JOURNAL OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 34, Number 2, April 2021, Pages 475–503
https://doi.org/10.1090/jams/960
Article electronically published on January 20, 2021

HYPERTRANSCENDENCE AND LINEAR DIFFERENCEEQUATIONSBORIS ADAMCZEWSKI, THOMAS DREYFUS, AND CHARLOTTE HARDOUIN

1. Introduction
Number theorists have to face the following challenge. On the one hand, theﬁelds of rational and algebraic numbers are too poor, while, on the other hand, theﬁelds of real and complex numbers seem far too large (uncountable). An attempt
to classify transcendental numbers leads to the introduction of the ring of periods
P (see [KZ01]) and to the following classiﬁcation:
Q ⊂ Q ⊂ P ⊂ C.
The main features of P are that most of classical mathematical constants (e.g., π,log 2, ζ(3), Γ(1/3)3 ...)1 belong to it, and that P is a countable set whose elements
contain only a ﬁnite amount of information and can be identiﬁed in an algorithmic
way. After the pioneering works of Cauchy and Riemann, analytic and holomorphic functions ﬁgure prominently in mathematics. As already observed by Hilbert[Hil02], their study gives rise to a similar diﬃculty: the ﬁelds of rational and algebraic functions are too limited, but the ring C[[x]] is huge. In order to overcome thisdeﬁciency, he suggested to classify transcendental functions according to whether
they are holonomic (i.e., satisfy a linear diﬀerential equation with polynomial coefﬁcients), diﬀerentially algebraic (i.e., satisfy an algebraic diﬀerential equation), or
not. Hence, we obtain the following classiﬁcation:(1.1)

Rat ⊂ Alg ⊂ Hol ⊂ Dif ⊂ C[[x]] .

Functions that are not diﬀerentially algebraic are said to be hypertranscendental.
Note that, if we restrict our attention to power series with algebraic coeﬃcients,
the rings Hol and Dif become countable, and, again, their elements contain only aﬁnite amount of information and can be identiﬁed in an algorithmic way. Theserings play, in the setting of holomorphic functions, a role similar to the one playedby P in the setting of complex numbers. The classiﬁcation (1.1) is also signiﬁcantin enumerative combinatorics. To some extent, the nature of a generating series
reﬂects the underlying structure of the objects it counts (see the discussion in[BM06]). An appealing example, where all cases of (1.1) appear at once, is given
Received by the editors October 7, 2019, and, in revised form, June 22, 2020.
2020 Mathematics Subject Classiﬁcation. Primary 39A06, 12H05.
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under the Grant Agreement No
648132, and from the ANR De rerum natura project (ANR-19-CE40-0018).
1 However, there should be some exceptions. For instance, it is conjectured that e is not aperiod.
c
2021
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476

BORIS ADAMCZEWSKI ET AL.

by the study of generating series associated with lattice walks (see, for instance,[DHRS18] and the references therein).
Hilbert also observed that the class of diﬀerentially algebraic functions misses
important functions coming from number theory. Here are few examples. The veryﬁrst comes from a classical result of Hölder [Höl87] stating that the gamma function
Γ(x) is hypertranscendental. Furthermore, it follows from the identity
 πx 
ζ(1 − x)ζ(x) = 2(2π)x−1 Γ(1 − x) sin
2
that the Riemann zeta function is also hypertranscendental.
In an other direction,
nMoore [Moo96] has shown that f1 (x) = n≥0 x2 is hypertranscendental. A result
reproved and extended later by Mahler [Mah30a] in connection with the so-called
Mahler’s method in transcendental number theory. More recently, Hardouin andSinger [HS08] proved hypertranscendence for some q-hypergeometric series, such as
f2 (x) :=∞
(1 − a)2 (1 − aq)2 · · · (1 − aq n−1 )2n=0(1 − q)2 (1 − q 2 )2 · · · (1 − q n )2

xn ,

∗

where q ∈ C is not a root of unity, a ∈ q Z and a2 ∈ q Z . Most proofs about
hypertranscendence turn out to be based on the fact that the function under consideration satisﬁes a functional equation of a diﬀerent type (i.e., not diﬀerential).
For instance, the three results we just mentioned are respectively based on thefollowing linear diﬀerence equations:Γ(x + 1) = xΓ(x) , f1 (x2 ) = f1 (x) − x ,and2ax − 2x−1
f2 (qx) + 2
f2 (x) = 0 .
2
a x−1
a x−1
Broadly speaking, satisfying both a linear diﬀerence equation and an algebraic differential equation would enforce too much symmetry for a transcendental function.
In this paper, we follow this motto:(A) A solution to a linear diﬀerence equation should be either rational or hypertranscendental.
Of course, this must be taken with a pinch
For instance, log(x), exp(x), of salt.
−n(n−1)/2 nqx are all diﬀerentiallyand the Jacobi theta function θq (x) =n∈Z
algebraic, but they also satisfy the following simple linear diﬀerence equations:f2 (q 2 x) −log(xp ) = p log x , exp(x + 1) = e exp(x) , θq (qx) = qxθq (x) .
Nevertheless, we obtain in this paper the ﬁrst complete results about hypertranscendence of solutions to general linear diﬀerence equations associated with the shiftoperator x → x + h (h ∈ C∗ ), the q-diﬀerence operator x → qx (q ∈ C∗ not a rootof unity), and the Mahler operator x → xp (p ≥ 2 integer). The only restrictionis that we constrain our solutions to be expressed as (possibly ramiﬁed) Laurent
series in the variable x (or in the variable 1/x in some special case associated withthe shift operator). For a precise statement, we refer the reader to our main result,
Theorem 1.2, which conﬁrms guess (A) for these three operators.
1.1. Statement of our main result. All along this paper, our framework consists
in a tower of ﬁeld extensions C ⊂ K ⊂ F0 ⊂ F with the following properties.

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HYPERTRANSCENDENCE AND LINEAR DIFFERENCE EQUATIONS

477• The ﬁeld K is equipped with an automorphism ρ and a derivation2 ∂.
• The automorphism ρ extends to F0 and F .
• The derivation ∂ extends to F , but F0 is not necessarily closed under ∂.
Our aim is to prove that, if f ∈ F0 is solution to a linear ρ-equation with coeﬃcients
in K, then either f ∈ K or f is hypertranscendental over K. Speciﬁcally, we
consider the following four situations, which we refer to as Cases S0 , S∞ , Q, andM, respectively.
dCase S0 . In this case, we let K = C(x), ∂ = dx
, and ρ denote the automorphismof K deﬁned by
ρ(f (x)) = f (x + h), h ∈ C∗ .
The automorphism ρ and the derivation ∂ naturally extend to the of ﬁeld F :=Mer(C) of meromorphic functions on C. Let K0 := C(x, exp(∗x)) denote the ﬁeld
generated over C(x) by the functions exp(x),  ∈ C. In this setting, we let F0 be
any ﬁeld extension of C(x) on which ρ extends and satisfying the following three
conditions.
(i) F0 ⊂ Mer(C).
(ii) {f ∈ F0 | ρ(f ) = f } = C.
(iii) F0 ∩ K0 = C(x).

Remark 1.1. For h = 1, let us see that we can choose F0 = C(x, Γ(x)), where Γ(x)is the Gamma function. The functional equation Γ(x + 1) = xΓ(x) shows that
ρ extends to F0 and it is well-known that Γ ∈ Mer(C). Thus, (i) holds. Now,
let β ∈ F0 be such that ρ(β) = β. Writing β as a rational function in Γ with
coeﬃcients in C(x), and then using the functional equation satisﬁed by Γ and thefact that Γ is transcendental over C(x), we can deduce that β ∈ C. Hence (ii)holds. Finally, to prove (iii) it is suﬃcient to show that Γ is transcendental overK0 . The latter property follows from classical asymptotics showing that, for every
α(x) ∈ K0 , we have limx→+∞ |Γ(x)/α(x)| = +∞. Thus, Case S0 of Theorem 1.2
is a generalization of Hölder’s theorem.
Case S∞ . In this case, we consider K, ρ, and ∂ as in Case S0 . We note that the
automorphism ρ and the derivation ∂ naturally extend to the of ﬁeld of Laurent
series C((x−1 )), and we set F0 = F = C((x−1 )).
Case Q. In this case, we let K = j≥1 C(x1/j ) denote the ﬁeld of ramiﬁed rational
functions. We also use the notation C(x1/∗ ) for this ﬁeld. Given q ∈ C∗ that is nota root of unity, we let ρ denote the automorphism of K deﬁned bydρ(f (x)) = f (qx), ∂ = x ,dxand we let F0 = F = j≥1 C((x1/j )) denote the ﬁeld of Puiseux series. We also use
the notation C((x1/∗ )) for this ﬁeld.
Case M. In this case, we let K = C(x1/∗ ) and, given a natural number p ≥ 2, we
let ρ denote the automorphism of K deﬁned by
ρ(f (x)) = f (xp ) ,dand F0 = F = C((x1/∗ )).
and we set ∂ = x dx
2 A derivation ∂ on K is a map from K into itself satisfying ∂(a + b) = ∂(a) + ∂(b) and∂(ab) = ∂(a)b + a∂(b) for every a, b ∈ K.

