)

###### Abstract

We solve a long standing problem in the theory of Siegel’s $E$-functions,
initiated by Lang for Bessel’s function $J_{0}$ in the 60’s and considered in
full generality by G. Chudnovsky in the 80’s: we prove that irrational values
taken at rational points by $E$-functions with rational Taylor coefficients
have irrationality exponent equal to 2. This result had been obtained before
by Zudilin under stronger assumptions on algebraic independence of
$E$-functions, satisfied by $J_{0}$ but not by all hypergeometric
$E$-functions for instance.

##  1 Introduction

In rational approximation, a landmark result is Roth’s theorem [28], proved in
1955: if $\xi\in\mathbb{R}\setminus\mathbb{Q}$ is an algebraic number, then

$\forall\varepsilon>0\quad\exists
c>0\quad\forall(p,q)\in\mathbb{Z}\times\mathbb{N}^{*}\quad\quad\left|\xi-\frac{p}{q}\right|\geq\frac{c}{q^{2+\varepsilon}}.$
(1.1)

This measure of irrationality is optimal in the following sense: it would be
false with $\varepsilon=0$ and $c=1$, as can be proved using continued
fractions or Dirichlet’s pigeonhole principle. With respect to Lebesgue
measure, almost all $\xi\in\mathbb{R}\setminus\mathbb{Q}$ satisfy (1.1). A
folklore belief is that classical transcendental constants coming from
analysis should satisfy (1.1); but this is known for very few of them, even
amongst the most important ones. For instance, (1.1) is known for $\xi=\pi$
only in a weaker form, with 2 in the exponent of $q$ replaced with $7.1033$
(Zeilberger-Zudilin [32]); in other words, the irrationality exponent of $\pi$
is at most $7.1033$. For $\log(2)$ the situation is analogous: its
irrationality exponent is at most $3.5746$ (Marcovecchio [25]). In both cases,
this exponent is currently best known and it comes after many successive
improvements.

The situation is different for the values of the exponential function: it is
well known that (1.1) holds with $\xi=e^{r}$ for any $r\in\mathbb{Q}^{*}$ (see
the introduction of [4] or [22]). This result opens the way to a possible
generalization to $E$-functions, a class of functions defined by Siegel [30]
in 1929 to extend Diophantine properties of the exponential function. In this
paper $\overline{\mathbb{Q}}$ is seen as a subset of $\mathbb{C}$.

######  Definition 1.

A power series
$F(z)=\sum_{n=0}^{\infty}a_{n}z^{n}/n!\in\overline{\mathbb{Q}}[[z]]$ is an
$E$-function if

$(i)$ $F$ is solution of a non-zero linear differential equation with
coefficients in $\overline{\mathbb{Q}}(z)$,

and there exists $C>0$ such that:

$(ii)$ for any $\sigma\in\textup{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ and
any $n\geq 0$, $|\sigma(a_{n})|\leq C^{n+1}$.

$(iii)$ there exists a sequence of integers $(d_{n})_{n\geq 0}$, with $1\leq
d_{n}\leq C^{n+1}$ for any $n\geq 0$, such that $d_{n}a_{m}$ are algebraic
integers for all $0\leq m\leq n$.

Siegel’s original definition is more general: in $(ii)$ and $(iii)$, he
required bounds of the form “for all $\varepsilon>0$, $\ldots\leq
n!^{\varepsilon}$ for any $n\geq N(\varepsilon)$” instead of “$\ldots\leq
C^{n+1}$ for any $n\geq 0$”. $E$-functions in the sense of Definition 1 shall
be called $E$-functions in the strict sense. Note that $(i)$ implies that the
$a_{n}$’s all lie in a certain number field $\mathbb{K}$. An $E$-function is
an entire function; it is transcendental unless it is a polynomial.

Amongst the simplest examples of $E$-functions, we mention
$\exp(z)=\sum_{n=0}^{\infty}z^{n}/n!$ and Bessel’s function
$J_{0}(z):=\sum_{n=0}^{\infty}(iz/2)^{2n}/n!^{2}$. Both are specializations of
the generalized hypergeometric function

${}_{p}F_{q}\left[\begin{matrix}a_{1},\ldots,a_{p}\\\
b_{1},\ldots,b_{q}\end{matrix}\,;z\right]:=\sum_{n=0}^{\infty}\frac{(a_{1})_{n}\cdots(a_{p})_{n}}{(1)_{n}(b_{1})_{n}\cdots(b_{q})_{n}}z^{n}$

where $q\geq p\geq 0$ and we define the Pochhammer symbol
$(a)_{n}:=a(a+1)\cdots(a+n-1)$ for $n\geq 1$, $(a)_{0}:=1$. Siegel has proved
that when $q\geq p\geq 0$, the parameters $a_{j}$ and $b_{j}$ are in
$\mathbb{Q}$ (with the restriction that $b_{j}\notin\mathbb{Z}_{\leq 0}$ so
that $(b_{j})_{n}\neq 0$ for all $n\geq 0$) and $c\in\overline{\mathbb{Q}}$,
then ${}_{p}F_{q}[a_{1},\ldots,a_{p};b_{1},\ldots b_{q};cz^{q-p+1}]$ is an
$E$-function. However, the $\overline{\mathbb{Q}}$-algebra generated by such
hypergeometric $E$-functions is not large enough to contain all $E$-functions,
as recently shown by Fresán-Jossen [18]. We also point out that values of
$E$-functions at algebraic points are closely related to exponential periods,
see [19].

The Diophantine theory of the values taken at algebraic points by
$E$-functions has a long history with classical results due, in chronological
order, to Siegel, Shidlovskii, Nesterenko, André and Beukers in particular. We
refer to their original works [30, 29, 26, 2, 8] as well as to [9, 16, 33] for
precise statements of these results and others.

Our purpose is to prove the following result, namely: (1.1) holds for all
irrational values of $E$-functions with rational Taylor coefficients at
rational points.

######  Theorem 1.

Let $f\in\mathbb{Q}[[z]]$ be an $E$-function and let $r\in\mathbb{Q}$. Then
either $f(r)\in\mathbb{Q}$, or for any $\varepsilon>0$, there exists a
constant $c>0$ depending on $f,r,\varepsilon$ such that for any
$p\in\mathbb{Z}$ and any $q\in\mathbb{N}^{*}$, we have

$\left|f(r)-\frac{p}{q}\right|\geq\frac{c}{q^{2+\varepsilon}}.$ (1.2)

We shall first give the proof of Theorem 1 for $E$-functions in the strict
sense, using in particular various lemmas in Zudilin’s paper [33] and results
by André and Beukers in the theory of $E$-operators for $E$-functions in the
strict sense. Then we shall explain in §3.4 the changes that must be made to
the proof to obtain Theorem 1 for $E$-functions in Siegel’s sense.

Theorem 1 was announced in 1984 by Chudnovsky [10, p. 1926, Theorem 1 and
Corollary] for $E$-functions in $\mathbb{Q}[[z]]$ in Siegel’s original sense
and in a stronger form, namely a measure of linear independence. However, as
explained in [11, p. 245], the proof contains a gap in the zero estimate.
Zudilin [33] succeeded in filling this gap, thereby proving Theorem 1 (in an
even more precise form, namely with a decreasing function of $q$ instead of
$\varepsilon$), but under additional assumptions on $f$ and $r$ and for
$E$-functions in the strict sense. Precisely, let $m$ be the minimal order of
a non-trivial inhomogeneous differential equation with coefficients in
$\mathbb{Q}(z)$ satisfied by $f$. Then Zudilin assumes that either $m\leq 2$
(Corollary 1 on page 583) or that $f,f^{\prime},\ldots,f^{(m-1)}$ are
algebraically independent over $\mathbb{Q}(z)$ (Corollary 1 on page 557). In
both cases, he also assumes that $r$ is not a singularity of the equation. In
the present paper, we use a different approach to zero estimates (see below),
which enables us to prove Theorem 1 without any additional assumption on $f$.

As just explained, the interest of Theorem 1 is that it applies to all
$E$-functions with rational coefficients, for instance to all generalized
hypergeometric $E$-functions of the form
${}_{p}F_{q}[a_{1},\ldots,a_{p};b_{1},\ldots b_{q};cz^{q-p+1}]$ with $q\geq
p\geq 0$, $c\in\mathbb{Q}$ and rational parameters $a_{j}$ and $b_{j}$,
whereas Zudilin applies his results to them only under some assumptions on the
parameters, in particular to $J_{0}(z)={}_{1}F_{2}[1;1,1;(iz/2)^{2}]$. The
$E$-function $g(z):={}_{1}F_{2}[1/2;1/3,2/3;z^{2}]$ does not fall under the
scope of Zudilin’s results because a minimal inhomogeneous differential
equation satisfied by $g$ is
$9z^{2}g^{\prime\prime\prime}(z)+9zg^{\prime\prime}(z)-(36z^{2}+1)g^{\prime}(z)-36zg(z)=0$
of order $m=3$ while
$g(z)^{2}-g^{\prime}(z)^{2}/4+9z^{2}(4g(z)-g^{\prime\prime}(z))^{2}/4=1$; on
the other hand, $g(r)\notin\mathbb{Q}$ for all $r\in\mathbb{Q}^{*}$ (see [17,
§7] for details) so that (1.2) holds for all these values. See below for non-
trivial examples of transcendental ${}_{1}F_{1}$ series with rational
parameters taking a rational value at a non-zero rational point, showing that
it is not possible to exclude a priori the “$f(r)\in\mathbb{Q}$” possibility
even in the hypergeometric case.

Another interesting example is the generating $E$-function of Apéry numbers
$\mathcal{A}_{2,2}(z):=\sum_{n=0}^{\infty}(\sum_{k=0}^{n}\binom{n}{k}^{2}\binom{n+k}{n}^{2})z^{n}/n!$.
We proved in [16] that for any $r\in\mathbb{Q}^{*}$, $\mathcal{A}_{2,2}(r)$
has irrationality exponent at most 5 (because we are not able to check that
the additional conditions of Zudilin’s result are met). Using Theorem 1 we
deduce the optimal measure (1.2) immediately for $\mathcal{A}_{2,2}(r)$.

A different problem, very interesting but not studied in this paper, is to
know in the setting of Theorem 1 whether $f(r)$ is rational or not when
$f\in\mathbb{Q}[[z]]$ is transcendental (if $f$ is a polynomial, obviously
$f(r)\in\mathbb{Q}$ for all $r\in\mathbb{Q}$). Beukers’ linear independence
theorem [8, Corollary 1.4] provides a sufficient condition: given
$\sum_{j=0}^{m}p_{j}(z)f^{(j)}(z)=q(z)$ a non-trivial inhomogeneous equation
satisfied by $f$ transcendental of minimal order $m\geq 1$ with polynomial
coefficients, if $r\in\mathbb{Q}^{*}$ is such that $p_{m}(r)\neq 0$ then
$f(r)\notin\mathbb{Q}$. If $p_{m}(r)=0$, it can be decided algorithmically
whether $f(r)\in\mathbb{Q}$ or not, see [1, 9]. Of course there are trivial
examples of transcendental $E$-functions such that $f(r)\in\mathbb{Q}$ for
some $r\in\mathbb{Q}^{*}$, for instance $(z-1)e^{z}$ at $z=1$. But there are
also non-trivial examples such as ${}_{1}F_{1}[5;7/3;-2/3]=5/27$ and
${}_{1}F_{1}[6;-2/5;-12/5]=1309/625$; see [9, §4.2] for more “exotic”
evaluations of hypergeometric $E$-functions at rational points.

As a consequence of [14, Theorem 4], if $f$ is an $E$-fonction with rational
Taylor coefficients such that $f(1)=e^{\alpha}$ for some
$\alpha\in\overline{\mathbb{Q}}$, then $\alpha\in\mathbb{Q}$. (The proof of
this result is an immediate adaptation of that of [13, Proposition 2], which
is due to the referee of that paper.) In particular, Theorem 1 does not apply
directly to numbers such as $e^{\sqrt{2}}$, for which (1.2) is conjectural.
(111Note however that Zudilin’s theorem can be applied to the $E$-function
$f(x):=e^{\sqrt{2}x}+e^{-\sqrt{2}x}\in\mathbb{Q}[[x]]$: the number $f(1)$ has
irrationality exponent 2, so that $e^{\sqrt{2}}$ has irrationality exponent at
most 4, which seems to be the best known upper bound (see [22] and [16] for
proofs of the upper bounds 12 and 8 respectively).)

It would be very interesting to generalize Theorem 1 to any $E$-function $f$
in $\mathbb{K}[[z]]$ evaluated at any $\alpha\in\mathbb{K}^{*}$, where
$\mathbb{K}$ is a given fixed number field of degree $d$ over $\mathbb{Q}$. To
our knowledge, as a consequence of [16, Theorem 1], the current best upper
bound in this generality for the irrationality exponent of $f(\alpha)$ (when
it is irrational) is $d(m+1)^{d}$ where $m\geq 1$ is the minimal order of a
non-trivial inhomogeneous differential equation with coefficients in
$\overline{\mathbb{Q}}(z)$ satisfied by $f$. Note however that if $\mathbb{K}$
is an imaginary quadratic number field, this exponent is 2 because all our
arguments go through using the modulus of complex numbers instead of the
absolute value of real numbers in the Diophantine construction in §§3.2, 3.3
and 3.4. Moreover, under further assumptions on $f$ and $\alpha$ (similar to
Zudilin’s), this bound can be largely improved using the Lang-Galochkin
transcendence measure; see [11, Theorem 5.29 and remarks]. Even more
specifically, Kappe [22] has obtained the upper bound $4d^{2}-2d$ for the
irrationality exponent of $e^{\alpha}$ for all
$\alpha\in\overline{\mathbb{Q}}^{*}$ of degree $d$.

The proof of Theorem 1 is based on Chudnovsky’s construction, i.e. on graded
Padé approximation. Actually Zudilin used the same construction as Chudnovsky
(and he provided all necessary details); so do we. The first new feature of
Theorem 1 is that we have do not have to exclude the case where $r$ is a
singularity of a differential system, because we use (variants of) results of
André and Beukers on $E$-operators and $E$-functions. The second, and main,
new feature is that no assumption is needed on $f$ in Theorem 1. Actually
Zudilin had to make very strong assumptions in order to prove the zero
estimate (namely, that a matrix consisting of values of polynomials has
maximal rank – see Proposition 3 in §3.2 below). In the present paper, we
prove this result using the multiplicity estimate of [15], which relies on the
approach of Bertrand-Beukers [6] and Bertrand [5], generalized in [12]. The
important feature of this multiplicity estimate is that it takes into account
the possibility that some exceptional solutions of the underlying differential
system have identically zero remainders. Since we make no assumption on $f$ in
Theorem 1, this could a priori happen and it would make it impossible to prove
that the matrix we are interested in has maximal rank. We prove in §3.6 that
there is no such exceptional solution, using both the multiplicity estimate of
[15] and an independent technical result (see §4 below). Let us also mention
that all the non-explicit constants in the proofs of these intermediate
results are effective and could in principle be computed, and consequently the
same can be said of the constant $c$ in Theorem 1.

The structure of this paper is as follows. We gather in §2 the auxiliary
results we shall need: a property of the minimal inhomogeneous differential
equation of an $E$-function and a desingularization lemma, adapted from
results of André and Beukers respectively. We also state in §2.3 the
multiplicity estimate proved in [15]. Then §3 is devoted to the proof of
Theorem 1, using a technical result stated and proved in §4. The case of
$E$-functions in Siegel’s sense is dealt with in §3.3.

##  2 Prerequisites

To begin with, we recall that the Nilsson class at $0$ is the set of finite
sums

$f(z)=\sum_{e\in\mathbb{C}}\sum_{i\in\mathbb{N}}\lambda_{i,e}\,h_{i,e}(z)z^{e}(\log
z)^{i}$

where $\lambda_{i,e}\in\mathbb{C}$, and $h_{i,e}$ is holomorphic at $0$. If
such a function $f(z)$ is not identically zero, we may assume that
$h_{i,e}(0)\neq 0$ for any $i$, $e$; then the generalized order of $f$ at $0$,
denoted by ${\rm ord}_{0}f$, is the minimal real part of an exponent $e$ such
that $\lambda_{i,e}\neq 0$ for some $i$. When $Y:\mathbb{C}\to\mathbb{C}^{q}$
is a vector-valued fonction, it is said to be Nilsson at 0 if its $q$
coordinates are.