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In all cases, ∂ is a derivation on K and F . An element f in F is said to be
diﬀerentially algebraic or ∂-algebraic over K if there exists n ∈ N such that thefunctions f, ∂(f ), . . . , ∂ (n) (f ) are algebraically dependent over K. Otherwise, f issaid to be hypertranscendental over K. Our main result reads as follow.
Theorem 1.2. Let K, F0 , and ρ be deﬁned as in Cases S0 , S∞ , Q, and M, andlet n be a positive integer. Let f ∈ F0 be solution to the ρ-linear diﬀerence equationof order n(1.2)

ρn (y) + an−1 ρn−1 (y) + · · · + a1 ρ(y) + a0 y = 0 ,where a0 , . . . , an−1 ∈ K. Then either f ∈ K or f is hypertranscendental over K.
On the road to Theorem 1.2, there is already a whole bunch of partial results
and a number of them are used in our proof. These results can roughly be divided
into three diﬀerent types.
• Those concerned with aﬃne equations of order one, that is, with equationsof the form ρ(y) = ay +b. References include [Höl87,Moo96,Mah30a,Nis84,
Ran92, Ish98, Har08, HS08, Ngu12].
• Those concerned with equations whose diﬀerence Galois group is large (e.g.,
simple, semisimple, reductive). References include [HS08, DHR18, DHR16,
AS17, ADR18]. In particular, such results have nice applications to equations of order 2.
• Those dealing with general equations, but reaching only nonholonomicity
instead of hypertranscendence (see Theorem 1.3 and Remark 1.4). References include [Ram92, BG93, Béz94, SS19].
Among these results, we quote the following one, which may be considered as aﬁrst step towards a proof of Theorem 1.2 (see Remark 1.4 for precise attribution).
Theorem 1.3. In each of the cases S0 , S∞ , Q, and M, the following property
holds. Any element of F0 that satisﬁes both a ρ-linear equation and a ∂-linear
equation with coeﬃcients in K belongs to K.
Remark 1.4. Theorem 1.3 can be rephrased by saying that, if f is as in Theorem 1.2,
then either f ∈ K or f is not holonomic. Note that recently the authors of [SS19]
give a uniform proof for all cases of Theorem 1.3, see Corollaries 3 and 5 therein.
In each case, we also mention the original reference. Case S0 is due to Bézivin
and Gramain [BG93]. Case S∞ is due to Schäfke and Singer [SS19, Corollary 5].
Furthermore, in Cases Q and M, a change of variable of the form z = x1/ ,  ∈ N∗ ,can be used to reduce the situation to the case where K = C(x) and F0 = C((x)).
In the latter situation, Case Q is due to Ramis [Ram92], while Case M is due toBézivin [Béz94].
1.2. Strategy of proof. Elements of the rings Rat, Alg, and Hol are wellunderstood. We have recurrence formulas for their coeﬃcients, precise asymptotics, and a good understanding of their ﬁnitely many singularities. In contrast,
our knowledge about diﬀerentially algebraic functions is much more limited (see, forinstance, the survey [Rub89]). In fact, such functions can behave quite wildly. For n2
example, the function
x is diﬀerentially algebraic and admits the unit circle
as a natural boundary3 . Thus, proving hypertranscendence requires more involved
3 Incidentally, this example shows that the ”outrageous conjecture” suggested at the end of −n2
is transcendental.
[Rub89] is false. Indeed, it follows from a theorem of Nesterenko that2

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HYPERTRANSCENDENCE AND LINEAR DIFFERENCE EQUATIONS

479

tools, and is substantially more diﬃcult than proving irrationality, transcendence,or nonholonomicity. In this direction, Hardouin and Singer [HS08] built a new
Galois theory of diﬀerence equations, which is designed to measure the diﬀerential dependencies among solutions to linear diﬀerence equations. In particular, itapplies to hypertranscendence. We refer the reader to [DV12] and [Har16] for an
introduction to this topic. Our proof of Theorem 1.2 relies on this theory.
Let us brieﬂy describe our strategy. Since f satisﬁes a linear diﬀerence equation,
it is classical to associate with this equation its diﬀerence Galois group, which is alinear algebraic group that encodes the algebraic relations between the solutions tothis equation. The more involved parametrized Galois theory developed in [HS08]
attaches to any linear diﬀerence equation, a geometric object, the parametrized Galois group, whose structure encodes the algebraic relations satisﬁed by the solutions
and their derivatives. This is not a linear algebraic group anymore, but a linear
diﬀerential group (see Section 2.2). Nevertheless, both groups are strongly related
and, viewed on a suitable ﬁeld extension, the latter is Zariski dense in the former(see Proposition 2.5). In this framework, we can conclude that f is hypertranscendental when the corresponding parametrized Galois group is large. Furthermore,
there are several results, culminating in [AS17], showing that if the classical diﬀerence Galois group is large, then both groups are the same when viewed on a suitableﬁeld extension. We emphasize that such results follow a strategy initiated in [HS08]
and developed further in [DHR18]. It combines a fundamental result about classiﬁcation of diﬀerential algebraic subgroups of semisimple algebraic groups by Cassidy[Cas72, Cas89], parametrized Galois correspondence, and Theorem 1.3.
We prove Theorem 1.2 by induction on the order n of the ρ-linear equation
satisﬁed by f . We ﬁrst prove the result for aﬃne equations of order one. Then weshow that, iterating the associated linear system if necessary, one can reduce thesituation to the case where the associated diﬀerence Galois group G is connected.
Now, if G acts irreducibly on Cn , we show how to reduce the situation to thecase where G is semisimple. In that case, the diﬀerence Galois group is large
enough and a recent result of Arreche and Singer [AS17] allows us to conclude
that f is hypertranscendental. Finally, if G acts reducibly on Cn , we assume thatf is diﬀerentially algebraic and we show how to construct from f a diﬀerentiallyalgebraic function g that satisﬁes a linear equation of smaller order. Furthermore,this construction ensures that g ∈ K if and only if f ∈ K. Applying the induction
assumption to g, we deduce that f ∈ K, as wanted. Thus, once the case of order
one equation is solved, the proof really makes a dichotomy between irreducible andreducible diﬀerence Galois groups.
1.3. An application to number theory. At the end of the 1920s, Mahler [Mah29,
Mah30b,Mah30a] introduced a new method for proving transcendence and algebraic
independence of values at algebraic points of analytic functions satisfying diﬀerent types of functional equations associated with the operator x → xp . Mahler’s
method aims at transferring results about the algebraic independence over Q(x)of such functions to the algebraic independence over Q of their values at algebraic
points. A major result in this direction is Nishioka’s theorem [Nis90], which provides an analogue of the Siegel-Shidlovskii theorem for linear Mahler systems. Let
us also mention that signiﬁcant progress in this theory has been made recently[Phi15, AF17, AF20, AF18]. Combining Case M of Theorem 1.2 with Nishioka’s
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480

BORIS ADAMCZEWSKI ET AL.