Thoughout the paper, we shall denote by $M_{n}(F)$ and $M_{n,m}(F)$ the sets
of $n\times n$ matrices and of $n\times m$ matrices over a given field $F$,
typically $\mathbb{C}$ or $\mathbb{C}(z)$.

###  2.1 Inhomogeneous differential equation of minimal order satisfied by an
$E$-function

Let $f$ be a transcendental $E$-function with coefficients in a number field
$\mathbb{K}$. Consider $f_{j}(z)=f^{(j-1)}(z)$ for any $j\geq 1$, and denote
by $m\geq 1$ the largest integer such that $1$, $f_{1}(z)$, …, $f_{m}(z)$ are
linearly independent over $\mathbb{K}(z)$. Then $f_{m+1}(z)=f^{(m)}(z)$ is a
$\mathbb{K}(z)$-linear combination of these functions: $f$ is solution of a
inhomogeneous linear differential equation of order $m$. This equation has
minimal order amongst all inhomogeneous linear differential equations
satisfied by $f$. Note that we consider homogeneous equations to be special
cases of inhomogeneous equations for which the constant term is equal to 0,
and it may happen that an inhomogeneous equation of minimal order for one of
its solutions is in fact homogeneous.

######  Proposition 1.

The point 0 is either a regular point, or a regular singularity, of any
inhomogeneous linear differential equation of minimal order satisfied by a
transcendental $E$-function.

André has proved this result for the homogeneous linear differential equation
of minimal order (and even a more precise one, see [2, Théorème de pureté, p.
706]). In this proposition, it is understood that an inhomogeneous equation is
regular or regular singular at $0$ if the companion differential system
$Y^{\prime}=AY$ with solution ${}^{t}(1,f,f^{\prime},\ldots,f^{(m-1)})$ is.

###### Proof.

Let $f$ be a transcendental $E$-function solution of a non-trivial homogeneous
differential equation $Ly=0$ of minimal order $r\geq 1$, where
$L\in\mathbb{C}(z)[d/dz]$. A minimality argument implies that a non-trivial
minimal inhomogeneous equation satisfied by $f$ has order $r$ or $r-1$ (see
[1, §4]).

In the former case, this inhomogeneous equation is then simply equal to $L$ up
a non-zero rational function factor, and the claim follows because, by André’s
above-mentioned theorem, $0$ is a regular point or a regular singularity of
$L$.

We now deal with the case of order $r-1$ which is more complicated. In this
situation, there exists $R\in\mathbb{C}(z)\setminus\\{0\\}$ such that
$L^{*}R=0$, where $L^{*}$ is the adjoint of $L$ (222Given a differential
operator $L=\sum_{j=0}^{r}q_{j}(d/dz)^{j}\in\mathbb{C}(z)[d/dz]$, its adjoint
$L^{*}\in\mathbb{C}(z)[d/dz]$ is defined by
$L^{*}y=\sum_{j=0}^{r}(-1)^{j}(q_{j}y)^{(j)}$; see [27, p. 38].) and a minimal
inhomogeneous equation satisfied by $f$ is of the form

$\sum_{j=0}^{r-1}p_{j}(z)f^{(j)}(z)=c$ (2.1)

where

$\frac{d}{dz}\left(\sum_{j=0}^{r-1}p_{j}(z)\frac{d^{j}}{dz^{j}}\right)=RL.$
(2.2)

The rational function $R$ can be explicitly determined from $L$, and then Eq.
(2.2) enables to determine suitable $p_{j}\in\mathbb{C}(z)$ from $R$ and $L$
(first $p_{r-1}$, then $p_{r-2}$, etc). Finally, the constant $c$ is computed
by determining the constant term in the Laurent series expansion of the left-
hand side of (2.1). See [9, §2.4] for more details. By minimality of $L$,
$c\neq 0$ and without loss of generality we can assume that $c=1$. Since $f$
is transcendental, we have $r\geq 2$. Let $f_{2},\ldots,f_{r}$ be other local
solutions of $Ly=0$ at $z=0$ such that $f_{1}:=f,f_{2},\ldots,f_{r}$ make up a
$\mathbb{C}$-basis of the vector space of solutions of $Ly=0$: by André’s
theorem, each $f_{k}$ is in the Nilsson class at $z=0$ because $0$ is at worst
a regular singularity of $L$. From Eq. (2.2), we observe that

$\frac{d}{dz}\left(\sum_{j=0}^{r-1}p_{j}(z)f_{k}^{(j)}(z)\right)=R(z)Lf_{k}(z)=0\quad\mbox{for
any }k\in\llbracket 1,r\rrbracket$

so that

$\sum_{j=0}^{r-1}p_{j}(z)f_{k}^{(j)}(z)=c_{k}$

for some $c_{k}\in\mathbb{C}$ (and $c_{1}=1$). Up to reordering the basis
$f_{1},f_{2},\ldots,f_{r}$ and multiplying each $f_{k}$ by a non-zero
constant, we can and shall assume without loss of generality that
$f_{1},\ldots,f_{s}$ are such that $c_{k}=1$ for $k=1,\ldots,s$ and
$f_{s+1},\ldots,f_{r}$ are such that $c_{k}=0$ for $k=s+1,\ldots,r$, for some
$s\in\llbracket 1,r\rrbracket$. We now write the inhomogeneous equation
$\sum_{j=0}^{r-1}p_{j}(z)y^{(j)}(z)=1$ satisfied by $f$ as a companion
differential system $Y^{\prime}=AY$ where $A\in M_{r}(\mathbb{C}(z))$, with
the vector solution ${}^{t}(1,f,\ldots,f^{(r-2)})$. From what precedes, it
turns out that in fact

$U:=\left(\begin{matrix}1&\cdots&1&0&\cdots&0\\\
f_{1}&\cdots&f_{s}&f_{s+1}&\cdots&f_{r}\\\
\vdots&\cdots&\vdots&\cdots&\vdots&\cdots\\\
f_{1}^{(r-2)}&\cdots&f_{s}^{(r-2)}&f_{s+1}^{(r-2)}&\cdots&f_{r}^{(r-2)}\end{matrix}\right)$

is a fundamental matrix solution of $Y^{\prime}=AY$. (By definition, if $s=r$,
there are only 1’s on the first line of $U$). Indeed, the columns of $U$ are
solutions of $Y^{\prime}=AY$ and they are $\mathbb{C}$-linearly independent
because on the second line $f_{1},\ldots,f_{r}$ are $\mathbb{C}$-linearly
independent, so that $U$ is invertible. Since the entries of $U$ are in the
Nilsson class at $z=0$, it follows from [31, p. 81] that $0$ is a regular
point or a regular singular point of $Y^{\prime}=AY$. ∎

###  2.2 A version of Beukers’ desingularization lemma

Beukers’ desingularization lemma [8, Theorem 1.5] is very useful when dealing
with $E$-functions, since it enables one to get rid of all non-zero
singularities of the underlying (homogeneous) differential equation. In this
section we state and prove a non-homogeneous version of this result that
incorporates several additional features: the coefficients lie in a fixed
number field (as in [16, Proposition 2]), and two useful properties are
preserved (the value at 1 of the first $E$-function, and the property that $0$
is a regular singularity). These properties will be very important in the
proof of Theorem 1.

######  Proposition 2.

Let $g_{1}$, …, $g_{m}$ be $E$-functions with coefficients in a number field
$\mathbb{K}$, such that $1$, $g_{1}$, …, $g_{m}$ are linearly independent over
$\mathbb{C}(z)$. Assume that the column vector $(1,g_{1},\ldots,g_{m})$ is
solution of a first-order differential system $Y^{\prime}=SY$ with $S\in
M_{m+1}(\mathbb{K}(z))$.

Then there exist $E$-functions $f_{1}$, …, $f_{m}$ with coefficients in
$\mathbb{K}$ such that:

  * •

The functions 1, $f_{1}$, …, $f_{m}$ are linearly independent over
$\mathbb{C}(z)$.

  * •

There exist polynomials $Q_{j,l}(z)\in\mathbb{K}[z]$ such that
$g_{j}(z)=\sum_{l=0}^{m}Q_{j,l}(z)f_{l}(z)$ for any $j\in\llbracket
1,m\rrbracket$, where we let $f_{0}(z)=1$.

  * •

The column vector $(1,f_{1},\ldots,f_{m})$ is solution of a first-order
differential system $Y^{\prime}=\widetilde{S}Y$ with $\widetilde{S}\in
M_{m+1}(\mathbb{K}[z,1/z])$.

  * •

If $g_{1}(1)$ is transcendental then $g_{1}(1)=f_{1}(1)$.

  * •

If 0 is a regular singularity of the system $Y^{\prime}=SY$, then it is also a
regular singularity of the system $Y^{\prime}=\widetilde{S}Y$.

###### Proof.

It follows closely that of Beukers, or more precisely the version over a
number field $\mathbb{K}$ given in [16, Proposition 2] (see also [9]). This
approach consists in finitely many steps. At each step, one obtains a
$\mathbb{K}$-linear combination of $1$, $g_{1}$, …, $g_{m}$ that vanishes at
some non-zero algebraic point $\alpha$; then one replaces one of the functions
by this linear combination, and divides by $z-\alpha$ if $\alpha\in\mathbb{K}$
(by the minimal polynomial of $\alpha$ over $\mathbb{K}$ in the general case).
The point is that the above-mentioned linear combination is never just the
function 1, since 1 does not vanish at $\alpha$. Therefore the function 1 can
be preserved at each step. Moreover it is clear from the proof that 0 being a
regular singularity holds throughout the procedure. This proves Proposition 2,
except for the property $g_{1}(1)=f_{1}(1)$. However, if
$g_{1}(1)=\sum_{l=0}^{m}Q_{1,l}(1)f_{l}(1)$ is transcendental then there
exists $l_{0}\in\llbracket 1,m\rrbracket$ such that $Q_{1,l_{0}}(1)\neq 0$.
Replacing $f_{l_{0}}(z)$ with $\sum_{l=0}^{m}Q_{1,l}(1)f_{l}(z)$ does not
change the other properties of the functions $f_{1}$, …, $f_{m}$, and provides
$g_{1}(1)=f_{l_{0}}(1)$. Up to a permutation of $f_{1}$, …, $f_{m}$ this
concludes the proof of Proposition 2. ∎

###  2.3 A multiplicity estimate with possibly zero remainders

In this section we state the multiplicity estimate our proof relies on, namely
Theorem A. It is a special case of [15, Theorem 3] because the vanishing
orders are considered at only one point (namely 0), and the polynomials are
evaluated at some $\alpha\neq 0$ which is not a singularity of the
differential system $Y^{\prime}=AY$.

Let $q\geq 1$, $A\in M_{q}(\mathbb{C}(z))$, $n\geq 0$, and
$P_{1},\ldots,P_{q}\in\mathbb{C}[z]$ be such that $\deg P_{i}\leq n$ for any
$i$. We identify tuples in $\mathbb{C}^{q}$ with column matrices in
$M_{q,1}(\mathbb{C})$. Then with any solution $Y=(y_{1},\ldots,y_{q})$ of the
differential system $Y^{\prime}=AY$ is associated a remainder $R(Y)$ defined
by

$R(Y)(z)=\sum_{i=1}^{q}P_{i}(z)y_{i}(z).$

The derivatives of such a remainder can be written as [29, Chapter 3, §4]

$R(Y)^{(k-1)}(z)=\sum_{i=1}^{q}P_{k,i}(z)y_{i}(z),$ (2.3)

where the rational functions $P_{k,i}\in\mathbb{C}(z)$ are defined for $k\geq
1$ and $1\leq i\leq q$ by

$\left(\begin{array}[]{c}P_{k,1}\\\ \vdots\\\
P_{k,q}\end{array}\right)=\left(\frac{{\rm d}}{{\rm
d}z}+\,{}^{t}A\right)^{k-1}\left(\begin{array}[]{c}P_{1}\\\ \vdots\\\
P_{q}\end{array}\right).$ (2.4)

We consider the matrix $M(z)=(P_{k,i}(z))_{1\leq i,k\leq q}\in
M_{q}(\mathbb{C}(z))$; obviously the poles of the coefficients $P_{k,i}$ of
$M$ are amongst those of $A$.

The main new feature of the multiplicity estimate proved in [15] is that it
takes into account the possibility that $R(Y)(z)$ is identically zero for some
non-zero solutions $Y(z)$ of the differential system $Y^{\prime}=AY$. To state
this result, we denote by $\varrho\geq 0$ the dimension of the
$\mathbb{C}$-vector space of solutions $Y$ such that $R(Y)(z)$ is identically
zero.

######  Theorem A.

There exists a positive constant $c_{1}$, which depends only on $A$, with the
following property. Let $(Y_{j})_{j\in J}$ be a family of solutions of
$Y^{\prime}=AY$ such that the functions $R(Y_{j})$, $j\in J$, are
$\mathbb{C}$-linearly independent and belong to the Nilsson class at $0$.
Assume that

$\sum_{j\in J}{\rm ord}_{0}(R(Y_{j}))\geq(n+1)(q-\varrho)-\tau$ (2.5)

for some $\tau\in\mathbb{Z}$. Then:

  * •

We have $\tau\geq-c_{1}$.

  * •

If $0\leq\tau\leq n-c_{1}$ then for any $\alpha\in\mathbb{C}^{*}$ which is not
a singularity of the differential system $Y^{\prime}=AY$, the matrix
$(P_{k,i}(\alpha))_{1\leq i\leq q,1\leq k<\tau+c_{1}}\in
M_{q,\tau+c_{1}-1}(\mathbb{C})$ has rank at least $q-\varrho$.

In this setting, under the assumptions of $(ii)$, the matrix
$(P_{k,i}(\alpha))$ has rank equal to $q-\varrho$. Indeed, there exist
$\varrho$ $\mathbb{C}$-linearly independent solutions $Y$ such that $R(Y)$ is
identically zero. For each of them, we have
$\sum_{i=1}^{q}P_{k,i}(\alpha)y_{i}(\alpha)=R(Y)^{k-1}(\alpha)=0$ for any
$k\geq 1$: since $\alpha$ is not a singularity, this provides $\varrho$
linearly independent linear relations between the columns of the matrix
$(P_{k,i}(\alpha))$.

We remark that Theorem A would not hold if the linear independence assumption
were on the $Y_{j}$ rather than on the $R(Y_{j})$. Indeed, for instance if
$R(Y_{j})$ were identically zero for some $j\in J$, then ${\rm
ord}_{0}(R(Y_{j}))$ would be infinite and Eq. (2.5) would hold for any
$\tau\in\mathbb{Z}$. Moreover, solutions $Y$ such that $R(Y)$ is identically
zero are taken advantage of in Theorem A: each such solution (in a linearly
independent family) provides the same benefit in Eq. (2.5) as an additional
function $Y_{j}$ such that $R(Y_{j})$ would vanish to order $n+1$. This
property will be used in a crucial way in the proof of Theorem 1 (see §3.6).

######  Remark 1.

Following the proof [15] of Theorem A and using the results of [7] shows that
$c_{1}$ can be effectively computed in terms of $A$.

##  3 Proof of the main result

This section is devoted to the proof of Theorem 1, using a technical result
that will proved later in §4.

In §3.1 we introduce the notation and setting of the proof, applying the
results of §§2.1 and 2.2. Then in §3.2 we give construction and properties of
graded Padé approximants, including the zero estimate (namely Proposition 3).
Admitting this result, we prove Theorem 1 stated in the introduction in §3.3,
and its generalization to an $E$-function in Siegel’s orginal sense, with
arbitrary Taylor coefficients in $\overline{\mathbb{Q}}$, in §3.4 (namely
Theorem LABEL:theo:main2 stated where).

The rest of the paper is devoted to the proof of Proposition 3, carried out in
§3.6 using the differential system considered in §3.5 and the technical result
proved in §4.

###  3.1 Setting, notations and parameters

In this section we describe the setting of the proof of Theorem 1. We use the
results of §§2.1 and 2.2 to obtain a family of linearly independent
$E$-functions, solution of a first order differential system with no non-zero
finite singularity. Then we introduce (essentially) the same notation and
parameters as in Zudilin’s proof.

To prove Theorem 1 we start with an $E$-function $g(z)$ with coefficients in
$\mathbb{Q}$, and $r\in\mathbb{Q}^{*}$. We assume that $g(r)$ is irrational;
then $g(r)$ is transcendental by [14, Theorem 4]. Considering $g(rz)$ instead
of $g$, we may assume that $r=1$.

As in §2.1 we consider a (possibly) inhomogeneous linear differential equation
of minimal order satisfied by the transcendental $E$-function $g$. Proposition
1 asserts that 0 is (at worst) a regular singularity of this equation. Viewing
this equation as a differential system of order one satisfied by the column
vector $(1,g,g^{\prime},\ldots,g^{(m-1)})$, we apply Proposition 2 and obtain
$E$-functions 1, $f_{1}$, …, $f_{m}$ with coefficients in $\mathbb{Q}$,
linearly independent over $\mathbb{C}(z)$, such that $f_{1}(1)=g(1)$ is the
number we are interested in to prove Theorem 1. The important point is that
$(f_{1},\ldots,f_{m})$ is a solution of a first-order inhomogeneous
differential system

$f^{\prime}_{l}(z)=S_{l,0}(z)+\sum_{j=1}^{m}S_{l,j}(z)f_{j}(z)\mbox{ for any
}l\in\llbracket 1,m\rrbracket$ (3.1)

with $S_{l,j}(z)\in\mathbb{Q}[z,1/z]$: the only possible finite singularity of
this system is zero, and it is regular (in case it is a singularity).

We consider multi-indices $\kappa\in\mathbb{N}^{m}$ and sums over such multi-
indices. In such a sum, whenever an index $\kappa$ belongs to $\mathbb{Z}^{m}$
but not to $\mathbb{N}^{m}$ (i.e., has at least one negative component), the
term corresponding to this index will be considered as 0. For
$\kappa=(\kappa_{1},\ldots,\kappa_{m})\in\mathbb{N}^{m}$, we write
$|\kappa|=\kappa_{1}+\ldots+\kappa_{m}$.

We denote by $(e_{1},\ldots,e_{m})$ the canonical basis of $\mathbb{Z}^{m}$,
i.e. $e_{i}=(0,\ldots,0,1,0,\ldots,0)$ where the $i$-th coordinate is equal to
1. We let

$\Omega=\\{\kappa\in\mathbb{N}^{m},\,N-1\leq|\kappa|\leq
N\\},\quad\Theta=\\{\kappa\in\mathbb{N}^{m},\,|\kappa|=N\\},$

and

$\omega={\rm
Card}\,\Omega=\binom{N+m-2}{m-1}+\binom{N+m-1}{m-1},\quad\theta={\rm
Card}\,\Theta=\binom{N+m-1}{m-1}.$

We remark, for future reference, that

$\frac{\omega}{\theta}=2-\frac{m-1}{N+m-1}=1+\frac{N}{N+m-1}.$

We also fix a bijective map $\llbracket 1,\omega\rrbracket\to\Omega$ so that
indices $\kappa\in\Omega$ can be seen as integers between 1 and $\omega$; for
instance a family $(x_{\omega})_{\omega\in\Omega}\in\mathbb{C}^{\Omega}$ can
also be seen as a tuple in $\mathbb{C}^{\omega}$, or as a column matrix in
$M_{\omega,1}(\mathbb{C})$.

Let us introduce now the parameters that will be used in the construction. Let
$N$ be sufficiently large with respect to $f_{1}$, …, $f_{m}$, and let
$\eta>0$ be a real number such that

$\eta\leq\frac{1}{3(N+m-1)}.$ (3.2)

This number $\eta$ plays the role of the real denoted by $\varepsilon$ in
[33]. Let $M$ be sufficiently large with respect to $f_{1}$, …, $f_{m}$, $N$,
and $\eta$; consider

$K=\Big{\lfloor}\frac{(\omega-\eta)M}{\theta}\Big{\rfloor}.$ (3.3)

######  Remark 2.

These parameters are the same as the ones used by Zudilin, except that he
assumes that equality holds in Eq. (3.2), and gives an explicit value for $M$
in terms of $N$. The reason for this difference is that we have not computed
explicitly the constant $C_{4}$ in Proposition 3 (which depends on $N$). This
is useless for our purpose but it could be done (see Remark 3); it would lead
to explicit values of $\eta$ and $M$, and then to an explicit value of the
constant $c$ in Theorem 1.

###  3.2 Construction and properties of graded Padé approximants

In this section, we state the construction and properties of graded Padé
approximants, sketched by Chudnovsky [10] and proved in detail by Zudilin
[33]. Apart from the zero estimate (namely Proposition 3 below), the results
are exactly the same as in Zudilin’s paper.

To motivate this construction, let us explain it with different notations in
the case $m=2$. We shall construct polynomials
$A_{0},\ldots,A_{N},B_{0},\ldots,B_{N-1}$ such that
$A_{i}(z)+B_{i-1}(z)f_{1}(z)+B_{i}(z)f_{2}(z)$ vanishes with high multiplicity
at 0, for any $i\in\llbracket 0,N\rrbracket$. The point here is that $B_{-1}$
and $B_{N}$ are considered to be identically zero, so that for $i=N$ the
function $A_{N}(z)+B_{N-1}(z)f_{1}(z)$ vanishes to high order at 0, and is
therefore presumably small at $z=1$.

Let us come back to our general setting now. Recall that the parameters are
given by Eqs. (3.2) and (3.3); they are the same as in [33], except that
$\eta$ and $M$ are not fixed in terms of $N$. The following construction is
exactly [33, Lemma 1.1].

######  Lemma 1.

There exist polynomials $P_{\kappa}(z)$ of degree less than $M$, for
$\kappa\in\Omega$, not all zero, such that

${\rm ord}_{0}\Big{(}P_{\kappa}(z)+\sum_{j=1}^{m}P_{\kappa-
e_{j}}(z)f_{j}(z)\Big{)}\geq K\quad\mbox{ for any }\kappa\in\Theta$

and

$\pi_{\kappa,\nu}\in\mathbb{Z},\quad|\pi_{\kappa,\nu}|\leq C_{0}^{\omega
M/\eta}$

for any $\kappa\in\Omega$ and any $\nu\in\llbracket 0,M-1\rrbracket$, where
the coefficients $\pi_{\kappa,\nu}$ are defined by

$P_{\kappa}(z)=\sum_{\nu=0}^{M-1}\frac{\pi_{\kappa,\nu}}{\nu!}z^{\nu}\quad\mbox{
for any }\kappa\in\Omega.$

Here and below, we denote by $C_{0}$, $C_{1}$, …, $C_{4}$ positive constants
that depend only on $f_{1}$, …, $f_{m}$ (except that $C_{4}$ depends also on
$N$).

Now recall that all coefficients $S_{l,j}(z)$ of the differential system (3.1)
belong to $\mathbb{Q}[z,1/z]$, i.e. that this system has no non-zero finite
singularity. Therefore denoting by $T(z)$ the least common denominator of the
$S_{l,j}(z)$, we have

$T(z)=z^{i}\mbox{ for some $i\in\mathbb{N}$, and
}T(z)S_{l,j}(z)\in\mathbb{Q}[z]\mbox{ for any }l,j.$ (3.4)

As in [33, Eq. (1.8)] we define recursively polynomials $P^{[k]}_{\kappa}(z)$,
for $k\geq 1$ and $\kappa\in\Omega$, by letting
$P^{[1]}_{\kappa}(z)=P_{\kappa}(z)$ and for any $k\geq 1$ and any
$\kappa\in\Omega$,

$\displaystyle P^{[k+1]}_{\kappa}(z)$ $\displaystyle=$ $\displaystyle
T(z)\Big{(}\frac{{\rm d}}{{\rm
d}z}P^{[k]}_{\kappa}(z)+(|\kappa|+1-N)\sum_{l=1}^{m}S_{l,0}(z)P^{[k]}_{\kappa-
e_{l}}(z)$
$\displaystyle\quad\quad\quad\quad\quad\quad-\sum_{l=1}^{m}\sum_{j=1}^{m}(\kappa_{j}-\delta_{l,j}+1)S_{l,j}(z)P^{[k]}_{\kappa-
e_{l}+e_{j}}(z)\Big{)}.$

We recall that $\delta_{l,j}$ is Kronecker’s symbol, and whenever $\kappa-
e_{l}\in\mathbb{Z}^{m}$ (resp. $\kappa-e_{l}+e_{j}$) has a negative
coefficient, the corresponding term is omitted. The only difference with [33,
Eq. (1.8)] is a shift in the index $k$: our $P^{[k]}_{\kappa}(z)$ is denoted
by $P^{[k-1]}_{\kappa}(z)$ in [33]. The connection of this definition of
$P^{[k]}_{\kappa}(z)$ with a differential system will be explained in §3.5
below.

The following result is part $(a)$ of [33, Lemma 1.3], with coefficients
$\pi_{k,\kappa,\nu}$ defined by

$P^{[k]}_{\kappa}(z)=\sum_{\nu=0}^{M+t(k-1)-1}\frac{\pi_{k,\kappa,\nu}}{\nu!}z^{\nu}\mbox{
for any }\kappa\in\Omega\mbox{ and any }k\geq 1,$

and

$t=\max\Big{(}\deg T(z),\max_{1\leq l\leq m}\max_{0\leq j\leq
m}\deg(T(z)S_{l,j}(z))\Big{)}.$

######  Lemma 2.

For any $\kappa\in\Omega$, any $k\geq 1$ and any $\nu\in\llbracket
0,M+t(k-1)-1\rrbracket$, we have $\pi_{k,\kappa,\nu}\in\mathbb{Z}$, and

$|\pi_{k,\kappa,\nu}|\leq C_{0}^{\omega M/\eta}M^{C_{2}\eta M}\quad\mbox{ if
}k<C_{1}\eta M.$

The following result is the special case $l^{*}=1$ and $\alpha=1$ of part
$(b)$ of [33, Lemma 1.3].

######  Lemma 3.

For any $k\geq 1$ such that $k<C_{1}\eta M$, we have

$\Big{|}P^{[k]}_{(N,0,0,\ldots,0)}(1)+P^{[k]}_{(N-1,0,0,\ldots,0)}(1)f_{1}(1)\Big{|}\leq
C_{0}^{\omega M/\eta}M^{C_{2}\eta M}C_{3}^{M}M^{-K}.$

The main point of the present paper is the following result. It is proved in
[33, Lemma 3.5] with an explicit value of $C_{4}$, namely $\omega$, under the
assumption that $f_{1}$, …, $f_{m}$ are algebraically independent.

######  Proposition 3.

There exists a constant $C_{4}$, which depends on $f_{1}$, …, $f_{m}$ and on
$N$ (but not on $M$ or $\eta$), such that the matrix
$(P^{[k]}_{\kappa}(1))_{\kappa\in\Omega,1\leq k\leq\lfloor 2\eta
M\rfloor+C_{4}}$ has rank $\omega$.

######  Remark 3.

We shall prove that $C_{4}$ can (in principle) be computed effectively in
terms of $f_{1}$, …, $f_{m}$ and $N$.

Proposition 3 will be proved in §3.6, using the differential system considered
in §3.5 and Theorem 2 proved in §4. In the next two sections, we admit
Proposition 3 and deduce from it the results announced in the introduction.

###  3.3 Proof of Theorem 1 for $E$-functions in the strict sense

Let us now prove Theorem 1 for $E$-functions in the strict sense, following
[33, pp. 581–583] but without explicit expressions for $\eta$ and $M$.

Starting wih an $E$-function $g\in\mathbb{Q}[[z]]$ and $r\in\mathbb{Q}$ such
that $g(r)\not\in\mathbb{Q}$, we construct $f_{1}$, …, $f_{m}$ as in §3.1 so
that $f_{1}(1)=g(r)$. Let $\varepsilon>0$; we may assume that $\varepsilon$ is
sufficiently small in terms of $f_{1}$, …, $f_{m}$. We choose $N=\lfloor
m/\varepsilon\rfloor+1$ so that

$N\geq m/\varepsilon$

and $N$ can be made sufficiently large in terms of $f_{1}$, …, $f_{m}$. We
recall that $\omega/\theta=1+\frac{N}{N+m-1}$, so that

$\Big{(}1+\frac{m}{N}\Big{)}\Big{(}1-\frac{\omega}{\theta}\Big{)}<-1.$

Using this bound and the fact that $C_{2}$, $t$, $\omega$ and $\theta$ depend
only on $N$ and $f_{1}$, …, $f_{m}$, we may choose $\eta>0$ sufficiently small
(with respect to $N$, $f_{1}$, …, $f_{m}$) so that Eq. (3.2) holds and

$\Big{(}1+\frac{m}{N}\Big{)}\Big{(}1+2t\eta+C_{2}\eta-\frac{\omega-\eta}{\theta}\Big{)}<-(1+2t\eta+C_{2}\eta).$
(3.6)

In what follows, we assume $M$ to be sufficiently large in terms on $\eta$,
$N$, $f_{1}$, …, $f_{m}$; we shall denote by $C_{j}(\eta,N)$ positive
constants that depend on $\eta$, $N$, $f_{1}$, …, $f_{m}$.

Since the matrix $(P^{[k]}_{\kappa}(1))_{\kappa\in\Omega,1\leq k\leq\lfloor
2\eta M\rfloor+C_{4}}$ of Proposition 3 has rank $\omega$, the submatrix
consisting in the columns indexed by $\kappa=(N,0,\ldots,0)$ and
$\kappa=(N-1,0,\ldots,0)$ has rank 2. This provides positive integers
$k_{1},k_{2}<2\eta M+C_{4}$ such that

$\det\left(\begin{matrix}P^{[k_{1}]}_{(N,0,0,\ldots,0)}(1)&P^{[k_{1}]}_{(N-1,0,0,\ldots,0)}(1)\\\
P^{[k_{2}]}_{(N,0,0,\ldots,0)}(1)&P^{[k_{2}]}_{(N-1,0,0,\ldots,0)}(1)\end{matrix}\right)\neq
0.$ (3.7)

For any $j\in\\{1,2\\}$, let

$p_{j}=-(M+tk_{j})!\,P^{[k_{j}]}_{(N,0,0,\ldots,0)}(1)\quad\mbox{ and }\quad
q_{j}=(M+tk_{j})!\,P^{[k_{j}]}_{(N-1,0,0,\ldots,0)}(1).$

Lemma 2 yields $p_{j},q_{j}\in\mathbb{Z}$, and also

$\displaystyle|q_{j}|$ $\displaystyle\leq$ $\displaystyle(M+t(2\eta
M+C_{4}))!\,eC_{0}^{\omega M/\eta}M^{C_{2}\eta M}$ (3.8) $\displaystyle\leq$
$\displaystyle(M(1+2t\eta)+tC_{4}))^{tC_{4}}\,(M(1+2t\eta))!\,eC_{0}^{\omega
M/\eta}M^{C_{2}\eta M}$ $\displaystyle\leq$ $\displaystyle
C_{5}(\eta,N)^{M}\,M^{(1+2t\eta+C_{2}\eta)M}$

provided $M\geq C_{6}(\eta,N)$, and Lemma 3 yields in the same way (using Eq.
(3.3))

$\displaystyle|q_{j}f_{1}(1)-p_{j}|$ $\displaystyle\leq$
$\displaystyle(M+t(2\eta M+C_{4}))!\,C_{0}^{\omega M/\eta}M^{C_{2}\eta
M}C_{3}^{M}M^{-K}$ (3.9) $\displaystyle\leq$ $\displaystyle
C_{7}(\eta,N)^{M}\,M^{(1+2t\eta+C_{2}\eta-\frac{\omega-\eta}{\theta})M}$

if $M\geq C_{8}(\eta,N)$.