[AF18], we can deduce the following general result. We recall that f (x) ∈ Q[[x]] isa p-Mahler function if there exist polynomials a0 (x), . . . , an (x) ∈ Q[x], not all zero,such thatna0 (x)f (x) + a1 (x)f (xp ) + · · · + an (x)f (xp ) = 0 .
We simply say that f is a Mahler function if it is p-Mahler for some integer p ≥ 2.
According to Theorem 1.2, a Mahler function is thus either rational or hypertranscendental. We recall that an irrational Mahler function is meromorphic on the
open unit disc and admits the unit circle as a natural boundary.
Theorem 1.5. Let f (x) ∈ Q[[x]] be a Mahler function that is not rational, let r bea positive integer, and let K be a compact subset of the open unit disc. Then, for all
but ﬁnitely many algebraic numbers α ∈ K, the complex numbers f (α), f  (α), . . . ,f (r) (α) are algebraically independent over Q.
Results of this type have been ﬁrst obtained by Mahler [Mah30a] for solutions
to aﬃne order one equations. The main feature of Theorem 1.5 is that it applies
to any irrational Mahler function. It is also almost best possible in the sense that
one cannot avoid the possibility of ﬁnding inﬁnitely many exceptions in the wholeopen unit disc. For instance, the transcendental 2-Mahler function∞n(1 − 3x2 )f (x) =
n=0nvanishes at all algebraic numbers α such that α2 = 1/3 for some integer n ≥ 0.
This shows that, even for r = 1, the exceptional set can depend on K. However, if
K is ﬁxed, we do not know whether the exceptional set always remains ﬁnite whenr tends to inﬁnity.
1.4. Organization of the paper. This article is organized as follows. In Section
2, we provide a short introduction to diﬀerence Galois theory and to parametrized
diﬀerence Galois theory, following [vdPS97, HS08]. In Section 3, we describe, foreach Cases S0 , S∞ , Q, and M, the suitable framework that is needed to use theparametrized diﬀerence Galois theory of [HS08]. Several auxiliary results are gathered in Section 4. They concern algebraic groups, the behavior of the diﬀerence
Galois group when considered over diﬀerent ﬁeld extensions, and Picard-Vessiot
extensions associated with iterated systems. Finally, Section 5 is devoted to theproof of Theorem 1.2.
2. Difference and parametrized difference Galois theories
In this section, we provide a short introduction to diﬀerence Galois theory andto parametrized diﬀerence Galois theory.
2.1. Diﬀerence Galois theory. We ﬁrst recall some notation, as well as classicalresults, concerning diﬀerence Galois theory. We refer the reader to [vdPS97] formore details.
2.1.1. Notation in operator algebra. A diﬀerence ring is a pair (R, ρ) where R isa ring and ρ is a ring automorphism of R. If R is a ﬁeld, then (R, ρ) is called adiﬀerence ﬁeld or a ρ-ﬁeld. An ideal I of R such that ρ(I) ⊂ I is called a diﬀerence
ideal or a ρ-ideal. We say that the diﬀerence ring (R, ρ) is simple if the only ρ-idealsof R are {0} and R. Two diﬀerence rings (R1 , ρ1 ) and (R2 , ρ2 ) are isomorphic if

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HYPERTRANSCENDENCE AND LINEAR DIFFERENCE EQUATIONS

481

there exists a ring isomorphism ϕ between R1 and R2 such that ϕ ◦ ρ1 = ρ2 ◦ ϕ. A
diﬀerence ring (S, ρ ) is a diﬀerence ring extension of (R, ρ) if S is a ring extensionof R and if ρ|R = ρ. In this case, we usually keep on denoting ρ by ρ. When R
is a ρ-ﬁeld, we say that S is a R-ρ-algebra. Two diﬀerence ring extensions (R1 , ρ)and (R2 , ρ) of the diﬀerence ring (R, ρ) are isomorphic over (R, ρ) if there exists a
diﬀerence ring isomorphism ϕ from (R1 , ρ) to (R2 , ρ) such that ϕ|R = IdR . The
ring of constants of the diﬀerence ring (R, ρ) is deﬁned byRρ := {f ∈ R | ρ(f ) = f } .
If Rρ is a ﬁeld, it is called the ﬁeld of constants. If there is no risk of confusion,
we usually simply say that R, instead of (R, ρ), is a diﬀerence ring (or a diﬀerence
ﬁeld, or a diﬀerence ring extension...).
2.1.2. Diﬀerence equations and linear diﬀerence systems. Let (K, ρ) be a diﬀerenceﬁeld. A linear ρ-equation of order n over K is an equation of the form
(2.1)

L(y) := ρn (y) + an−1 ρn−1 (y) + · · · + a0 y = 0 ,with a0 , ..., an−1 ∈ K. If a0 = 0, this relation can be written in matrix form as(2.2)

ρY = AL Y

⎞
···
0
⎜
.. ⎟
..
⎜ 0
.
0
1
. ⎟
⎟
⎜
⎟ ∈ GLn (K) .
⎜
.
.
.
.
:=
AL
..
..
..
⎜ ..
0 ⎟
⎟
⎜
⎝ 0
0
···
0
1 ⎠
−a0 −a1 · · · · · · −an−1
The matrix AL is called the companion matrix associated with Equation (2.1). It
is often more convenient to use the notion of linear diﬀerence system, that is, ofsystem of the form
where

(2.3)

⎛

0

1

0

ρ(Y ) = AY, with A ∈ GLn (K).

We recall that two diﬀerence systems ρ(Y ) = AY and ρ(Y ) = BY with A, B ∈GLn (K) are said to be equivalent over K if there exists a gauge transformation
T ∈ GLn (K) such that B = ρ(T )AT −1 . In that case, ρ(Y ) = AY if and only if
ρ(T Y ) = B(T Y ).
Remark 2.1. By [HS99, Theorem B.2.], if K contains a nonperiodic element withrespect to ρ, then the cyclic vector lemma holds, and any linear diﬀerence system is
equivalent to one given by a companion matrix associated with some linear equation.
Let L be a ρ-ﬁeld extension of K and let (f0 , f1 , . . . , fn−1 ) ∈ Ln be a nonzero
solution to a linear system ρ(Y ) = AY , with A ∈ GLn (K). Then each coordinate
fj satisﬁes some linear ρ-equation over K of order at most n.
2.1.3. Picard-Vessiot rings and diﬀerence Galois groups. A Picard-Vessiot ring for(2.3) over a diﬀerence ﬁeld (K, ρ) of characteristic zero is a K-ρ-algebra satisfying
the following three properties.
(1) There exists U ∈ GLn (R) such that ρ(U ) = AU . Such a matrix U is calleda fundamental matrix.
(2) R is generated as a ring over K by the coordinates of U and by det(U )−1 ,that is, R = K[U, det(U )−1 ].

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(3) R is a simple diﬀerence ring.
A Picard-Vessiot ring is not always an integral domain. However, it is a direct
sum of integral domains which are transitively permuted by ρ. More precisely, by[vdPS97, Corollary 1.16], there exist r ≥ 1 and orthogonal idempotent elements
e0 , . . . , er−1 , such thatR = e0 R ⊕ · · · ⊕ er−1 R ,where ei R is an integral domain and ρ(ei ) = ei+1modr . We recall that PicardVessiot rings always exist. Indeed, it is suﬃcient to endow the polynomial K1] with a structure of K-ρ-algebra by setting ρ(X) = AX. Then,
algebra K[X, det(X)11], the quotient R = K[X, det(X)]/M is a
for any maximal ρ-ideal M of K[X, det(X)Picard-Vessiot ring. However, this construction does not guarantee that the ring of
ρ-constants has not grown. This justiﬁes the introduction of the more convenient
notion of Picard-Vessiot extension.
A Picard-Vessiot extension Q for (2.3) over a diﬀerence ﬁeld (K, ρ) of characteristic zero is a K-ρ-algebra Q satisfying the following properties.
(1) There exists U ∈ GLn (Q) such that ρ(U ) = AU . Such a matrix U is calleda fundamental matrix.
(2) Q is a pseudoﬁeld extension of K, that is, there exist a positive integer r,
orthogonal idempotent elements e0 , . . . , er−1 of Q, and a ﬁeld extension L
of K such that Q = e0 L ⊕ . . . ⊕ er−1 L and ρ(ei ) = ei+1modr .
(3) Q is the smallest pseudoﬁeld extension of K containing U .
(4) Qρ = K ρ .
When Q is a ﬁeld, we say that Q is a Picard-Vessiot ﬁeld extension. By [vdPS97,§1.1], if K ρ is algebraically closed, then there exists a Picard-Vessiot extension
and two Picard-Vessiot extensions are isomorphic as K-ρ-algebras. The relation
between Picard-Vessiot-rings and Picard-Vessiot-extensions is given by Proposition
2.2, which is a straightforward consequence of [OW15, Proposition 2.5 and Corollary
2.6].
Proposition 2.2. If K ρ is an algebraically closed ﬁeld of characteristic zero, thenthe following properties hold.
• Let Q be a Picard-Vessiot extension over K and let us deﬁne1] ⊂ Q, where U is a fundamental matrix. Then R is a
R := K[U, det(U)Picard-Vessiot ring.
• If R is a Picard-Vessiot ring over K then the total quotient ring4 of R is a
Picard-Vessiot extension.
From now on, we assume that K is a ρ-ﬁeld such that k := K ρ is an algebraicallyclosed ﬁeld of characteristic zero. Let Q be a Picard-Vessiot extension over K. The
diﬀerence Galois group Gal(Q/K) of (2.3) over K is deﬁned as the group of K-ρalgebra automorphisms of Q:
Gal(Q/K) := {σ ∈ Aut(Q/K) | ρ ◦ σ = σ ◦ ρ}.
For any fundamental matrix U ∈ GLn (Q), an easy computation shows thatU −1 σ(U ) ∈ GLn (k) for all σ ∈ Gal(Q/K). By [vdPS97, Theorem 1.13], the faithful
4 We recall that, given a ring R, its total quotient ring is deﬁned as the localization of R at themultiplicative set formed by all nonzero divisors of R.