Now let $p\in\mathbb{Z}$ and $q\in\mathbb{N}^{*}$; upon changing the constant
$c$ in Theorem 1 (since $f_{1}(1)\not\in\mathbb{Q}$), we may assume that $|p|$
and $q$ are sufficiently large (with respect to $\eta$, $N$, $f_{1}$, …,
$f_{m}$, since these quantities have been chosen in terms of $f_{1}(1)$ and
$\varepsilon$ only). We choose for $M$ the least integer such that

$C_{7}(\eta,N)^{M}\,M^{(1+2t\eta+C_{2}\eta-\frac{\omega-\eta}{\theta})M}\leq\frac{1}{2q}.$
(3.10)

This integer exists because we have assumed $\eta>0$ sufficiently small in
terms of $N$, $f_{1}$, …, $f_{m}$, so that
$1+2t\eta+C_{2}\eta-\frac{\omega-\eta}{\theta}<0$; moreover $M$ can be made
large enough (in terms of $\eta$, $N$, $f_{1}$, …, $f_{m}$) by assuming that
$q$ is. Then Eq. (3.9) yields

$q\,|q_{j}f_{1}(1)-p_{j}|\leq 1/2\mbox{ for any }j\in\\{1,2\\}.$

Now Eq. (3.7) yields $\det\left(\begin{matrix}p_{1}&q_{1}\\\
p_{2}&q_{2}\end{matrix}\right)\neq 0$, so that $(p,q)$ is non-collinear to at
least one of the $(p_{j},q_{j})$, $j\in\\{1,2\\}$. For this index $j$ we have

$\det\left(\begin{matrix}p&p_{j}\\\
q&q_{j}\end{matrix}\right)\in\mathbb{Z}\setminus\\{0\\}.$ (3.11)

This determinant is also equal to

$\det\left(\begin{matrix}p-qf_{1}(1)&p_{j}-q_{j}f_{1}(1)\\\
q&q_{j}\end{matrix}\right),$

so that

$\displaystyle|q_{j}|\,|qf_{1}(1)-p|$
$\displaystyle=\Big{|}\det\left(\begin{matrix}p&p_{j}\\\
q&q_{j}\end{matrix}\right)-q\,(q_{j}f_{1}(1)-p_{j})\Big{|}$ $\displaystyle\geq
1-q\,|q_{j}f_{1}(1)-p_{j}|\geq 1-1/2=1/2$

and, using Eqs. (3.8) and (3.6):

$\displaystyle|qf_{1}(1)-p|$
$\displaystyle\geq\frac{1}{2|q_{j}|}\geq\frac{1}{2}C_{5}(\eta,N)^{-M}\,M^{-(1+2t\eta+C_{2}\eta)M}$
$\displaystyle\geq\Big{[}C_{7}(\eta,N)^{M-1}\,(M-1)^{(1+2t\eta+C_{2}\eta-\frac{\omega-\eta}{\theta})(M-1)}\Big{]}^{1+m/N}$

provided $M\geq C_{9}(\eta,N)$. Since $M$ is the least integer such that Eq.
(3.10) holds, we deduce that

$|qf_{1}(1)-p|>\Big{(}\frac{1}{2q}\Big{)}^{1+m/N}.$

Since $m/N\leq\varepsilon$ this concludes the proof of Theorem 1 for
$E$-functions in the strict sense.

###  3.4 Proof of Theorem 1 for $E$-functions in Siegel’s sense

In this section, we explain the changes that must be done to obtain Theorem 1
to any $E$-function $f$ in Siegel’s original sense.

$\bullet$ Firstly, in the proof of Theorem 1 for $E$-functions in the strict
sense, we use various results of André and Beukers that they have proved only
for $E$-functions in the strict sense (using the theory of $E$-operators due
to the former). Since then, all these results have been proved to hold
verbatim for $E$-functions in Siegel’s sense by Lepetit [24], completing the
results already given in [3, pp. 746–747].

$\bullet$ Secondly, in §3.2 we use verbatim Zudilin’s estimates that he has
also proved only for $E$-functions in $\mathbb{Q}[[z]]$ in the strict sense.
Let us mention the changes that must be made to his lemmas to deal with
Siegel’s $E$-functions in $\mathbb{Q}[[z]]$. We recall that the archimedean
and non-archimedean bounds on the Taylor coefficients of Siegel’s
$E$-functions are of the form “for all $\varepsilon^{\prime}>0$, $\ldots\leq
n^{\varepsilon^{\prime}n}$ for all $n\geq N(\varepsilon^{\prime})$”. In Lemma
1, this changes the quantity $C_{0}^{\omega M/\eta}$ by
$M^{\omega\varepsilon^{\prime}M/\eta}$, where $\varepsilon^{\prime}>0$ is
fixed and independent of the other parameters but arbitrarily small, and
$M\geq M_{0}(\varepsilon^{\prime})$. The same remark applies in Lemma 2, where
$C_{0}^{\omega M/\eta}M^{C_{2}\eta M}$ becomes
$M^{\omega\varepsilon^{\prime}M/\eta+C_{2}\eta M}$, and in Lemma 3, where
$M^{-K}$ reads $M^{-K(1-\varepsilon^{\prime})}$ and the constant $C_{3}$ is
also possibly changed but it still does not depend on $M$. With these
estimates, we conclude the proof as that of Theorem 1 for $E$-functions in the
strict sense because $\varepsilon^{\prime}$ can be taken arbitrarily small
provided $M$ is assumed to be large enough, which can be assumed as in §3.3.

The rest of the present paper is devoted to a proof of Proposition 3, which
has been admitted in §§3.3 and 3.4.

###  3.5 Differential system

In this section we define a matrix $A\in M_{\omega}(\mathbb{Q}(z))$ and
consider the differential system $Y^{\prime}=AY$, of which solutions will be
constructed in Proposition 4. As stated in §3.1 a bijective map $\llbracket
1,\omega\rrbracket\to\Omega$ is fixed, so that a solution $Y$ is a vector
$(y_{\kappa}(z))$ indexed by $\kappa\in\Omega$. Here and below, we identify
tuples in $\mathbb{C}^{q}$ with column matrices in $M_{q,1}(\mathbb{C})$.

We shall also relate the notation $P^{[k]}_{\kappa}$ of §3.2 to the
$P_{k,\kappa}$ of §2.3; in what follows we will use mostly the notation of
§2.3, including

$R(Y)(z)=\sum_{\kappa\in\Omega}P_{\kappa}(z)y_{\kappa}(z)$ (3.12)

when $Y=(y_{\kappa}(z))$ is a solution of the differential system
$Y^{\prime}=AY$.

The matrix $A=(A_{\lambda,\kappa}(z))_{\lambda,\kappa\in\Omega}\in
M_{\omega}(\mathbb{Q}(z))$ that we consider is defined (in terms of the
coefficients $S_{l,j}(z)$ of the differential system (3.1)) by

$A_{\lambda,\kappa}(z)=\begin{cases}-\lambda_{j}S_{l,j}(z)\;\mbox{ if
$\kappa=\lambda-e_{j}+e_{l}$ for some $j,l\in\llbracket 1,m\rrbracket$ with
$j\neq l$,}\\\ -\sum_{j=1}^{m}\lambda_{j}S_{j,j}(z)\;\mbox{ if
$\kappa=\lambda$,}\\\ S_{j,0}(z)\;\mbox{ if $\kappa=\lambda+e_{j}$ and
$|\lambda|=N-1$,}\\\ 0\;\mbox{ otherwise,}\end{cases}$ (3.13)

as in [33, Eq. (3.2)]. We recall from §3.1 that all rational functions
$S_{l,j}(z)$, and therefore all coefficients of $A$, belong to
$\mathbb{Q}[z,1/z]$. With this definition, Eq. (3.2) reads

$\left(\begin{array}[]{c}P^{[k+1]}_{1}(z)\\\ \vdots\\\
P^{[k+1]}_{\omega}(z)\end{array}\right)=T(z)\left(\frac{{\rm d}}{{\rm
d}z}+\,{}^{t}A(z)\right)\left(\begin{array}[]{c}P^{[k]}_{1}(z)\\\ \vdots\\\
P^{[k]}_{\omega}(z)\end{array}\right).$

Except for the multiplicative factor $T(z)$ (used to ensure that all
$P^{[k]}_{\kappa}(z)$ are polynomials), this is the same recurrence relation
as the one used in §2.3 to define the rational functions $P_{k,\kappa}(z)$
(see Eq. (2.4)); notice that $P_{k,\kappa}(z)\in\mathbb{Q}[z,1/z]$ since $A\in
M_{\omega}(\mathbb{Q}[z,1/z])$. Using the fact that
$P_{1,\kappa}=P^{[1]}_{\kappa}=P_{\kappa}$ by definition, we obtain by
induction that

$\left(\begin{array}[]{c}P^{[k]}_{1}(z)\\\ \vdots\\\
P^{[k]}_{\omega}(z)\end{array}\right)=T(z)^{k-1}\left(\begin{array}[]{c}P_{k,1}(z)\\\
\vdots\\\
P_{k,\omega}(z)\end{array}\right)+\sum_{k^{\prime}=1}^{k-1}U_{k,k^{\prime}}(z)\left(\begin{array}[]{c}P_{k^{\prime},1}(z)\\\
\vdots\\\ P_{k^{\prime},\omega}(z)\end{array}\right)$

with rational functions $U_{k,k^{\prime}}(z)\in\mathbb{Q}[z,1/z]$, since
$T(z)$ and all coefficients of $A(z)$ belong to $\mathbb{Q}[z,1/z]$. Now
recall from Eq. (3.4) that $T(z)=z^{i}$ for some $i\in\mathbb{N}$, so that
$T(1)=1$ and

${\rm rk}(P^{[k]}_{\kappa}(1))_{\kappa\in\Omega,1\leq k\leq k_{0}}={\rm
rk}(P_{k,\kappa}(1))_{\kappa\in\Omega,1\leq k\leq k_{0}}$ (3.14)

for any $k_{0}\geq 1$. This equality will be used at the end of §3.6 to prove
Proposition 3, since Theorem A yields a lower bound on ${\rm
rk}(P_{k,\kappa}(1))_{\kappa\in\Omega,1\leq k\leq k_{0}}$.

The end of this section is devoted to the proof of the following result;
notice that parts $(i)$ and $(ii)$ are essentially proved in [33, pp.
575–576]. For $\kappa\in\Theta$ we define
$Z_{\kappa}=(z_{\kappa,\lambda})_{\lambda\in\Omega}\in\mathbb{C}^{\omega}$ by

$z_{\kappa,\lambda}=\left\\{\begin{array}[]{l}1\mbox{ if $\lambda=\kappa$,}\\\
f_{j}(1)\mbox{ if $\lambda=\kappa-e_{j}$ for some $j\in\llbracket
1,m\rrbracket$,}\\\ 0\mbox{ otherwise.}\end{array}\right.$

######  Proposition 4.

There exist solutions $Y_{\kappa}(z)$ of the differential system
$Y^{\prime}=AY$, for $\kappa\in\Theta$, such that:

  * $(i)$

For any $\kappa\in\Theta$, $Y_{\kappa}(1)=Z_{\kappa}$.

  * $(ii)$

The functions $Y_{\kappa}(z)$, $\kappa\in\Theta$, are linearly independent
over $\mathbb{C}$.

  * $(iii)$

For any $\kappa\in\Theta$, we have $R(Y_{\kappa})(z)=O(z^{K-cN})$ as $z\to 0$,
where $c>0$ is a constant that depends only on $f_{1}$, …, $f_{m}$.

  * $(iv)$

For any $\kappa\in\Theta$, the function $R(Y_{\kappa})(z)$ belongs to the
Nilsson class at 0.

######  Remark 4.

The constant $c$ in part $(iii)$ can be made effective, using the results of
[7].

###### Proof.

Consider the differential system

$a^{\prime}_{k}(z)=-S_{1,k}(z)a_{1}(z)-\ldots-S_{m,k}(z)a_{m}(z)\mbox{ for any
}k\in\llbracket 1,m\rrbracket.$ (3.15)

Since the system (3.1) of §3.1 has no non-zero finite singularity, all
rational functions $S_{\ell,k}(z)$ belong to $\mathbb{Q}[z,1/z]$ and the
system (3.15) has no non-zero finite singularity. In particular, there exists
a fundamental matrix of solutions $(\varphi_{k,l}(z))_{1\leq k,l\leq m}$ such
that $\varphi_{k,l}(1)$ is equal to the Kronecker symbol $\delta_{k,l}$. Let
$\varrho_{1},\ldots,\varrho_{m}$ be independent variables, and put
$a_{k}(z)=\sum_{l=1}^{m}\varrho_{l}\varphi_{k,l}(z)$ for $k\in\llbracket
1,m\rrbracket$. Then $(a_{1}(z),\ldots,a_{m}(z))$ is a solution of the system
(3.15).

Consider the vector
$\overline{Y}(z)=(\overline{y}_{\lambda}(z))_{\lambda\in\Omega}$ defined by:

$\overline{y}_{\lambda}(z)=\begin{cases}a_{1}(z)^{\lambda_{1}}\ldots
a_{m}(z)^{\lambda_{m}}\;\mbox{ if }|\lambda|=N,\\\
a_{1}(z)^{\lambda_{1}}\ldots
a_{m}(z)^{\lambda_{m}}(1+a_{1}(z)f_{1}(z)+\ldots+a_{m}(z)f_{m}(z))\;\mbox{ if
}|\lambda|=N-1.\end{cases}$ (3.16)

Each of these functions is a polynomial in the variables
$\varrho_{1},\ldots,\varrho_{m}$, with coefficients that depend on $z$; all
monomials that appear in this expression have total degree $N-1$ or $N$.
Therefore we have

$\overline{Y}(z)=\sum_{\kappa\in\Omega}\varrho_{1}^{\kappa_{1}}\ldots\varrho_{m}^{\kappa_{m}}Y_{\kappa}(z)$
(3.17)

and this expression defines functions $Y_{\kappa}(z)$ independent from
$\varrho_{1},\ldots,\varrho_{m}$. We shall be interested in these functions
only when $\kappa\in\Theta$, i.e. $|\kappa|=N$.

To prove part $(i)$, we deduce from $\varphi_{k,l}(1)=\delta_{k,l}$ that
$a_{k}(1)=\varrho_{k}$, and Eq. (3.16) yields

$\overline{y}_{\lambda}(1)=\begin{cases}\varrho_{1}^{\lambda_{1}}\ldots\varrho_{m}^{\lambda_{m}}\;\mbox{
if }|\lambda|=N,\\\
\varrho_{1}^{\lambda_{1}}\ldots\varrho_{m}^{\lambda_{m}}(1+\varrho_{1}f_{1}(1)+\ldots+\varrho_{m}f_{m}(1))\;\mbox{
if }|\lambda|=N-1.\end{cases}$ (3.18)

Given $\kappa\in\Theta$, we write
$Y_{\kappa}(1)=(z_{\kappa,\lambda})_{\lambda\in\Omega}$. Then Eq. (3.17) shows
that $z_{\kappa,\lambda}$ is the coefficient of
$\varrho_{1}^{\kappa_{1}}\ldots\varrho_{m}^{\kappa_{m}}$ in the expression of
$\overline{y}_{\lambda}(1)$. Using Eq. (3.18), we obtain

$z_{\kappa,\lambda}=\begin{cases}1\;\mbox{ if }\lambda=\kappa,\\\
f_{j}(1)\;\mbox{ if }\lambda=\kappa-e_{j}\mbox{ for some }j\in\llbracket
1,m\rrbracket,\\\ 0\;\mbox{ otherwise. }\end{cases}$

By definition of $Z_{\kappa}$, this means $Y_{\kappa}(1)=Z_{\kappa}$ and
concludes the proof of part $(i)$.

Part $(ii)$ follows easily from part $(i)$. Indeed we consider the matrix
$M\in M_{\omega,\theta}(\mathbb{C})$ with columns $Z_{\kappa}$,
$\kappa\in\Theta$. We may assume that the bijective map $\llbracket
1,\omega\rrbracket\to\Omega$ we have chosen in §3.1 maps $\llbracket
1,\theta\rrbracket$ to $\Theta$: it allows us to identify $\Theta$ and
$\llbracket 1,\theta\rrbracket$. Then by definition on the $Z_{\kappa}$, we
have $M=\left(\begin{array}[]{c}I\\\ M^{\prime}\end{array}\right)$ for some
matrix $M^{\prime}\in M_{\omega-\theta,\theta}(\mathbb{C})$, where $I\in
M_{\theta}(\mathbb{C})$ is the identity matrix. Therefore $M$ has rank
$\theta$, and the vectors $Z_{\kappa}$ are linearly independent over
$\mathbb{C}$. Using part $(i)$, this concludes the proof of $(ii)$.

Let us prove parts $(iii)$ and $(iv)$ now. For brevity we let
$\rho^{\kappa}=\rho_{1}^{\kappa_{1}}\ldots\rho_{m}^{\kappa_{m}}$ and define
$a(z)^{\lambda}$ in an analogous way. We have:

$\displaystyle\sum_{\kappa\in\Omega}\rho^{\kappa}R(Y_{\kappa})(z)$
$\displaystyle=R(\overline{Y})(z)=\sum_{\lambda\in\Omega}P_{\lambda}(z)\overline{y}_{\lambda}(z)\mbox{
using Eqns.\leavevmode\nobreak\ \eqref{eqdefR} and \eqref{eqdefycapa}}$
$\displaystyle=\sum_{\lambda\in\Theta}P_{\lambda}(z)a(z)^{\lambda}+\sum_{\lambda\in\Omega\setminus\Theta}P_{\lambda}(z)a(z)^{\lambda}\Big{(}1+\sum_{j=1}^{m}a_{j}(z)f_{j}(z)\Big{)}\;\mbox{
by Eq.\leavevmode\nobreak\ \eqref{eqdefylb}}$
$\displaystyle=\sum_{\lambda\in\Theta}a(z)^{\lambda}\Big{(}P_{\lambda}(z)+\sum_{j=1}^{m}P_{\lambda-
e_{j}}(z)f_{j}(z)\Big{)}+\sum_{\lambda\in\Omega\setminus\Theta}P_{\lambda}(z)a(z)^{\lambda}.$

Now recall that $a_{k}(z)=\sum_{l=1}^{m}\varrho_{l}\varphi_{k,l}(z)$. In the
previous expression, we fix $\kappa\in\Theta$ and identify the coefficients of
$\rho^{\kappa}$ in both sides. Since the second term of the right hand side is
homogeneous of degree $N-1$, whereas $|\kappa|=N$, it does not contribute and
we have

$R(Y_{\kappa})(z)=\sum_{\lambda\in\Theta}b_{\lambda,\kappa}(z)\Big{(}P_{\lambda}(z)+\sum_{j=1}^{m}P_{\lambda-
e_{j}}(z)f_{j}(z)\Big{)}$ (3.19)

where $b_{\lambda,\kappa}(z)$ is the coefficient of $\rho^{\kappa}$ in the
expansion of $a(z)^{\lambda}$. This coefficient $b_{\lambda,\kappa}(z)$ is an
explicit homogeneous polynomial of degree $N$ (with constant integer
coefficients) in the functions $\varphi_{k,l}(z)$.

Now denote by $Z^{\prime}=SZ$ the differential system (3.1) of §3.1. It has at
worst a regular singularity at 0, and therefore admits a fundamental matrix of
solutions $M(z)$ with coefficients in the Nilsson class at 0. Then
${}^{t}M(z)^{-1}$ is a fundamental matrix of solutions of the (dual)
differential system $Y^{\prime}=-\,{}^{t}SY$. Removing the first coordinate of
the solutions yields a fundamental matrix of solutions of the differential
system (3.15), all of which coefficients are in the Nilsson class at 0.
Accordingly all $\varphi_{k,l}$ and all $b_{\lambda,\kappa}$ belong to the
Nilsson class at 0, and so does $R(Y_{\kappa})$ using Eq. (3.19): this proves
part $(iv)$. Moreover, there exists a constant $c$, which depends only on this
system, such that $\varphi_{k,l}(z)=O(z^{-c})$ as $z\to 0$. Therefore we have
$b_{\lambda,\kappa}(z)=O(z^{-cN})$ for any $\lambda,\kappa\in\Theta$. Using
Eq. (3.19) and Lemma 1 this concludes the proof of part $(iii)$.

To conclude the proof of Proposition 4, let us prove that
$Y^{\prime}_{\kappa}=AY_{\kappa}$ for any $\kappa\in\Theta$. To begin with,
Eq. (3.16) yields for any $\lambda\in\Theta$:

$\displaystyle\overline{y}_{\lambda}^{\prime}(z)$
$\displaystyle=-\sum_{j=1}^{m}\lambda_{j}a(z)^{\lambda-
e_{j}}\sum_{l=1}^{m}S_{l,j}(z)a_{l}(z)\;\mbox{ using Eq.\leavevmode\nobreak\
\eqref{eqsysdiffnv}}$
$\displaystyle=-\sum_{j=1}^{m}\sum_{l=1}^{m}\lambda_{j}S_{l,j}(z)a(z)^{\lambda-
e_{j}+e_{l}}$
$\displaystyle=\sum_{\kappa\in\Omega}A_{\lambda,\kappa}(z)\overline{y}_{\kappa}(z)\;\mbox{
using Eq.\leavevmode\nobreak\ \eqref{eqdefA}.}$

To prove the same formula for $\lambda\in\Omega\setminus\Theta$, we notice
that, using Eqns. (3.15) and (3.1):

$\displaystyle\frac{{\rm d}}{{\rm d}z}\Big{(}1+$
$\displaystyle\sum_{p=1}^{m}a_{p}(z)f_{p}(z)\Big{)}=\sum_{p=1}^{m}a^{\prime}_{p}(z)f_{p}(z)+\sum_{l=1}^{m}a_{l}(z)f^{\prime}_{l}(z)$
$\displaystyle=-\sum_{p=1}^{m}\sum_{l=1}^{m}S_{l,p}(z)a_{l}(z)f_{p}(z)+\sum_{l=1}^{m}a_{l}(z)(S_{l,0}(z)+\sum_{p=1}^{m}S_{l,p}(z)f_{p}(z))$
$\displaystyle=\sum_{l=1}^{m}S_{l,0}(z)a_{l}(z).$

This gives for any $\lambda\in\Omega\setminus\Theta$, using Eqns. (3.16) and
(3.15):

$\displaystyle\overline{y}_{\lambda}^{\prime}(z)$
$\displaystyle=a(z)^{\lambda}\sum_{p=1}^{m}S_{p,0}(z)a_{p}(z)+\sum_{j=1}^{m}\lambda_{j}a(z)^{\lambda-
e_{j}}a^{\prime}_{j}(z)(1+\sum_{p=1}^{m}a_{p}(z)f_{p}(z))$
$\displaystyle=\sum_{p=1}^{m}S_{p,0}(z)a(z)^{\lambda+e_{p}}-\sum_{j=1}^{m}\lambda_{j}a(z)^{\lambda-
e_{j}}\sum_{l=1}^{m}S_{l,j}(z)a_{l}(z)(1+\sum_{p=1}^{m}a_{p}(z)f_{p}(z))$
$\displaystyle=\sum_{p=1}^{m}S_{p,0}(z)\overline{y}_{\lambda+e_{p}}(z)-\sum_{j=1}^{m}\sum_{l=1}^{m}\lambda_{j}S_{l,j}(z)\overline{y}_{\lambda-
e_{j}+e_{l}}(z)$
$\displaystyle=\sum_{\kappa\in\Omega}A_{\lambda,\kappa}(z)\overline{y}_{\kappa}(z)\;\mbox{
using Eq.\leavevmode\nobreak\ \eqref{eqdefA}.}$

This concludes the proof of Proposition 4. ∎

###  3.6 Application of the multiplicity estimate: proof of Proposition 3

In this section we prove Proposition 3 stated in §3.2, as consequence of
Theorem A stated in §2.3 (using also Theorem 2 that will be stated and proved
in §4). We apply Theorem A in the setting of §3.5, namely with $q=\omega$, the
differential system $Y^{\prime}=AY$ where $A$ is defined by Eq. (3.13),
$n=M-1$ and the polynomials $P_{\kappa}(z)$, $z\in\Omega$, defined in Lemma 1.
All vector spaces, dimensions and other notions of linear algebra are over
$\mathbb{C}$.

We denote by $\mathcal{R}$ the space of solutions $Y$ of $Y^{\prime}=AY$ such
that $R(Y)(z)$ is identically zero, so that $\varrho=\dim\mathcal{R}$ with the
notation of Theorem A. The difficult part is to prove that
$\mathcal{R}=\\{0\\}$; this is a necessary condition for the matrix of
Proposition 3 to be of maximal rank (see the remark after Theorem A). Before
going into details, let us explain our strategy by studying the influence on
Eq. (2.5) of Theorem A of a non-zero solution $Y\in\mathcal{R}$: it changes
$\varrho$ to $\varrho+1$, and therefore decreases the right hand side by
$n+1$. Accordingly it has the same effect on Eq. (2.5) as an additional
solution $Y_{j}$ that would vanish at 0 to order $n+1$. This is optimal in the
general setting, and it was sufficient in the application we had in [15], but
it isn’t for our purposes. Indeed, if a non-zero solution $Y\in\mathcal{R}$ is
amongst the solutions $Y_{\kappa}$ of Proposition 4, then it cannot be
included in the family $(Y_{j})$ of Theorem A because in that theorem the
functions $R(Y_{j})$ have to be linearly independent over $\mathbb{C}$. In Eq.
(2.5) of Theorem A this results in a loss of ${\rm ord}_{0}R(Y)\geq K-cN$ in
the left hand side; as explained above we gain $n+1$ because
$Y\in\mathcal{R}$, but the gain does not compensate the loss because $K-cN$ is
much bigger than $n+1=M$ (recall from Eq. (3.3) that $K/M$ is very close to
$\omega/\theta=2-\frac{m-1}{N+m-1}>1$). The bound on $\tau$ such that Eq.
(2.5) holds is not good enough, and we cannot complete the proof unless we
manage to gain more. Such a gain is provided by Theorem 2 proved in §4: the
solutions $Y\in{\rm Span}\\{Y_{\kappa}\\}$ such that $R(Y)=0$ are balanced by
additional solutions $Y\not\in{\rm Span}\\{Y_{\kappa}\\}$ such that $R(Y)=0$,
that count for $n+1$ in Eq. (2.5). If $\mathcal{R}\neq\\{0\\}$, the overall
gain is even slightly bigger than the loss, so that we deduce a too good
estimate on $\tau$, in contradiction with part $(i)$ of Theorem A. Therefore
$\mathcal{R}=\\{0\\}$, and part $(ii)$ of Theorem A enables us to deduce
Proposition 3.

To work out the proof of Proposition 3, we let $\mathcal{F}={\rm
Span}\\{Y_{\kappa},\,\kappa\in\Theta\\}$ where the solutions $Y_{\kappa}$ are
given by Proposition 4. They are in the Nilsson class at 0, and are linearly
independent over $\mathbb{C}$ so that $\dim\mathcal{F}=\theta$.

We denote by $J$ a maximal subset of $\Theta$ such that $\mathcal{R}\cap{\rm
Span}\\{Y_{\kappa},\,\kappa\in J\\}=\\{0\\}$. Then we have
$\mathcal{F}=(\mathcal{F}\cap\mathcal{R})\oplus{\rm
Span}\\{Y_{\kappa},\,\kappa\in J\\}$ so that ${\rm
Card}\,J=\theta-\dim(\mathcal{F}\cap\mathcal{R})$.

With this definition, the functions $R(Y_{\kappa})$, $\kappa\in J$, are
linearly independent over $\mathbb{C}$: if $\sum_{\kappa\in
J}\lambda_{\kappa}R(Y_{\kappa})=0$ with $\lambda_{\kappa}\in\mathbb{C}$, then
$\sum_{\kappa\in J}\lambda_{\kappa}Y_{\kappa}\in\mathcal{R}\cap{\rm
Span}\\{Y_{\kappa},\,\kappa\in J\\}=\\{0\\}$ so that $\lambda_{\kappa}=0$ for
any $\kappa$.

Therefore we may apply Theorem A with

$\tau=(\omega-\varrho)M-\sum_{\kappa\in J}{\rm
ord}_{0}(R(Y_{\kappa}))\leq(\omega-\varrho)M-(K-cN)(\theta-\dim(\mathcal{F}\cap\mathcal{R}))$
(3.20)

using assertion $(iii)$ of Proposition 4 and the equality ${\rm
Card}\,J=\theta-\dim(\mathcal{F}\cap\mathcal{R})$.

Now consider the map ${\rm ev}_{1}:Y\mapsto Y(1)$, from the space of solutions
of $Y^{\prime}=AY$ to $\mathbb{C}^{\omega}$. It is bijective because 1 is not
a singularity of this differential system, defined by (3.13) with
$S_{l,j}(z)\in\mathbb{Q}[z,1/z]$ for any $l,j$. We let $F={\rm
ev}_{1}(\mathcal{F})$; this space is spanned by the vectors $Z_{\kappa}$,
$\kappa\in\Theta$, by Proposition 4 $(i)$.

We claim that $R={\rm ev}_{1}(\mathcal{R})$ is defined over $\mathbb{Q}$.
Indeed given a solution $Y=(y_{\kappa}(z))_{\kappa\in\Omega}$ of
$Y^{\prime}=AY$, we have $Y(1)\in R$ if, and only if, the function
$\sum_{\kappa\in\Omega}P_{\kappa}(z)y_{\kappa}(z)$ is identically zero. This
means that all derivatives of this function vanish at 1, i.e.
$\sum_{\kappa\in\Omega}P_{k,\kappa}(1)y_{k}(1)=0$ for any $k\geq 1$ using Eq.
(2.3). This set of linear equations with rational coefficients
$P_{k,\kappa}(1)$ defines the subspace $R$.

By construction (see §3.1) the functions 1, $f_{1}(z)$, …, $f_{m}(z)$ are
linearly independent over $\mathbb{C}(z)$, and they make up a vector solution
of a differential system of order 1 with coefficients in $\mathbb{Q}[z,1/z]$,
given by Eq. (3.1). Since $1$ is not a singularity of this system, Beukers’
refinement [8, Corollary 1.4] of the Siegel-Shidlovskii theorem shows that the
values at 1 of these functions are linearly independent over
$\overline{\mathbb{Q}}$. Therefore Theorem 2 applies with $\xi_{1}=f_{1}(1)$,
…, $\xi_{m}=f_{m}(1)$.

To begin with, assume that $R\neq\\{0\\}$ and $R\neq\mathbb{C}^{\Omega}$. Then
Theorem 2 yields

$\dim R>\Big{(}2-\frac{m-1}{N+m-1}\Big{)}\dim(F\cap R).$

Since both sides of the inequality are integer multiples of $\frac{1}{N+m-1}$,
we obtain

$\displaystyle\dim R$
$\displaystyle\geq\Big{(}2-\frac{m-1}{N+m-1}\Big{)}\dim(F\cap
R)+\frac{1}{N+m-1}$ $\displaystyle=\frac{\omega}{\theta}\dim(F\cap
R)+\frac{1}{N+m-1}.$

We have $\dim R=\dim\mathcal{R}=\varrho$ and $\dim(F\cap
R)=\dim(\mathcal{F}\cap\mathcal{R})$, so that Eq. (3.20) yields

$\tau\leq\Big{(}1-\frac{\varrho}{\omega}\Big{)}\omega
M-(K-cN)\theta\Big{(}1-\frac{\varrho}{\omega}+\frac{1}{\omega(N+m-1)}\Big{)}.$

Now recall that $\omega M-K\theta\leq 2\eta M$ using Eq. (3.3), and
$\varrho,\theta\leq\omega$, so that

$\displaystyle\tau$
$\displaystyle\leq\Big{(}1-\frac{\varrho}{\omega}\Big{)}(2\eta
M+cN\theta)-\frac{(\omega-2\eta)M-cN\theta}{\omega(N+m-1)}$ $\displaystyle\leq
2\eta
M+cN\theta-\frac{\Big{(}1-\frac{2\eta}{\omega}\Big{)}M}{N+m-1}+\frac{cN}{N+m-1}$
$\displaystyle\leq-\frac{\Big{(}\frac{1}{3}-\frac{2\eta}{\omega}\Big{)}M}{N+m-1}+c(1+N\theta)<-c_{1}$

since $\eta\leq\frac{1}{3(N+m-1)}$ and $M$ has been chosen large enough with
respect to $N$, $m$ and $\eta$; the crucial point here is that the constants
$c$ of Proposition 4 and $c_{1}$ of Theorem A depend only on $N$ and $f_{1}$,
…, $f_{m}$ but not on $M$. This upper bound on $\tau$ contradicts conclusion
$(i)$ of Theorem A.