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HYPERTRANSCENDENCE AND LINEAR DIFFERENCE EQUATIONS

483

representation
Gal(Q/K) → GLn (k)σ→ U −1 σ(U )identiﬁes Gal(Q/K) with a linear algebraic subgroup G ⊂ GLn (k). Choosing
another fundamental matrix of solutions U leads to a conjugate representation.
2.1.4. Torsor and algebraic relations. A fundamental result in diﬀerence Galois theory ([vdPS97, Theorem 1.13]) says that the Picard-Vessiot ring R is the coordinate
ring of a Gal(Q/K)-torsor over K. Thereby, the diﬀerence Galois group controls
the algebraic relations satisﬁed by the solutions to the underlying linear system.
As a corollary of this structure of Gal(Q/K)-torsor, one obtains the fundamental
equality
dimk Gal(Q/K) = trdegK Q ,where we let trdegK Q denote the transcendence degree of the ﬁeld extension L/K.
Here, the ﬁeld L is deﬁned as in (2) of the deﬁnition of a Picard-Vessiot extension.
2.1.5. The Galois correspondence. The Galois correspondence for linear diﬀerence
systems can be summarized as follows (see [vdPS97, Theorem 1.29]).
Theorem 2.3. Let Q be a Picard-Vessiot extension over K. Let R denote the setof K-ρ-algebras F such that F ⊂ Q and such that every nonzero divisor of F isa unit of F . Let G denote the set of algebraic subgroups of Gal(Q/K). Then thefollowing properties hold.
- For any F ∈ R, the set
G(Q/F ) := {σ ∈ Gal(Q/K) | σ(f ) = f,

∀f ∈ F }is an algebraic subgroup of Gal(Q/K).
- For any H ∈ G, the set
QH := {f ∈ Q | σ(f ) = f, ∀σ ∈ H}
belongs to R.
- The mapsr:

R→G
F → G(Q/F )andg: G→RH → QH

are each other’s inverses.
Remark 2.4. In the case where R is an integral domain, Q is a diﬀerence ﬁeld,
and Theorem 2.3 provides a correspondence between the diﬀerence subﬁelds of Q
containing K, on the one hand, and the algebraic subgroups of Gal(Q/K), on theother hand.
2.2. Parametrized diﬀerence Galois theory. We recall here some notation andresults concerning parametrized diﬀerence Galois theory. We refer the reader to[HS08] for more details.

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2.2.1. Notation in operator algebra. A diﬀerential ring is a pair (R, δ) where R isa ring and δ is a derivation on R, that is, a homomorphism of the additive groupfrom R into itself satisfying the Leibniz rule:δ(ab) = δ(a)b + aδ(b) .
If R is a ﬁeld, then (R, δ) is called a diﬀerential ﬁeld or a δ-ﬁeld. The ring of
constants of R is deﬁned byRδ = {f ∈ R | δ(f ) = 0} .
Let L be a δ-ring extension of a δ-ring R. Given a subset S of L, we let R{S}δ
denote the smallest R-δ-algebra generated by S. A ﬁeld K endowed with a structureof both diﬀerence and diﬀerential ﬁeld is called a (ρ, δ)-ﬁeld if ρ and δ commute,that is, ifρ ◦ δ(f ) = δ ◦ ρ(f )for every f ∈ K.
The study of algebraic structures of diﬀerence and diﬀerential ﬁelds is the main
object of the so-called diﬀerential and diﬀerence algebra. We refer the interested
reader to the founding book of Kolchin ([Kol73]) for further details concerning
diﬀerential algebra and to the book of Cohn for further details concerning diﬀerence
algebra ([Coh65]). It is worth mentioning that these two setting are very diﬀerent.
In order to avoid many analogous deﬁnitions, we use the following convention: we
add an operator preﬁx to an algebraic attribute to signify the compatibility of thealgebraic structure with respect to the operator. For instance, a (ρ, δ)-ring is aring equipped with ρ and δ, a (ρ, δ)-ﬁeld extension K ⊂ L is a ﬁeld extension suchthat the ﬁelds L and K are (ρ, δ)-ﬁelds and such that the action of ρ and δ arecompatible with the ﬁeld extension, and so on.
2.2.2. (ρ, δ)-Picard-Vessiot ring and the parametrized diﬀerence Galois group. Let
 We consider the diﬀerence system be a (ρ, δ)-ﬁeld and let A ∈ GLn (K).
K
(2.4)

ρ(Y ) = AY.

 is a (ρ, δ)-ring extension R
 of KA (ρ, δ)-Picard-Vessiot ring for (2.4) over K
satisfying the following three properties.
 such that ρ(U ) = AU . Such a matrix U is called(1) There exists U ∈ GLn (R)a fundamental matrix.
 is generated as a δ-ring over K
 by the coordinates of U and by det(U )−1 ,(2) R = K{U,that is, R
det(U )−1 }δ .
 are {0} and R.
(3) R is a simple (ρ, δ)-ring, that is, the only (ρ, δ)-ideals of R
A (ρ, δ)-Picard-Vessiot ring is not always an integral domain but it is a direct sumof integral domains closed under δ and transitively permuted by ρ. A construction
similar to that of Section 2.1.3 shows that (ρ, δ)-Picard-Vessiot rings always exist.
Again, this construction does not ensure that the ring of ρ-constants has not grown.
This justiﬁes the introduction of the more convenient notion of (ρ, δ)-Picard-Vessiot
extension.
 for (2.4) over K
 is a K-ρ-δ-algebraA (ρ, δ)-Picard-Vessiot extension Q
satisfying
the following properties.
 such that ρ(U ) = AU . Such a matrix U is called(1) There exists U ∈ GLn (Q)a fundamental matrix.

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HYPERTRANSCENDENCE AND LINEAR DIFFERENCE EQUATIONS

485

 is a pseudo δ-ﬁeld extension of K, that is, there exist a positive integer r,(2) Q and L
 a δ-ﬁeld extension
orthogonal idempotent elements e0 , . . . , er−1 of Q,
of K such that Q = e0 L ⊕ . . . ⊕ er−1 L and ρ(ei ) = ei+1modr .
 is equal to KU , that is, the smallest pseudo δ-ﬁeld extension of K(3) Q
containing U .
ρ = K ρ.
(4) Q is a ﬁeld, we say that Q
 is a (ρ, δ)-Picard-Vessiot ﬁeld extension.
When Qρ
By [Wib12, Corollary 9], if K is algebraically closed, then there exists a (ρ, δ) for (2.4) over K.
 The smallest K-(ρ,δ)-subalgebra
Picard-Vessiot extension Q
1