Therefore we have $R=\\{0\\}$ or $R=\mathbb{C}^{\Omega}$. The latter means
that for any solution $Y=(y_{\kappa}(z))_{\kappa\in\Omega}$ of the
differential system $Y^{\prime}=AY$, the function
$R(Y)(z)=\sum_{\kappa\in\Omega}P_{\kappa}(z)y_{\kappa}(z)$ is identically
zero. This is impossible: there exist $\kappa_{0}\in\Omega$ and
$z_{0}\in\mathbb{C}^{*}$ such that $P_{\kappa_{0}}(z_{0})\neq 0$, and since
$z_{0}$ is not a singularity there exists a solution $Y$ such that
$y_{\kappa_{0}}(z_{0})=1$ and $y_{\kappa}(z_{0})=0$ for any
$\kappa\neq\kappa_{0}$.

Accordingly we have proved that $R=\\{0\\}$, so that
$\mathcal{F}\cap\mathcal{R}=\mathcal{R}=\\{0\\}$ and Eq. (3.20) yields

$\tau\leq\omega M-(K-cN)\theta\leq 2\eta M+cN\theta$

using Eq. (3.3). Therefore $\tau\leq\lfloor 3\eta M\rfloor\leq M-c_{1}$ since
$M$ has been chosen sufficiently large with respect to $N$, $f_{1}$, …,
$f_{m}$ and $\eta$. Increasing $\tau$ if necessary, we may assume $\tau\geq 0$
so that Theorem A yields ${\rm rk}(P_{k,\kappa}(1))_{\kappa\in\Omega,1\leq
k\leq\lfloor 2\eta M\rfloor+c_{1}-1}=\omega$. Using Eq. (3.14) proved in §3.5,
this concludes the proof of Proposition 3.

##  4 The key technical result

In this section we state and prove Theorem 2, a key ingredient in the proof of
Proposition 3 given in §3.6. The motivation and application of this result is
explained at the beginning of §3.6. We believe that analogous auxiliary
results might be useful when the multiplicity estimate of [15] is applied in
settings where $R(Y)$ may vanish identically for non-zero solutions $Y$.

Theorem 2 is an independent result, for which we need only the following
notations. We consider integers $m,N\geq 1$ and complex numbers
$\xi_{1},\ldots,\xi_{m}$; we assume that $1$, $\xi_{1}$, …, $\xi_{m}$ are
linearly independent over $\overline{\mathbb{Q}}$. As in §3.1 we let
$\Omega=\\{\kappa\in\mathbb{N}^{m},\,N-1\leq|\kappa|\leq N\\}$ where
$|\kappa|=\kappa_{1}+\ldots+\kappa_{m}$, and $\omega={\rm
Card}\,\Omega=\binom{N+m-2}{m-1}+\binom{N+m-1}{m-1}$; we denote by
$(e_{j})_{1\leq j\leq m}$ the canonical basis of $\mathbb{Z}^{m}$, and by
$(E_{\kappa})_{\kappa\in\Omega}$ that of $\mathbb{C}^{\Omega}$. In other
words, we have
$E_{\kappa}=(\delta_{\kappa,\kappa^{\prime}})_{\kappa^{\prime}\in\Omega}$
where $\delta_{\kappa,\kappa^{\prime}}$ is the Kronecker’s symbol. We consider

$Z_{\kappa}=E_{\kappa}+\sum_{j=1}^{m}\xi_{j}E_{\kappa-e_{j}}\mbox{ for any
}\kappa\in\Theta=\\{\kappa\in\mathbb{N}^{m},\,|\kappa|=N\\}$

with the convention that $E_{\kappa-e_{j}}=0$ if $\kappa-
e_{j}\in\mathbb{Z}^{m}$ has at least one negative component (namely, if
$\kappa_{j}=0$); with $\xi_{j}=f_{j}(1)$ these are exactly the vectors
$Z_{\kappa}$ defined before Proposition 4 in §3.5. We denote by $F$ the
subspace of $\mathbb{C}^{\Omega}$ generated by these vectors, namely

$F={\rm Span}\\{Z_{\kappa},\,\kappa\in\Theta\\}.$

It is not difficult (see the proof of $(ii)$ in Proposition 4 above) to see
that the $Z_{\kappa}$ are linearly independent, so that $\dim
F=\binom{N+m-1}{m-1}$, denoted by $\theta$.

######  Theorem 2.

Let $R$ be a vector subspace of $\mathbb{C}^{\Omega}$ defined over
$\overline{\mathbb{Q}}$. Then we have:

$\dim R\geq\Big{(}2-\frac{m-1}{N+m-1}\Big{)}\dim(F\cap R),$

and equality holds if and only if $R=\\{0\\}$ or $R=\mathbb{C}^{\Omega}$.

By defined over $\overline{\mathbb{Q}}$, or rational over
$\overline{\mathbb{Q}}$, we mean that $R$ has a $\mathbb{C}$-basis consisting
in vectors in $\overline{\mathbb{Q}}^{\Omega}$. This is equivalent to the
existence of a system of linear equations with coefficients in
$\overline{\mathbb{Q}}$ that defines $R$. We point out that an inequality on
dimensions, such as the one of Theorem 2, is reminiscent of the notion of
$(e,j)$-irrationality introduced in [20] and [21].

###  4.1 Two lemmas

The data of Theorem 2 depend only on $m,N\geq 1$ and on
$\xi=(\xi_{1},\ldots,\xi_{m})\in\mathbb{C}^{m}$; in the proof we shall often
deduce Theorem 2 for some triples $(m,N,\xi)$ from the same statement for
other triples. It will always be assumed, implicitly or explicitly, that $1$,
$\xi_{1}$, …, $\xi_{m}$ are linearly independent over $\overline{\mathbb{Q}}$.
We first prove two lemmas; recall that we identify tuples in $\mathbb{C}^{m}$
to column matrices in $M_{m,1}(\mathbb{C})$.

######  Lemma 4.

Let $m,N\geq 1$ and $\xi=(\xi_{1},\ldots,\xi_{m})\in\mathbb{C}^{m}$, with $1$,
$\xi_{1}$, …, $\xi_{m}$ linearly independent over $\overline{\mathbb{Q}}$. Let
$A\in{\rm GL}_{m}(\overline{\mathbb{Q}})$; define
$\xi^{\prime}=(\xi^{\prime}_{1},\ldots,\xi^{\prime}_{m})$ by
$\xi^{\prime}=A\,\xi$.

If Theorem 2 holds for $(m,N,\xi)$ with any subspace $R$ defined over
$\overline{\mathbb{Q}}$ of a given dimension $\varrho$, then it also does for
$(m,N,\xi^{\prime})$.

###### Proof.

Decomposing $A$ into a product of simpler matrices, we may restrict to the
following 3 cases.

$\bullet$ Case 1: $\xi^{\prime}_{j}=\xi_{\sigma(j)}$ for any $j\in\llbracket
1,m\rrbracket$, with $\sigma\in\mathfrak{S}_{m}$. In this case Lemma 4 is
obvious, by permuting the coordinates in $\mathbb{N}^{m}$ and accordingly in
$\mathbb{C}^{\Omega}$.

$\bullet$ Case 2: $\xi^{\prime}_{j_{0}}=\lambda\xi_{j_{0}}$ for some
$\lambda\in\overline{\mathbb{Q}}^{*}$ and $j_{0}\in\llbracket 1,m\rrbracket$,
and $\xi^{\prime}_{j}=\xi_{j}$ for any $j\in\llbracket
1,m\rrbracket\setminus\\{j_{0}\\}$. Let
$f:\mathbb{C}^{\Omega}\to\mathbb{C}^{\Omega}$ be the (bijective) linear map
defined by $f(E_{\kappa})=\lambda^{\kappa_{j_{0}}}E_{\kappa}$ for any
$\kappa\in\Omega$. We denote by $(Z_{\kappa})$ and $F$ (resp.
$(Z^{\prime}_{\kappa})$ and $F^{\prime}$) the data associated with $\xi$
(resp. $\xi^{\prime}$). Then we have for any $\kappa\in\Theta$:

$f(Z^{\prime}_{\kappa})=f(E_{\kappa})+\sum_{j=1}^{m}\xi^{\prime}_{j}f(E_{\kappa-
e_{j}})\\\
=\lambda^{\kappa_{j_{0}}}E_{\kappa}+\lambda\xi_{j_{0}}\lambda^{\kappa_{j_{0}}-1}E_{\kappa-
e_{j_{0}}}+\sum_{j\neq j_{0}}\xi_{j}\lambda^{\kappa_{j_{0}}}E_{\kappa-
e_{j}}=\lambda^{\kappa_{j_{0}}}f(Z_{\kappa})$

so that $f(F^{\prime})=F$. Let $R^{\prime}$ be a subspace of
$\mathbb{C}^{\Omega}$ defined over $\overline{\mathbb{Q}}$, with $\dim
R^{\prime}=\varrho$. Taking $R=f(R^{\prime})$ we have $\dim R=\dim
R^{\prime}$, $\dim(F\cap R)=\dim(F^{\prime}\cap R^{\prime})$ and $R$ is
defined over $\overline{\mathbb{Q}}$. Therefore Theorem 2 applied to
$(m,N,\xi)$ with $R$ shows that Theorem 2 holds for $(m,N,\xi^{\prime})$ with
$R^{\prime}$.

$\bullet$ Case 3: $\xi^{\prime}_{j_{0}}=\xi_{j_{0}}+\xi_{j_{1}}$ with distinct
$j_{0},j_{1}\in\llbracket 1,m\rrbracket$, and $\xi^{\prime}_{j}=\xi_{j}$ for
any $j\in\llbracket 1,m\rrbracket\setminus\\{j_{0}\\}$. We consider the linear
map $f:\mathbb{C}^{\Omega}\to\mathbb{C}^{\Omega}$ given by

$f(E_{\kappa})=\sum_{t=0}^{\kappa_{j_{0}}}\binom{t+\kappa_{j_{1}}}{\kappa_{j_{1}}}E_{\kappa-
te_{j_{0}}+te_{j_{1}}}\mbox{ for any }\kappa\in\Omega.$

It is bijective because $f(E_{\kappa})-E_{\kappa}$ is a linear combination of
the $E_{\kappa^{\prime}}$ with $\kappa^{\prime}_{j_{0}}<\kappa_{j_{0}}$. For
any $\kappa\in\Theta$ we have:

$\displaystyle f($ $\displaystyle E_{\kappa-e_{j_{1}}})+f(E_{\kappa-
e_{j_{0}}})=\sum_{t=0}^{\kappa_{j_{0}}}\binom{t+\kappa_{j_{1}}-1}{\kappa_{j_{1}}-1}E_{\kappa-
te_{j_{0}}+(t-1)e_{j_{1}}}+\sum_{t=0}^{\kappa_{j_{0}}-1}\binom{t+\kappa_{j_{1}}}{\kappa_{j_{1}}}E_{\kappa-(t+1)e_{j_{0}}+te_{j_{1}}}$
$\displaystyle=\sum_{t=0}^{\kappa_{j_{0}}}\bigg{(}\binom{t+\kappa_{j_{1}}-1}{\kappa_{j_{1}}-1}+\binom{t+\kappa_{j_{1}}-1}{\kappa_{j_{1}}}\bigg{)}E_{\kappa-
te_{j_{0}}+(t-1)e_{j_{1}}}\;\mbox{($t^{\prime}=t+1$ in the second sum)}$
$\displaystyle=\sum_{t=0}^{\kappa_{j_{0}}}\binom{t+\kappa_{j_{1}}}{\kappa_{j_{1}}}E_{\kappa-
te_{j_{0}}+(t-1)e_{j_{1}}}$ (4.1)

so that

$\displaystyle f(Z^{\prime}_{\kappa})$
$\displaystyle=f(E_{\kappa})+\sum_{j=1}^{m}\xi_{j}f(E_{\kappa-
e_{j}})+\xi_{j_{1}}f(E_{\kappa-e_{j_{0}}})$
$\displaystyle=\sum_{t=0}^{\kappa_{j_{0}}}\binom{t+\kappa_{j_{1}}}{\kappa_{j_{1}}}\Big{(}E_{\kappa-
te_{j_{0}}+te_{j_{1}}}+\sum_{j\neq j_{1}}\xi_{j}E_{\kappa-
e_{j}-te_{j_{0}}+te_{j_{1}}}\Big{)}+\xi_{j_{1}}\Big{(}f(E_{\kappa-
e_{j_{1}}})+f(E_{\kappa-e_{j_{0}}})\Big{)}$
$\displaystyle=\sum_{t=0}^{\kappa_{j_{0}}}\binom{t+\kappa_{j_{1}}}{\kappa_{j_{1}}}Z_{\kappa-
te_{j_{0}}+te_{j_{1}}}\in F,\mbox{ using Eq. \eqref{eqaux}.}$

Therefore $f(F^{\prime})=F$, and we deduce the result as in Case 2. This
concludes the proof of Lemma 4.∎

######  Lemma 5.

Let $m,N\geq 1$. We consider the subspace

$K={\rm Span}\\{E_{\kappa},\,\kappa\in\Omega,\,\kappa_{m}=0\\}.$

Assume that Theorem 2 holds for any $\xi\in\mathbb{C}^{m}$ and any $R$ defined
over $\overline{\mathbb{Q}}$ such that

$\dim(F\cap R\cap K)\geq 2\dim(F\cap R)-\dim(R).$

Then Theorem 2 holds for any $\xi\in\mathbb{C}^{m}$ and any $R$ defined over
$\overline{\mathbb{Q}}$.

###### Proof.

We start with any $\xi=(\xi_{1},\ldots,\xi_{m})\in\mathbb{C}^{m}$ such that
$1$, $\xi_{1}$, …, $\xi_{m}$ are linearly independent over
$\overline{\mathbb{Q}}$. Let $R$ be a subspace of $\mathbb{C}^{\Omega}$
defined over $\overline{\mathbb{Q}}$, and put $\varrho=\dim R$. We shall
construct $g\in{\rm GL}(\mathbb{C}^{m})$ defined over $\overline{\mathbb{Q}}$
(i.e., whose matrix in the canonical basis of $\mathbb{C}^{m}$ has
coefficients in $\overline{\mathbb{Q}}$), such that $\xi^{\prime}=g(\xi)$
satisfies $\dim(F^{\prime}\cap R^{\prime}\cap K)\geq 2\dim(F^{\prime}\cap
R^{\prime})-\dim(R^{\prime})$ for any subspace $R^{\prime}$ of
$\mathbb{C}^{\Omega}$ defined over $\overline{\mathbb{Q}}$ of dimension
$\varrho$, where $F^{\prime}$ is associated with $\xi^{\prime}=g(\xi)$ as
before the statement of Theorem 2. Then Lemma 4 shows that if Theorem 2 holds
for $\xi^{\prime}$, then it does for $\xi$.

Denote by $\mathcal{V}_{0}$ the Zariski closure of $\\{\xi\\}$ in the affine
space $\mathbb{A}_{\overline{\mathbb{Q}}}^{m}$ over $\overline{\mathbb{Q}}$,
i.e. the smallest subset of $\mathbb{C}^{m}$, defined by polynomial equations
with coefficients in $\overline{\mathbb{Q}}$, that contains $\xi$. In more
concrete terms, $\mathcal{V}_{0}$ is the set of all
$(z_{1},\ldots,z_{m})\in\mathbb{C}^{m}$ such that $P(z_{1},\ldots,z_{m})=0$
for any $P\in\overline{\mathbb{Q}}[X_{1},\ldots,X_{m}]$ such that
$P(\xi_{1},\ldots,\xi_{m})=0$. Since $\xi_{1}$ is transcendental,
$\mathcal{V}_{0}$ has dimension at least 1. There exists an algebraic curve
$\mathcal{C}$, defined by polynomial equations with coefficients in
$\overline{\mathbb{Q}}$, contained in $\mathcal{V}_{0}$ but in no hypersurface
defined over $\overline{\mathbb{Q}}$ of degree less than or equal to $\theta$
(with possible exceptions for such hypersurfaces that contain
$\mathcal{V}_{0}$).

There exists $j_{0}\in\llbracket 1,m\rrbracket$ such that the $j_{0}$-th
projection $\mathcal{C}\to\mathbb{C}$,
$(\chi_{1},\ldots,\chi_{m})\mapsto\chi_{j_{0}}$, has infinite image; then this
image contains all real numbers greater than some $M_{0}$. Parametrizing a
branch of $\mathcal{C}$, we obtain algebraic functions over
$\overline{\mathbb{Q}}(z)$, denoted by $\chi_{1}(z),\ldots,\chi_{m}(z)$, such
that $(\chi_{1}(a),\ldots,\chi_{m}(a))\in\mathcal{C}$ for any real $a\geq
M_{0}$, and $|\chi_{j_{0}}(a)|\to\infty$ as $a\to+\infty$ with
$a\in\mathbb{R}$. Their asymptotic behaviour as $a\to+\infty$ is given by
$\chi_{j}(a)\sim\varpi_{j}^{0}a^{d_{j}}$ with
$\varpi_{j}^{0}\in\overline{\mathbb{Q}}^{*}$ and $d_{j}\in\mathbb{Q}$; we have
$d_{j_{0}}>0$. Let $D=\max(d_{1},\ldots,d_{m})>0$, and put
$\varpi_{j}=\varpi_{j}^{0}$ for any $j\in\llbracket 1,m\rrbracket$ such that
$d_{j}=D$, and $\varpi_{j}=0$ otherwise. In this way, for any $j\in\llbracket
1,m\rrbracket$ we have

$\varpi_{j}=\lim_{a\in\mathbb{R},\,a\to+\infty}a^{-D}\chi_{j}(a)\quad\mbox{
with }D>0.$ (4.2)

We can now construct a bijective linear map
$g:\mathbb{C}^{m}\to\mathbb{C}^{m}$ defined over $\overline{\mathbb{Q}}$ such
that $g(\varpi_{1},\ldots,\varpi_{m})=(0,\ldots,0,1)$. We let
$\xi^{\prime}=(\xi^{\prime}_{1},\ldots,\xi^{\prime}_{m})=g(\xi)$; then $1$,
$\xi^{\prime}_{1}$, …, $\xi^{\prime}_{m}$ are linearly independent over
$\overline{\mathbb{Q}}$. We denote by $F^{\prime}$, $\mathcal{C}^{\prime}$,
$\varpi^{\prime}_{j}$, …the objects defined as above, starting from
$\xi^{\prime}$ instead of $\xi$. Then we may choose
$\mathcal{C}^{\prime}=g(\mathcal{C})$,
$(\chi^{\prime}_{1}(z),\ldots,\chi^{\prime}_{m}(z))=g(\chi_{1}(z),\ldots,\chi_{m}(z))$
so that $D^{\prime}=D$ and
$(\varpi^{\prime}_{1},\ldots,\varpi^{\prime}_{m})=g(\varpi_{1},\ldots,\varpi_{m})=(0,\ldots,0,1)$.
As explained at the beginning of the proof, we shall prove that $\xi^{\prime}$
satisfies the additional property $\dim(F\cap R\cap K)\geq 2\dim(F\cap
R)-\dim(R)$ for any subspace $R$ of $\mathbb{C}^{\Omega}$ of dimension
$\varrho$ defined over $\overline{\mathbb{Q}}$; here and below (until the end
of the proof), for simplicity we write $\xi$, $\mathcal{C}$,
$\mathcal{V}_{0}$, $F$, $R$, $\varpi_{j}$, …instead of $\xi^{\prime}$,
$\mathcal{C}^{\prime}$, $\mathcal{V}_{0}^{\prime}$, $F^{\prime}$,
$R^{\prime}$, $\varpi^{\prime}_{j}$, ….

We let $R$ be a subspace of $\mathbb{C}^{\Omega}$ of dimension $\varrho$
defined over $\overline{\mathbb{Q}}$, write

$d=\dim(F\cap R),\quad\quad d^{\prime}=\dim(F\cap R\cap K),$

and assume (by contradiction) that $d^{\prime}<2d-\varrho$.

For any $\chi=(\chi_{1},\ldots,\chi_{m})\in\mathbb{C}^{m}$, we denote by
$F_{\chi}$ the subspace defined exactly like $F$, except that $\chi_{1}$, …,
$\chi_{m}$ are used instead of $\xi_{1}$, …, $\xi_{m}$ to define the
$Z_{\kappa}$. We denote by $\mathcal{V}$ the set of all
$\chi\in\mathbb{C}^{m}$ such that

$\dim F_{\chi}=\theta,\quad\dim(F_{\chi}\cap R)=d,\quad\dim(F_{\chi}\cap R\cap
K)=d^{\prime},$

i.e. such that these dimensions are the same as for $\chi=\xi$.

We claim that we have inclusions

$(\mathbb{C}^{m}\setminus H_{1})\cap(\mathbb{C}^{m}\setminus
H_{2})\cap(\mathbb{C}^{m}\setminus H_{3})\cap H_{4}\cap\ldots\cap
H_{v}\subset\mathcal{V}\subset H_{4}\cap\ldots\cap H_{v}$ (4.3)

where $v\geq 3$ and $H_{1}$, …, $H_{v}$ are hypersurfaces of $\mathbb{C}^{m}$
of degree at most $\theta$ defined over $\overline{\mathbb{Q}}$, such that
$\mathcal{V}_{0}\not\subset H_{i}$ for any $i\in\llbracket 1,3\rrbracket$. We
recall that $\mathcal{V}_{0}$ is the Zariski closure of $\\{\xi\\}$ in
$\mathbb{A}_{\overline{\mathbb{Q}}}^{m}$, so that $\mathcal{V}_{0}\not\subset
H_{i}$ is equivalent to $\xi\not\in H_{i}$. Indeed we denote by
$Z(X_{1},\ldots,X_{m})\in
M_{\omega,\theta}(\overline{\mathbb{Q}}[X_{1},\ldots,X_{m}])$ the matrix of
which the columns are the coordinates of the $Z_{\kappa}$ in the canonical
basis of $\mathbb{C}^{\Omega}$ (for $\kappa\in\Theta$), in which $\xi_{j}$ is
replaced with $X_{j}$. Then $\dim F_{\chi}={\rm rk}(Z(\chi))$ is equal to
$\theta$ if, and only if, at least one minor of $Z(\chi)$ of size $\theta$ is
non-zero. The coefficients of $Z(X_{1},\ldots,X_{m})$ are polynomials of total
degree at most 1 in $X_{1}$, …, $X_{m}$, so each minor of size $\theta$ has
degree at most $\theta$. We choose a minor which is non-zero at $\xi$, and
denote by $H_{1}$ the hypersurface defined in $\mathbb{C}^{m}$ by the
vanishing of this minor.

Now we consider the matrix $S(X_{1},\ldots,X_{m})\in
M_{\omega,\theta+\varrho}(\overline{\mathbb{Q}}[X_{1},\ldots,X_{m}])$ of which
the first $\theta$ columns are those of $Z(X_{1},\ldots,X_{m})$, and the last
$\varrho$ columns belong to $\overline{\mathbb{Q}}^{\Omega}$ and make up a
basis of $R$ (which is possible since $R$ is defined over
$\overline{\mathbb{Q}}$). Assuming that $\dim F_{\chi}=\theta$, we have
$\dim(F_{\chi}\cap R)=d$ if, and only if, $\dim(F_{\chi}+R)=\theta+\varrho-d$;
this is equivalent to ${\rm rk}(S(\chi))=\theta+\varrho-d$. This condition can
be expressed as the vanishing of all minors of size $\theta+\varrho-d+1$, and
the non-vanishing of at least one minor of size $\theta+\varrho-d$. Again we
choose such a minor of size $\theta+\varrho-d$ that does not vanish at $\xi$,
and denote by $H_{2}$ the corresponding hypersurface (which has degree at most
$\theta$); we define $H_{4}$, $H_{5}$, …to be the hypersurfaces defined by the
vanishing of the minors of size $\theta+\varrho-d+1$. We proceed in the same
way with $R\cap K$ instead of $R$ to ensure that $\dim(F_{\chi}\cap R\cap
K)=d^{\prime}$. This concludes the proof of the claimed inclusions (4.3).

These inclusions imply that
$\mathcal{C}\setminus\mathcal{C}_{1}\subset\mathcal{V}$ for some finite set
$\mathcal{C}_{1}$. Indeed, we have $\xi\in\mathcal{V}\subset H_{i}$ for any
$i\in\llbracket 4,v\rrbracket$, so that
$\mathcal{C}\subset\mathcal{V}_{0}\subset H_{i}$ by definition of
$\mathcal{V}_{0}$, since $H_{i}$ is defined over $\overline{\mathbb{Q}}$. For
$i\in\llbracket 1,3\rrbracket$, we have $\mathcal{V}_{0}\not\subset H_{i}$ and
$H_{i}$ is a hypersurface of degree at most $\theta$, so that
$\mathcal{C}\not\subset H_{i}$ (by construction of $\mathcal{C}$) and
$\mathcal{C}\cap H_{i}$ is a finite set; taking for $\mathcal{C}_{1}$ the
union of these finite sets, Eq. (4.3) yields
$\mathcal{C}\setminus\mathcal{C}_{1}\subset\mathcal{V}$.

Since $\mathcal{C}_{1}$ is finite, there exists a real number $M_{1}\geq
M_{0}$ such that for any real $a\geq M_{1}$, the point
$\chi(a)=(\chi_{1}(a),\ldots,\chi_{m}(a))$ belongs to
$\mathcal{C}\setminus\mathcal{C}_{1}\subset\mathcal{V}$. We shall focus on
real algebraic values of $a\geq M_{1}$; since
$\chi(a)\in\mathcal{V}\cap\overline{\mathbb{Q}}^{m}$, the subspace
$F_{\chi(a)}$ is then defined over $\overline{\mathbb{Q}}$ and we have $\dim
F_{\chi(a)}=\theta$, $\dim(F_{\chi(a)}\cap R)=d$, $\dim(F_{\chi(a)}\cap R\cap
K)=d^{\prime}$. We fix such an $a$, denoted by $a_{0}$, and consider a
subspace $W$ defined over $\overline{\mathbb{Q}}$ such that

$\big{(}F_{\chi(a_{0})}\cap R\cap K\big{)}\oplus W=F_{\chi(a_{0})}\cap R.$
(4.4)

For any $a\in\overline{\mathbb{Q}}\cap\mathbb{R}$ such that $a\geq M_{1}$, we
have $\chi(a)\in\mathcal{V}$ so that

$\dim W+\dim\big{(}F_{\chi(a)}\cap R\big{)}=(d-d^{\prime})+d>\varrho=\dim R$

since we have assumed (by contradiction) that $d^{\prime}<2d-\varrho$.
Accordingly these subspaces of $R$ are not in direct sum: there exists
$u_{a}\in W\cap F_{\chi(a)}\cap R$ with $u_{a}\neq 0$. Since $W\cap
F_{\chi(a)}\cap R$ is defined over $\overline{\mathbb{Q}}$, we may assume that
$u_{a}\in\overline{\mathbb{Q}}^{\Omega}$. We write
$u_{a}=(u_{a,\kappa})_{\kappa\in\Omega}$ with
$u_{a,\kappa}\in\overline{\mathbb{Q}}$. These coordinates satisfy

$u_{a,\kappa}=\sum_{j=1}^{m}\chi_{j}(a)u_{a,\kappa+e_{j}}\mbox{ for any
}\kappa\in\Omega\setminus\Theta$ (4.5)

since all generators of $F_{\chi(a)}$ satisfy these linear equations.

Let $(T_{1},\ldots,T_{w})$ be a basis of $W$ consisting of vectors of
$\overline{\mathbb{Q}}^{\Omega}$, with $w=\dim W=d-d^{\prime}$. Since
$u_{a}\in W\cap\overline{\mathbb{Q}}^{\Omega}$ there exist
$\lambda_{a,1},\ldots,\lambda_{a,w}\in\overline{\mathbb{Q}}$, not all zero,
such that $u_{a}=\sum_{\ell=1}^{w}\lambda_{a,\ell}T_{\ell}$. Writing
$T_{\ell}=(t_{\ell,\kappa})_{\kappa\in\Omega}$ we have
$u_{a,\kappa}=\sum_{\ell=1}^{w}\lambda_{a,\ell}t_{\ell,\kappa}$ for any
$\kappa\in\Omega$. Using this into Eq. (4.5) yields

$\sum_{\ell=1}^{w}\lambda_{a,\ell}P_{\ell,\kappa}(a)=0\mbox{ for any
}\kappa\in\Omega\setminus\Theta,$ (4.6)

where

$P_{\ell,\kappa}(z)=-t_{\ell,\kappa}+\sum_{j=1}^{m}t_{\ell,\kappa+e_{j}}\chi_{j}(z)$

is a function algebraic over $\overline{\mathbb{Q}}(z)$. Let $P(z)$ denote the
matrix $(P_{\ell,\kappa}(z))_{\kappa\in\Omega\setminus\Theta,1\leq\ell\leq
w}$. For any $a\in\overline{\mathbb{Q}}\cap\mathbb{R}$ with $a\geq M_{1}$, Eq.
(4.6) shows that ${\rm rk}(P(a))<w$: all minors of size $w$ of the matrix
$P(z)$ vanish at $a$. Since these minors are functions algebraic over
$\overline{\mathbb{Q}}(z)$, they are identically zero: $P(z)$ has rank at most
$w-1$, as a matrix with coefficients in $\overline{\mathbb{Q}(z)}$. This
provides algebraic functions
$\mu_{1}(z),\ldots,\mu_{w}(z)\in\overline{\mathbb{Q}(z)}$, not all zero, such
that $\sum_{\ell=1}^{w}\mu_{\ell}(z)P_{\ell,\kappa}(z)=0$ for any
$\kappa\in\Omega\setminus\Theta$. In other words,

$\sum_{\ell=1}^{w}\mu_{\ell}(z)\Big{(}-t_{\ell,\kappa}+\sum_{j=1}^{m}t_{\ell,\kappa+e_{j}}\chi_{j}(z)\Big{)}=0\mbox{
for any }\kappa\in\Omega\setminus\Theta.$ (4.7)

Now as $z\to+\infty$ with $z\in\mathbb{R}$, each non-zero function
$\mu_{\ell}(z)$ has an asymptotic behavior given by
$\mu_{\ell}(z)\sim\mu_{\ell,0}z^{e_{\ell}}$ with
$\mu_{\ell,0}\in\overline{\mathbb{Q}}^{*}$ and $e_{\ell}\in\mathbb{Q}$; if
$\mu_{\ell}(z)$ is identically zero we put $e_{\ell}=-\infty$. Let
$e=\max(e_{1},\ldots,e_{m})$; for any $\ell$, we let
$\mu_{\ell,1}=\mu_{\ell,0}$ if $e_{\ell}=e$, and $\mu_{\ell,1}=0$ otherwise,
so that $\lim_{z\to+\infty}z^{-e}\mu_{\ell}(z)=\mu_{\ell,1}$. We recall that
Eq. (4.2) has a similar flavour, and that in this equation we have
$\varpi_{1}=\ldots=\varpi_{m-1}=0$, $\varpi_{m}=1$, since $\xi^{\prime}$
(denoted now by $\xi$) has been constructed for this purpose. Combining these
limits and letting $z\to+\infty$, with $z\in\mathbb{R}$, Eq. (4.7) yields
(since $D>0$)