K{U,
}δ of Q generated by a fundamental matrix and the inverse of its deterdet(U)minant is a (ρ, δ)-Picard-Vessiot ring. However, in order to ensure the uniqueness
of (ρ, δ)-Picard-Vessiot rings up to K-(ρ,δ)-algebra isomorphisms, one needs theρ to be δ-closed ([HS08, Proposition 6.16]). We recall that a diﬀerential ﬁeld
ﬁeld K(L, δ) is called diﬀerentially closed or δ-closed if, for every set of δ-polynomials F,
the system of δ-equations F = 0 has a solution in some δ-ﬁeld extension of L if
and only if it has a solution in L. Note that a δ-closed ﬁeld is algebraically closed.
Diﬀerentially closed ﬁelds are huge and, for instance, none of the function ﬁelds K
introduced in Section 1.1 satisﬁes the assumption that K ρ is diﬀerentially closed.
Nonetheless, working with δ-closed ﬁelds allows us to simplify many arguments.
 whose ﬁeld of ρ-constants is δ-closed,
Thus, we will embed K into a (ρ, δ)-ﬁeld K
and we will use some descent argument to go back to K if necessary (see Sections
3 and 4.2).
 denote a (ρ, δ)-ﬁeld whose ﬁeld of ρ-constants C := Kρ
From now on, we let Kis δ-closed. Let Q be a (ρ, δ)-Picard-Vessiot extension over K. The parametrized
 K)
 of (2.4) over (K, ρ, δ), also called the (ρ, δ)diﬀerence Galois group Galδ (Q/
Galois group, is deﬁned as the group of K-(ρ,δ)-algebra automorphisms of Q:



 K)
 := σ ∈ Aut(Q/ K)  ρ ◦ σ = σ ◦ ρ and δ ◦ σ = σ ◦ δ .
Galδ (Q/ K)
 is a geometric object that
The parametrized diﬀerence Galois group Galδ (Q/encodes the diﬀerential algebraic relations between the solutions to (2.4). See [HS08,
 K),
 the
Proposition 6.24] for more details. Roughly speaking, the larger Galδ (Q/ that hold among the elements of Q.
fewer δ-algebraic relations over K
For any fundamental matrix U ∈ GLn (Q), an easy computation shows that for any element σ ∈ Galδ (Q/ K).
 By [HS08, Proposition 6.18],
U −1 σ(U ) ∈ GLn (C)the faithful representationGalδ (Q/K)→ GLn (C)σ

→ U −1 σ(U ) K) with a linear diﬀerential algebraic subgroup H ⊂ GLn (C),
identiﬁes Galδ (Q/
i.e., a subgroup of GLn (C) that can be deﬁned as the vanishing set of polynomial
 Furthermore, choosing another fundamental
δ-equations with coeﬃcients in C.
matrix U leads to a conjugate representation.
2.2.3. Comparison between diﬀerence Galois groups and parametrized diﬀerence denote a (ρ, δ)-Picard-Vessiot ring for the system (2.4) over K,Galois groups. Let R denote the associated (ρ, δ)-Picard-Vessiot extension, and let U ∈ GLn (R)
 be
let Q

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486

BORIS ADAMCZEWSKI ET AL.

a fundamental matrix of solutions.
By [HS08, Proposition 6.21], R :=−1
 Let Q denote the PicardK[U, det(U ) ] is a Picard-Vessiot ring for (2.4) over K.
 and one can identify
Vessiot extension corresponding to R. Then we have Q ⊂ Q
δ  

 K)

Gal (Q/K) with a subgroup of Gal(Q/K) by restricting the elements of Galδ (Q/to Q. The following result provides a fundamental link between classical diﬀerence
Galois theory and its parametrized counterpart, see [HS08, Proposition 6.21].
 K)
 is aProposition 2.5. The parametrized diﬀerence Galois group Galδ (Q/
Zariski-dense subgroup of Gal(Q/K).

3. The parametrized framework
Let us go back to our K, F0 , F, ρ, ∂ corresponding to Cases S0 , S∞ , Q, and M(see Section 1.1). We ﬁrst observe that, in each case, K ρ = C is algebraically closed
and K has characteristic zero. Hence the theory of [vdPS97] described in Section
 of K, and
2.1 applies. In this section, we describe some suitable ﬁeld extensions K
 of C = F ρ , that allow us to apply the parametrized diﬀerence Galois theory ofC[HS08] described in Section 2.2.
We ﬁrst recall that any diﬀerential ﬁeld K has a diﬀerential closure, i.e., adiﬀerentially closed ﬁeld extension that can be embedded in every diﬀerentially denote a
closed ﬁeld extension of K. In Case S0 , we set δ := ∂, and we let C
δ-closure of Ch , the subﬁeld of Mer(C) formed by those meromorphic functions
that are h-periodic. Note that Ch is also a δ-ﬁeld for ρ and δ commute. Sometimes,h instead of C.
 In Cases S∞ and Q, we also set δ := ∂,we will use the notation C By [DHR18, Lemma 2.3],
 := C.
and we let C denote a δ-closure of C and we set C
in these three cases, the ﬁeld
 := Frac(K ⊗C C)K ρ = C.
is a (ρ, δ)-ﬁeld extension of K such that K
Case M is a bit more tricky. Indeed, in order to apply the theory of [HS08],
 such that δ and ρ commute. Following
we need to consider a derivation δ on K
[DHR18], we let log denote a transcendental element over K, and we deﬁne δ =log ×∂ to be the derivation that acts on K(log) byδ(log) = log and δ(xa ) = axa log .
Then we deﬁne a structure of ρ-ﬁeld on K(log) by setting ρ(log) = p log. We observe a δ-closure ofthat K(log)ρ = C and that ρ and δ commute. Finally, we consider C:=C such that ρ acts trivially on C. We set C
C and by [DHR18, Lemma 2.3], theﬁeld , := Frac(K(log) ⊗C C)K ρ = C.
is a (ρ, δ)-ﬁeld such that K
In all cases, K is a ρ-ﬁeld of characteristic zero such that K ρ = C, F is a ρ-ﬁeld
 is a (ρ, δ)-ﬁeld of characteristic zero, suchextension of K with C := F ρ , and Kρ where C
 is diﬀerentially closed. Furthermore, ρ and δ commute. The
 = C,
that K
following table summarizes the diﬀerent frameworks we consider in this paper.

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HYPERTRANSCENDENCE AND LINEAR DIFFERENCE EQUATIONS

Case
S0
S∞
Q
M

F
Mer(C)
C((x−1 ))C((x1/∗ ))C((x1/∗ ))

K
C(x)
C(x)
C(x1/∗ )C(x1/∗ )

C
Ch
C
C
C

∂

δ

d
dx
d
dx
d
x dx
d
x dx

d
dx
d
dx
d
x dx

d
log ×x dx


C

Ch

C

C

C

487


K

Ch (x)C(x) 1/∗ )
C(x
 1/∗ )(log)C(xRemark 3.1. Note that the ﬁeld of δ-constants of C is the algebraically closed ﬁeld
 is also C. At someC. Then, by [CS07, Lemma 9.3], the ﬁeld of δ-constants of C
places, we will consider some iteration of the operator ρ. In Case S0 , it might =C
h (x) of K, and then consider
happen that we start with the ρ-ﬁeld extension Krh = Cthis extension as a ρ -ﬁeld extension. Note that C
rh . Nonetheless, Ch is a
ρr -constant ﬁeld and all results of Section 2 can be applied to the (ρr , δ)-ﬁeld K.
4. Auxiliary results
In this section, we let (K, ρ) be a diﬀerence ﬁeld of characteristic zero and we
consider a ρ-linear diﬀerence system(4.1)

ρ(Y ) = AY, with A ∈ GLn (K) ,where k := K ρ is an algebraically closed ﬁeld.
4.1. Irreducible and reducible diﬀerence Galois groups. The strategy forproving Theorem 1.2 is very diﬀerent according to whether the diﬀerence Galois
group associated with the diﬀerence equation over K satisﬁed by f is irreducible ornot. By irreducible, we mean that the group acts irreducibly on kn ; see Deﬁnition
4.1. We point out that this is diﬀerent from saying that an algebraic group, viewed
as an algebraic variety, is irreducible.
Deﬁnition 4.1. Let G ⊂ GLn (k) be a group. We say that G is irreducible if it
acts irreducibly on kn , that is, if the only k-vector subspaces of kn invariant by G
are {0} and kn . We say that G is reducible otherwise. When G is an irreduciblegroup, we say that G is imprimitive if there exist an integer r ≥ 2, and V1 , . . . , Vr ,some k-vector spaces satisfying the following conditions.
(i) kn = V1 ⊕ · · · ⊕ Vr .
(ii) For every g ∈ G, the mapping Vi → g(Vi ) is a permutation of the set{V1 , . . . , Vr }.
We say that G is primitive otherwise.
The next result shows that a diﬀerence Galois group that is irreducible andimprimitive cannot be connected. We recall that an algebraic group G is connected
if it has no proper open subgroup with respect to the Zariski topology. The identity
component of G is the connected component of G containing the identity.
Lemma 4.2. Let G ⊂ GLn (k) be an algebraic group. If G is connected and irreducible, then G is primitive.
Proof. Let us assume by contradiction that G is imprimitive. Then there exist
an integer r ≥ 2 and some nonzero C-vector spaces V1 , . . . , Vr  kn such thatkn = V1 ⊕ · · · ⊕ Vr . Furthermore, the action induced by every element of G on kn
is a permutation of the sets {V1 , . . . , Vr }. Let GV1 denote the stabilizer of V1 in G.

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This is an algebraic subgroup of G. We claim that it has ﬁnite index in G. Let us
assume by contradiction that there are an inﬁnite number of cosets gi GV1 of GV1 in
G. Since the group of permutations of a set with r elements is ﬁnite, there exist two
distinct cosets g1 GV1 and g2 GV1 such that g1 and g2 induce the same permutationof the Vi ’s. In that case, g2−1 g1 stabilizes V1 and thus belongs to GV1 , providing acontradiction. This proves the claim. Now, since GV1 is a closed subgroup of G ofﬁnite index, it contains the identity component of G. The latter is equal to G for
G is assumed to be connected. This means that V1 is invariant under the action of
G. Hence, G is reducible. This provides a contradiction.
Now, let us recall the deﬁnition of a diﬀerence module; see [vdPS97, §1.4].
Deﬁnition 4.3. A K-diﬀerence module M is a pair (M, Σ) where M is a ﬁnitedimensional K-vector space and Σ is an additive endomorphism of M such that
Σ(λY ) = ρ(λ)Σ(Y ) for every Y ∈ M and λ ∈ K.
Let A ∈ GLn (K). The K-diﬀerence module MA attached to the system ρ(Y ) =AY is MA = (K n , ΣA ), where ΣA (Y ) = A−1 ρ(Y ). Given a monic linear diﬀerence
operator L = a0 y + a1 ρ(y) + · · · + ρn (y) with coeﬃcients ai in K and a0 = 0, one
can consider the corresponding linear diﬀerence system ρ(Y ) = AL Y , where AL is
the companion matrix of L. If K contains a nonperiodic element with respect to ρ,
then [HS99, Theorem B.2] asserts that any K-diﬀerence module M is isomorphic
to MAL for some monic linear diﬀerence operator L. This is equivalent to theexistence of a so-called cyclic vector in M, that is, a nonzero element e ∈ M suchthat the vectors e, Σ(e), . . . , Σn−1 (e) form a K-basis of M .
Lemma 4.4 characterizes linear diﬀerence systems whose diﬀerence Galois groupis reducible.
Lemma 4.4. Let G ⊂ GLn (k) be the diﬀerence Galois group of (4.1) over K and
let r be an integer with 0 < r < n. The following statements are equivalent.
• There exists a G-submodule5 of kn of dimension r over k.
• There exists a diﬀerence submodule of dimension r in the K-diﬀerence module MA associated with (4.1).
• There exists T ∈ Gln (K) such that


B1 B2
−1
ρ(T )AT =0 B3
with B1 ∈ GLr (K).
In particular, G is reducible if and only if the statements above hold for some r,with 0 < r < n. Furthermore, if the matrix A in (4.1) is the companion matrix ofa diﬀerence operator L, then L admits a nontrivial right factor if and only if G is
reducible.
Proof. Since the category of K-diﬀerence modules is a Tannakian category, theﬁrst two statements are nothing else than the usual Tannakian equivalence, see[And01, Théorème 3.2.1.1]. Completing a K-basis of a diﬀerence submodule M inen ) of K n , shows that
MA of dimension r, 0 < r < n, to a K-basis e =(e1 , . . . , 
B1 B2
the matrix of Σ in the basis e is of the form B =, with B1 ∈ GLr (K).
0 B3
5 kn is a G-module via G ⊂ Gl (k).
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HYPERTRANSCENDENCE AND LINEAR DIFFERENCE EQUATIONS

489

Let T ∈ GLn (K) denote the matrix associated with the change of basis from thecanonical basis f = (f1 , . . . , fn ) to e. Then
BT f = Be = Σ(e) = Σ(T f ) = ρ(T )Af.
This proves the equivalence between the second and the third statements.
Now, let us assume that A is the companion matrix of some diﬀerence operator L
and that there exists a nontrivial diﬀerence submodule N of MA . Let e be a cyclic
vector of MA such that L is the minimal monic linear diﬀerence operator annihilating e. Since N is a nontrivial diﬀerence submodule of MA , then dimK MA /N < nand there exists a nontrivial monic linear diﬀerence operator L1 of order smaller
than or equal to dimK MA /N such that L1 e ∈ N . The element L1 e is nonzero byminimality of L. Since N is a diﬀerence submodule of MA there exists a monic
linear diﬀerence operator L2 of order smaller than or equal to dimK N such that
L2 L1 (e) = 0. By minimality of L, this proves that L = L2 L1 , where L1 has ordersmaller than n. Conversely, if there is a nontrivial factorization L = L2 L1 , thenthe diﬀerence module corresponding to L1 is a nontrivial diﬀerence submodule of
M. Hence G is reducible.

4.2. Going up and down. In this section, we let K, F0 , ρ be deﬁned as in Section
 be deﬁned as in Section 3. Lemmas 4.5 and 4.6 compare the
1.1, and we let Kdiﬀerence Galois groups over K and over K.
We recall that a connected linear algebraic group G is said to be semisimple if it
is of positive dimension and its only solvable (or, equivalently, Abelian) connectedclosed normal subgroup is the trivial group.
 be the Galois group of ρ(Y ) =Lemma 4.5. Let A ∈ GLn (K) and let G (resp. G)AY over K (resp. K). The following properties hold.
 = G(C).
• In Cases S0 , S∞ , and Q, we have G
 = G(C).
• In Case M, if G is semisimple, then G
 of K is obtained by extension
Proof. In Cases S0 , S∞ , and Q, the ﬁeld extension K
 Then, [CHS08, Corollary 2.5] gives that theof the ﬁeld of constants C to C.
diﬀerence Galois group over K is just the extension of G to the C-points.
Let us deal with Case M. Let G denote the diﬀerence Galois group over K(log).
 = G(C).
As previously, we infer from [CHS08, Corollary 2.5] that G
Thus, itremains to prove that G = G when G is semisimple. By [DHR18, Proposition 1.6],
we can see G as a subgroup of G such that one of the following two situations
occurs.
(1) G = G.
(2) G is a normal subgroup of G and G/G  Gm the multiplicative group(C∗ , .).
By [Mil17, Corollary 21.50], a semisimple algebraic group is perfect, that is, equal
to its derived subgroup, see [Mil17, Deﬁnition 12.45]. Thus, a semisimple algebraic
group has no nontrivial abelian quotient. Hence, (2) cannot occur. We deduce that
G = G, which ends the proof.
 be the Galois group of ρ(Y ) =Lemma 4.6. Let A ∈ GLn (K) and let G (resp. G) is
AY over K (resp. K).
The group G is irreducible if and only if the group G
irreducible.

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490

BORIS ADAMCZEWSKI ET AL.

 = G(C) and the result
Proof. In Cases S0 , S∞ , and Q, Lemma 4.5 ensures that G
follows directly.
 = G(C),Let us deal with Case M. We infer from [CHS08, Corollary 2.5] that G
where we let G denote the Galois group of the system over K(log). Thus, it remains
to prove that G is irreducible if and only if G is irreducible. As seen in the proof of
Lemma 4.5, G is a subgroup of G. Thus, G is reducible as soon as G is reducible.
Conversely, let us assume that G is reducible. By the cyclic vector lemma over K,
the linear diﬀerence system (4.1) is equivalent to an equation
L := ρn + an−1 ρn−1 + · · · + a0 = 0 ,with ai ∈ K and a0 = 0. Since G is reducible, Lemma 4.4 gives that L admits afactorization over K(log). Multiplying L by some nonzero element if necessary, we
can get rid of the denominators. Then, there exists α ∈ K[log] such that
αL = (bk ρk + bk−1 ρk−1 + · · · + b0 )(cn−k ρn−k + cn−k−1 ρn−k−1 + · · · + c0 ) ,where bi , ci ∈ K[log], bk b0 cn−k c0 = 0, and 0 < k < n. Rewriting the equation interms of powers of log, one ﬁnds
⎞⎛
⎞
⎛