$\sum_{\ell=1}^{w}\mu_{\ell,1}t_{\ell,\kappa+e_{m}}=0\mbox{ for any
}\kappa\in\Omega\setminus\Theta.$

Let $T=\sum_{\ell=1}^{w}\mu_{\ell,1}T_{\ell}\in W\setminus\\{0\\}$, and write
$T=(t_{\kappa})_{\kappa\in\Omega}$. Then we have $t_{\kappa+e_{m}}=0$ for any
$\kappa\in\Omega\setminus\Theta$, i.e. $t_{\lambda}=0$ for any
$\lambda\in\Theta$ such that $\lambda_{m}\geq 1$. For any
$\lambda\in\Omega\setminus\Theta$ such that $\lambda_{m}\geq 1$, we obtain
$t_{\lambda}=\sum_{j=1}^{m}\chi_{j}(a_{0})t_{\lambda+e_{j}}=0$ since $T\in
W\subset F_{\chi(a_{0}}$ (see Eq. (4.5)). Therefore $T\in K$: this is a
contradiction with the definition (4.4) of $W$. This concludes the proof of
Lemma 5.∎

###  4.2 Proof of Theorem 2

We prove Theorem 2 by induction on $m+N$. Letting $m,N\geq 1$, if $m\geq 2$
(resp. $N\geq 2$) we may assume that Theorem 2 holds with $m-1$ instead of $m$
(resp. $N-1$ instead of $N$). We shall apply this idea using the linear map
$\pi:\mathbb{C}^{\Omega}\to\mathbb{C}^{\overline{\Omega}}$ defined by
$\pi(E_{\kappa})=E_{\kappa-e_{m}}$ for any $\kappa\in\Omega$; here we let

${\overline{\Omega}}=\\{\kappa\in\mathbb{N}^{m},\,N-2\leq|\kappa|\leq N-1\\}.$

As explained at the beginning of §4, we have $E_{\kappa-e_{m}}=0$ if, and only
if, $\kappa-e_{m}$ has a negative coordinate, i.e. $\kappa_{m}=0$. Therefore
the kernel of $\pi$ is

$K=\ker\pi={\rm Span}\\{E_{\kappa},\,\kappa\in\Omega,\,\kappa_{m}=0\\}$

with the same notation $K$ as in Lemma 5.

The sketch of the proof is the following. Using Theorem 2 with $m-1$ if $m\geq
2$, we shall prove that

$\dim(R\cap K)\geq\Big{(}2-\frac{m-2}{N+m-2}\Big{)}\dim(F\cap R\cap K).$ (4.8)

Then we shall use Theorem 2 with $N-1$ (if $N\geq 2$) to prove that

$\dim\pi(R)\geq\Big{(}2-\frac{m-1}{N+m-2}\Big{)}\dim\pi(F\cap R).$ (4.9)

At last we shall conclude the proof by combining these inequalities with the
one provided by Lemma 5.

Let us start by proving Eq. (4.8). It holds trivially if $m=1$ since in this
case, $K=\\{0\\}$. Therefore we may assume $m\geq 2$ and let
$\Omega^{\prime}=\\{\kappa\in\mathbb{N}^{m-1},\,N-1\leq|\kappa|\leq N\\}$,
$\xi^{\prime}=(\xi_{1},\ldots,\xi_{m-1})$, and $F^{\prime}$ be the subspace
defined from $(m-1,N,\xi^{\prime})$ as explained before the statement of
Theorem 2, spanned by vectors
$Z^{\prime}_{\kappa}\in\mathbb{C}^{\Omega^{\prime}}$ for
$\kappa\in\Theta^{\prime}=\\{\kappa\in\mathbb{N}^{m-1},\,|\kappa|=N\\}.$

Let $\iota$ denote the injective linear map
$\mathbb{C}^{\Omega^{\prime}}\to\mathbb{C}^{\Omega}$ defined by
$\iota(E_{\kappa})=E_{(\kappa,0)}$ for any
$\kappa=(\kappa_{1},\ldots,\kappa_{m-1})\in\Omega^{\prime}$, where
$(\kappa,0)$ stands for $(\kappa_{1},\ldots,\kappa_{m-1},0)$. We claim that
$\iota(F^{\prime})=F\cap K$. The inclusion $\iota(F^{\prime})\subset F\cap K$
is obvious since $\iota(Z^{\prime}_{\kappa})=Z_{(\kappa,0)}$ for any
$\kappa\in\Theta^{\prime}$. Conversely, let
$Z=\sum_{\kappa\in\Theta}\lambda_{\kappa}Z_{\kappa}\in F\cap K$, with complex
numbers $\lambda_{\kappa}$. For any $\kappa\in\Theta$, the coordinate of $Z$
on $E_{\kappa}$ (in the canonical basis of $\mathbb{C}^{\Omega}$) is equal to
$\lambda_{\kappa}$. Since $Z\in K$, we deduce that $\lambda_{\kappa}=0$ for
any $\kappa\in\Theta$ such that $\kappa_{m}\geq 1$. Therefore
$Z=\iota(\sum_{\kappa\in\Theta^{\prime}}\lambda_{(\kappa,0)}Z^{\prime}_{\kappa})\in\iota(F^{\prime})$
and the claim follows.

We may apply Theorem 2 with $(m-1,N,\xi^{\prime})$ to the subspace
$\iota^{-1}(R)$ which is defined over $\overline{\mathbb{Q}}$. This yields

$\dim\iota^{-1}(R)\geq\Big{(}2-\frac{m-2}{N+m-2}\Big{)}\dim(F^{\prime}\cap\iota^{-1}(R)).$
(4.10)

Since $\iota(F^{\prime})=F\cap K$ we have
$F^{\prime}\cap\iota^{-1}(R)=\iota^{-1}(F\cap R\cap K)$. Now the image of
$\iota$ is $K$, so that $\dim(F^{\prime}\cap\iota^{-1}(R))=\dim(F\cap R\cap
K)$ and $\dim\iota^{-1}(R)=\dim(R\cap K)$: Eq. (4.8) follows from Eq. (4.10).

We shall now prove Eq. (4.9). If $N=1$ it reads $\dim\pi(R)\geq\dim\pi(F\cap
R)$ and holds trivially. Let us assume $N\geq 2$. Recall that
${\overline{\Omega}}=\\{\kappa\in\mathbb{N}^{m},\,N-2\leq|\kappa|\leq N-1\\}$,
and denote by $\overline{Z}_{\kappa}$, for
$\kappa\in\overline{\Theta}=\\{\kappa\in\mathbb{N}^{m},\,|\kappa|=N-1\\}$, the
vectors constructed from $\xi$ with respect to $m$ and $N-1$. We write
$\overline{F}={\rm
Span}\\{\overline{Z}_{\kappa},\,\kappa\in\overline{\Theta}\\}$. It is clear
that for any $\kappa\in\Theta$ we have $\pi(Z_{\kappa})=\overline{Z}_{\kappa-
e_{m}}$ if $\kappa_{m}\geq 1$, and $\pi(Z_{\kappa})=0$ otherwise. Therefore
$\pi(F)=\overline{F}$, and Theorem 2 applied to $\pi(R)$ with $(m,N-1,\xi)$
yields

$\dim\pi(R)\geq\Big{(}2-\frac{m-1}{N+m-2}\Big{)}\dim(\pi(F)\cap\pi(R)).$
(4.11)

Since $\pi(F\cap R)\subset\pi(F)\cap\pi(R)$ this implies Eq. (4.9).

We may now conclude the proof of Theorem 2, since using Lemma 5 we may assume
that

$\dim R\geq 2\dim(F\cap R)-\dim(F\cap R\cap K).$ (4.12)

The restriction $\pi_{R}$ has kernel $R\cap K$ and image $\pi(R)$, so that
$\dim R=\dim(R\cap K)+\dim\pi(R)$. Similarly, $\dim(F\cap R)=\dim(F\cap R\cap
K)+\dim\pi(F\cap R)$. Therefore adding Eqs. (4.8) and (4.9) yields

$\dim R\geq\Big{(}2-\frac{m-1}{N+m-2}\Big{)}\dim(F\cap
R)+\frac{1}{N+m-2}\dim(F\cap R\cap K).$

Multiplying this equation by $\frac{N+m-2}{N+m-1}$, and adding Eq. (4.12)
divided by $N+m-1$, yields

$\dim R\geq\Big{(}2-\frac{m-1}{N+m-1}\Big{)}\dim(F\cap R).$ (4.13)

This concludes the proof of the inequality in Theorem 2.

Assume now that equality holds in Eq (4.13). If $m=1$ then $\dim F=\theta=1$,
$\omega=2$, and $R$ is either $\\{0\\}$ or $\mathbb{C}^{\Omega}$ since $F$ is
not defined over $\overline{\mathbb{Q}}$ (because $\xi_{1}$ is
transcendental). Assume now that $m\geq 2$, and notice that equality holds in
Eqs. (4.8), (4.9), (4.11) and (4.12). Using Theorem 2 with $m-1$ instead of
$m$, since equality holds in Eq. (4.8) we have either $R\cap K=\\{0\\}$ or
$R\cap K=K$. In the former case, equality in Eq. (4.12) implies $\dim
R=2\dim(F\cap R)$, which implies $\dim R=0$ since we have assumed that
equality holds in Eq (4.13). In the latter case, we use (if $N\geq 2$) Theorem
2 and the fact that equality holds in Eq. (4.11) to deduce a new alternative:
either $\pi(R)=\mathbb{C}^{{\overline{\Omega}}}$ or $\pi(R)=\\{0\\}$. In the
former case, since $R\cap K=K$ we obtain $R=\mathbb{C}^{\Omega}$. In the
latter case, we have $R=K$ so that Eq. (4.8) reads $\dim
R=\Big{(}2-\frac{m-2}{N+m-2}\Big{)}\dim(F\cap R)>0$. This contradiction with
equality in Eq. (4.13) concludes the proof of Theorem 2.

## References

  * [1] B. Adamczewski, T. Rivoal, Exceptional values of $E$-functions at algebraic points, Bull. Lond. Math. Soc. 50.4 (2018), 697–708.
  * [2] Y. André, Séries Gevrey de type arithmétique I. Théorèmes de pureté et de dualité, Ann. of Math. 151.2 (2000), 705–740.
  * [3] Y. André, Séries Gevrey de type arithmétique II. Transcendance sans transcendance, Ann. of Math. 151.2 (2000), 741–756.
  * [4] A. Baker, On some Diophantine inequalities involving the exponential function, Can. J. Math. 17 (1965), 616–626.
  * [5] D. Bertrand, Le théorème de Siegel-Shidlovsky revisité, in Number theory, Analysis and Geometry: in memory of Serge Lang (D. Goldfeld et al., eds), Springer, 2012, 51–67.
  * [6] D. Bertrand, F. Beukers, Équations différentielles linéaires et majorations de multiplicités, Ann. Sci. École Norm. Sup. 18.1 (1985), 181–192.
  * [7] D. Bertrand, V. Chirskii, J. Yebbou, Effective estimates for global relations on Euler-type series, Annales de la faculté des sciences de Toulouse 13.2 (2004), 241–260.
  * [8] F. Beukers, A refined version of the Siegel-Shidlovskii theorem, Ann. of Math. 163.1 (2006), 369–379.
  * [9] A. Bostan, T. Rivoal, B. Salvy, Minimization of differential equations and algebraic values of $E$-functions, preprint (2022), 48 pages, to appear in Math. Comp.
  * [10] G. V. Chudnovsky, On some applications of diophantine approximations, Proc. Nat. Acad. Sci. USA 81 (1984), 1926–1930.
  * [11] N. I. Feldman, Yu. V. Nesterenko, Transcendental Numbers, in Encyclopaedia of Mathematical Sciences, Vol. 44: Number Theory IV (Springer, 1998).
  * [12] S. Fischler, Shidlovsky’s multiplicity estimate and irrationality of zeta values, J. Austral. Math. Soc. 105.2 (2018), 145–172.
  * [13] S. Fischler, T. Rivoal, On the values of $G$-functions, Commentarii Math. Helv. 89.2 (2014), 313–341.
  * [14] S. Fischler, T. Rivoal, Arithmetic theory of $E$-operators, J. Éc. Polytech. – Mathématiques 3 (2016), 31–65.
  * [15] S. Fischler, T. Rivoal, Linear independence of values of $G$-functions, II: outside the disk of convergence, Ann. Math. Qué. 45 (2021), 53–93.
  * [16] S. Fischler, T. Rivoal, Values of $E$-functions are not Liouville numbers, J. Éc. Polytech. – Mathématiques 11 (2024), 1–18.
  * [17] S. Fischler, T. Rivoal, Effective algebraic independence of values of $E$-functions, preprint (2019), 20 pages, accepted for publication in Math. Z.
  * [18] J. Fresán, P. Jossen, A non-hypergeometric $E$-function, Ann. of Math. 194.3 (2021), 903–942.
  * [19] J. Fresán, Une introduction aux périodes, in Périodes et transcendance, Journées X-UPS 2019, 147 pages, to appear at Éditions de l’École polytechnique, Palaiseau.
  * [20] E. Joseph, On the approximation exponents for subspaces of $\mathbb{R}^{n}$, Mosc. J. Comb. Number Theory 11.1 (2022), 21–35.
  * [21] E. Joseph, Upper bounds and spectrum for approximation exponents for subspaces of $\mathbb{R}^{n}$, J. Théor. Nombres Bordeaux 34.3 (2022), 827–850.
  * [22] L.-C. Kappe, Zur Approximation von $e^{\alpha}$, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 9 (1966), 3–14.
  * [23] S. Lang, Report on Diophantine approximations, Bull. SMF 93 (1965), 177–192.
  * [24] G. Lepetit, Le théorème d’André-Chudnovsky-Katz au sens large, North-West. Eur. J. Math. 7 (2021), 83–149.
  * [25] R. Marcovecchio, The Rhin-Viola method for $\log(2)$, Acta Arith. 139.2 (2009), 147–184.
  * [26] Yu. V. Nesterenko, A. B. Shidlovskii, On the linear independence of values of $E$-functions, Sb. Math. 187 (1996), 1197–1211; translation from the russian Math. Sb. 187 (1996), 93–108.
  * [27] E. G. C. Poole, Introduction to the theory of linear differential equations, Dover Publications Inc., New York, 1960.
  * [28] K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2.1 (1955), 1–20, corrigendum p. 168.
  * [29] A. B. Shidlovskii, Transcendental numbers, de Gruyter Studies in Math. 12, de Gruyter, Berlin, 1989.
  * [30] C. Siegel, Über einige Anwendungen diophantischer Approximationen, vol. 1 S. Abhandlungen Akad., Berlin, 1929.
  * [31] M. van der Put, M. F. Singer, Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, vol. 328, Springer-Verlag, Berlin, 2003.
  * [32] D. Zeilberger, W. Zudilin, The irrationality measure of $\pi$ is at most $7.103205334137\ldots$, Mosc. J. Comb. Number Theory 9.4 (2020), 407–419.
  * [33] W. Zudilin, On rational approximations of values of a certain class of entire functions, Sb. Math. 186.4 (1995), 555–590; translation from the russian Mat. Sb. 186.4 (1995), 89–124.

Stéphane Fischler, Université Paris-Saclay, CNRS, Laboratoire de mathématiques
d’Orsay, 91405 Orsay, France.

Tanguy Rivoal, Université Grenoble Alpes, CNRS, Institut Fourier, CS 40700,
38058 Grenoble cedex 9, France.

Keywords: $E$-functions, Irrationality Measures, Graded Padé Approximants,
Roth’s Theorem.

MSC 2020: 11J82 (Primary), 11J91 (Secondary)