logi (αi L) = ⎝logκ Lκ ⎠ ⎝logj Dj ⎠ ,
i≥i0

κ≥κ0

j≥j0

where Lκ , Dj are linear diﬀerence operators over K, αi ∈ K, αi0 ∈ K ∗ , and Lκ0 :=tr
r=0 γr ρ and Dj0 are nonzero. Using the fact that log is transcendental over
K and ρ log = p log ρ, we deduce that i0 = κ0 + j0 . Considering the terms in
logi0 in both sides of the factorization, we deduce that L = α1i MDj0 where M =0t
j0 r rρ , leading to a nontrivial factorization of L over K. By Lemma 4.4,
r=0 γr p
we obtain that G is reducible, as wanted.
Now, we prove the following descent lemma.
 deﬁned as in Section 3, we have
Lemma 4.7. For F0 , K and K
 = K.
F0 ∩ K 1/∗ ). There exists a positive
Proof. Let us ﬁrst consider Case Q. Let a ∈ F0 ∩ C(x
1/r
1/r

integer r such that a ∈ C((x )) ∩ C(x ). It follows that there exists a positive
integer N such that xN a ∈ C[[x1/r ]]. Thus, xN a is a formal power series in x1/r with
 Thiscomplex coeﬃcients that represents a rational fraction with coeﬃcients in C.
property can be characterized by the vanishing of its associated Kronecker-Hankel
determinants (see, for instance, [Sal63, p. 5]). This condition does not depend on
the ﬁeld of coeﬃcients, so we deduce that xN a ∈ C(x1/r ). Hence, a ∈ K.
The proof of Case S∞ is entirely similar. For Case M, if a ∈ C((x1/∗ )) ∩
 1/∗ )(log), we use the fact that the logarithm is transcendental over C((x1/∗ )) toC(x 1/∗ ). Arguing as above, we can show that a ∈ K.
conclude that a ∈ C((x1/∗ )) ∩ C(xSince F0 ⊂ Mer(C), one
Now, let us deal with Case S0 . Let a ∈ F0 ∩ C(x).
can ﬁnd x0 ∈ C such that a ∈ C[[x − x0 ]]. Once again, using Kronecker-Hankel
determinants, one can deduce that a ∈ K.
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HYPERTRANSCENDENCE AND LINEAR DIFFERENCE EQUATIONS

491

4.3. Iterating the diﬀerence operator. For every positive integer , let us set
A[] := ρ−1 (A) × · · · × A ∈ GLn (K) .
Note that if U is a fundamental matrix of ρ(Y ) = AY , then it is also a fundamental
matrix of ρ (Y ) = A[] Y . In other words, a vector solution to the system ρ(Y ) = AY
is also a solution to the iterated system ρ (Y ) = A[] Y . In this section, we show that,
replacing the original system by a suitable iteration, we can reduce the situation to
the more convenient case where the corresponding Galois group is connected andthe Picard-Vessiot extension is a ﬁeld.
This iteration process might introduce some new constants. Lemma 4.8 showsthat this cannot happen when the ﬁeld of ρ-constants is algebraically closed.
Lemma 4.8. Let L be a ρ-ﬁeld such that Lρ = k, and r be a positive integer.
Then, any ρr -constant of L is algebraic of degree less than or equal to r over k. Inrparticular, if k is algebraically closed then Lρ = k.
Proof. Let a ∈ L be a ρr -constant. Then the polynomial P (X) = (X − a) · · · (X −ρr−1 (a)) belongs to Lρ [X] = k[X]. This proves that a is algebraic over k of degree
less than or equal to r.
One of the main diﬃculty of diﬀerence algebra is the control of the algebraic
diﬀerence ﬁeld extensions. This is the source of many technical diﬃculties andconnected to the fact that the theory ACFA is unstable (see [CH99]). The diﬀerenceﬁelds (K, ρ) introduced in Section 1.1 have the following very strong property withrespect to ﬁnite diﬀerence ﬁeld extensions, which will allow us to bypass thesediﬃculties.
Lemma 4.9. Let r be a positive integer. In each of the cases S0 , S∞ , Q, and M,
there are no proper ρr -ﬁeld extension of K of ﬁnite degree.
Proof. The results in [Hen97, Hen98, Roq18] cover all cases.

We are now ready to prove the following result.
Proposition 4.10. Let (K, ρ) be deﬁned as in Section 1.1 and let L be a ρ-ﬁeld
extension of K such that Lρ = C. Let (f1 , . . . , fn ) ∈ Ln be a nonzero solution to(4.1). Let G be the diﬀerence Galois group of (4.1) over K. There exists a positive
integer r such that the following properties hold.
(a) There exists a Picard-Vessiot ﬁeld extension Q for ρr (Y ) = A[r] Y over (K, ρr ),with a fundamental solution matrix U having (f1 , . . . , fn ) as ﬁrst column.
(b) The diﬀerence Galois group Gr := Gal(Q/K) of ρr (Y ) = A[r] Y over (K, ρr )coincides with the identity component of G and is therefore connected.
Though this result can be easily deduced from the proof of [DHR16, Proposition 4.6] in the particular case where the parametric operator is the identity, we
ﬁnd it more convenient for the reader to include the proof below.
Proof. (a) Since K(f1 , . . . , fn )ρ ⊂ Lρ = C, there exists a Picard-Vessiot ring R0 forρ(Y ) = AY over K(f1 , . . . , fn ). Since f1 , . . . , fn ∈ R0 , there exists a fundamental
matrix U for R0 having (f1 , . . . , fn ) as ﬁrst column. Let Q0 be the total quotientring of R0 . By Proposition 2.2, we ﬁnd Qρ0 = C.
Set R := K[U, 1/det(U )]. The diﬀerence ring (R, ρ) is a subring of the PicardVessiot ring (R0 , ρ). Since any element of R that is a zero divisor in R0 is a

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492

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zero divisor in R, see for instance [Roq18, Lemma 5], the total quotient ring Q of R
embeds in Q0 and consequently is a pseudoﬁeld (see [Wib10, Lemma 1.3.4]). Consequently C ⊂ Qρ ⊂ Qρ0 = C. Then Qρ = C, and (Q, ρ) is a Picard-Vessiot extension
for ρ(Y ) = AY over (K, ρ). Since K ρ = C is algebraically closed, the uniquenessof Picard-Vessiot extensions above K allows to conclude that Gal(Q/K) = G. We
have proved that we can embed any solution in L in a Picard-Vessiot extension
above K.
By Proposition 2.2, R is a Picard-Vessiot ring for ρ(Y ) = AY over (K, ρ). Let
e0 , . . . , er−1 denote the orthogonal idempotents relative to its ring structure. Notethat K[f1 , . . . , fn ] is an integral domain that is closed under ρ. For i = 0, . . . , r − 1,
let φi : R → ei R denote the projection from R to ei R. Note that φi (x) = xei .
Let us prove that the restriction to K[f1 , . . . , fn ] of this projection is injective forall i ∈ {0, . . . , r − 1}. Suppose to the contrary that there exist a nonzero elementx ∈ K[f1 , . . . , fn ] and i ∈ {0, . . . , r − 1} such that φi (x) = 0. Since ρ permutes
the integral domains ei R, we ﬁnd that xρ(x) · · · ρr−1 (x) = 0. The latter equality
provides a contradiction with the fact that K[f1 , . . . , fn ] is an integral domain which
is closed under the injective morphism ρ. Thus, ei K[f1 , . . . , fn ] ⊂ ei R is a copyof K[f1 , . . . , fn ] in ei R. Since K ρ = C is algebraically closed, Lemma 4.8 impliesrthat K ρ = C. By [vdPS97, Lemma 1.26], (ei R, ρr ) is a Picard-Vessiot ring forr
the ρ -system ρr (Y ) = A[r] Y and ei R is an integral domain. Its total quotientrring Q1 is a ﬁeld and, since K ρ is algebraically closed, Q1 is a Picard-Vessiot ﬁeld
extension by Proposition 2.2. By construction, Q1 contains f1 , . . . , fn , and we canthus choose a fundamental matrix having (f1 , . . . , fn ) as ﬁrst column.
(b) By [Roq18, Theorem 12], Gr = Gal(Q1 /K) is a normal algebraic subgroupof G and the quotient G/Gr is ﬁnite. To conclude, it remains to prove that Gr isconnected. Let G◦r denote its identity component. The Galois correspondence, see◦
Theorem 2.3, gives that the ρr -ﬁeld Q1 Gr is a ﬁnite extension of K. Lemma 4.9◦
implies that Q1 Gr = K and, applying the Galois correspondence again, we deducethat G◦r = Gr .
Remark 4.11. The proof of (b) shows that, if G is connected, then, for every positive
integer , the Galois group of ρ (Y ) = A[] Y over K coincides with G.
In the parametrized framework, we have the following similar result.
 with Lρ = C, and let
Proposition 4.12. Let L be a (ρ, δ)-ﬁeld extension of Kn(f1 , . . . , fn ) ∈ L be a nonzero solution to (4.1). Then there exist a positive
 for ρr (Y ) = A[r] Y
integer r and a parametrized Picard-Vessiot ﬁeld extension Q with a fundamental matrix U having (f1 , . . . , fn ) as ﬁrst column.
over K,
Proof. The proof is similar to the one of Proposition 4.10 and [DHR18, Lemma for C is algebraically closed.
 ρr = C3.7], just noting that, by Lemma 4.8, K
The following result is obtained by combining the two previous propositions.
Corollary 4.13. Let (f1 , . . . , fn ) ∈ F0n be a nonzero solution to (4.1) and let G
be the diﬀerence Galois group of (4.1). Then, there exists a positive integer r suchthat the following properties hold.
 for ρr (Y ) = A[r] Y(a) There exists a (ρr , δ)-Picard-Vessiot ﬁeld extension Qwith a fundamental matrix U having (f1 , . . . , fn ) as ﬁrst column over K
 =C
(where K
sh (x) for some positive integer s dividing r in Case S0 ).

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HYPERTRANSCENDENCE AND LINEAR DIFFERENCE EQUATIONS

493(b) The diﬀerence Galois group of ρr (Y ) = A[r] Y over K coincides with theidentity component of G.
Proof. By Proposition 4.10, there exists a positive integer s such that the diﬀerence
Galois group of ρs (Y ) = A[s] Y is equal to the identity component of G. In order
to apply Proposition 4.12 to the system ρs (Y ) = A[s] Y , we need to embed F0 into
 with Lρs = C.
 We proceed as follows.
a (ρs , δ)-ﬁeld extension L of K
• Case S0 . We let F1 denote the smallest (ρs , δ)-subﬁeld of Mer(C) containss
ing F0 and Csh . Since Mer(C)ρ = Csh , we ﬁnd that F1 ρ = Csh . Let denote a δ-closure of Csh , considered as a constant ρs -ﬁeld. The ﬁeldC and F1 are linearly disjoint above Csh (see [Wib10, Lemma
extensions C ⊗C F1
1.1.6]). Thereby, their compositum L is the fraction ﬁeld of C
sh(see [Bou90, A.V.13]). Arguing as in the proof of [DHR18, Lemma 2.3], we
obtain that L has the required properties.
• Cases S∞ and Q. We let L be the fraction ﬁeld of F0 ⊗C C.
1/∗
• Case M. We let L be the ﬁeld C((x ))(log).
 one can perApplying Proposition 4.12 to the system ρs (Y ) = A[s] Y over K,
form a second iteration to obtain a system ρsl (Y ) = A[sl] Y satisfying (a). By
Remark 4.11, the diﬀerence Galois group of ρsl (Y ) = A[sl] Y over K is the identity
component of G. This ends the proof.

5. Proof of Theorem 1.2
This section is devoted to the proof of our main result. Before proving Theorem 1.2 in full generality, we ﬁrst consider the following two particular cases.
- The function f is solution to an inhomogeneous equation of order one.
- The diﬀerence Galois group of the equation associated with f is both connected and irreducible.
All along this section, we keep on with the notation of Sections 1.1 and 3.
5.1. Aﬃne order one equations. Various hypertranscendence criteria for solutions to inhomogeneous order one equations have already been obtained, see for
instance [Ran92, Ish98, HS08, Ngu12]. We ﬁrst deduce from these criteria the following result.
Proposition 5.1. If f ∈ F0 is solution to an equation of the form ρ(f ) = af + b,
with a, b ∈ K, then either f is ∂-transcendental over K or f belongs to K.
Proof of Proposition 5.1. Let us ﬁrst note that if a = 0, then f = ρ−1 (b) ∈ K. We
can thus assume that a = 0. Furthermore, we observe that, in Cases Q and M,
we can use a change of variable of the form z = x1/ to reduce the situation to thecase where K = C(x). From now on, we thus assume that K = C(x). Let us also
assume that f is ∂-algebraic over K. It remains to prove that f ∈ K.
We ﬁrst note that Case M corresponds to [Ran92, Théorème 5.2]. Now, let us
consider Cases S0 , S∞ , and Q, which are partially covered by [HS08]. We recall thefollowing deﬁnition from [HS08, §6.1] (see also [vdPS97, §2]). We say that α ∈ K
is standard if, for all positive integers , α and ρ (α) have no common zero or pole.
By [HS08, Lemma 6.2], there exists a standard element a∗ and a nonzero e ∈ Ksuch that a = a∗ ρ(e)/e. It follows that f /e is solution to the equation
ρ(f /e) = a∗ f /e + b/ρ(e) .

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Set g = f /e and b̃ = b/ρ(e). Since e ∈ K, it suﬃces to prove that g ∈ K. Let us
note that g = f /e is ∂-algebraic over K.
The same argument as in [ADR18, Lemma 3.6] shows that there exists a nonzero
δ-algebraic solution of ρ(y) = a∗ y + b/ρ(e) in a (ρ, δ)-Picard-Vessiot ﬁeld extension
 In Case S0 , [HS08, Proposition 3.9] implies that a∗ ∈ C and there existsof K.
h ∈ K such that b̃ = ρ(h) − a∗ h. In Case S∞ , the same argument as in the proofof [HS08, Proposition 3.9] also yields the same conclusion. We obtain that
ρ(g) = a∗ g + b̃= a∗ g + ρ(h) − a∗ h .
Hence ρ(g − h) = a∗ (g − h). If g = h, then g ∈ K and we are done. Thus, we canassume that g = h. In that case, setting f˜ = g − h, we obtain ρ(∂(f˜)) = a∗ ∂(f˜). It
follows that ρ(∂(f˜)/f˜) = ∂(f˜)/f˜, and thus ∂(f˜)/f˜ ∈ C ⊂ K. Since f˜ ∈ F0 satisﬁes
both a linear ρ-equation and a linear ∂-equation with coeﬃcients in K, Theorem
1.3 implies that f˜ ∈ K. Since h ∈ K, we deduce that g ∈ K, as expected.
In Case Q, the same argument as in the proof of [HS08, Proposition 3.10] showsthat a∗ = cxα for some c ∈ C∗ and α ∈ Z, and one of the following conditions
holds.
(a) a∗ = q r for some r ∈ Z, and b̃ = ρ(h) − a∗ h + dxr for some h ∈ K andd ∈ C.
/ q Z and b̃ = ρ(h) − a∗ h for some h ∈ K.
(b) a∗ ∈In Case (a), we deduce that


g−h
g−h
(5.1)
ρ
+ dq −r ·=xr
xr
Since (g − h)/xr ∈ F0 , we can consider its expansion in Puiseux series. Since q is
not a root of unity, we easily deduce from Equation (5.1) that (g − h)/xr ∈ C ⊂ K.
Hence g ∈ K, as wanted.
/ q Z and b̃ = ρ(h) − a∗ h for some h ∈ K. It follows
In Case (b), we have that a∗ ∈that ρ(g − h) = cxα (g − h), where g − h ∈ F0 is a Puiseux series. Arguing as inCase (a), we obtain that α = 0 and g − h = c0 xr for some r ∈ Q and c0 ∈ C. Henceg ∈ K. This ends the proof.
For order one equations over K whose solutions do not necessarily belong to F0 ,one has the following criterion for diﬀerential algebraicity.
 with Lρ = C.
 Let f ∈ L∗
Lemma 5.2. Let L be a (ρ, δ)-ﬁeld extension of K∗
 if and only ifsuch that ρ(f ) = af with a ∈ K . Then, f is δ-algebraic over Kα ρ(g)∗∗
a = cx g for some c ∈ C , α ∈ Q, and g ∈ K . Moreover, α = 0 in Cases S0 ,S∞ , and M.
Proof. Cases S0 , S∞ , and Q can be deduced in a similar way to [HS08, Corollary 3.4] (resp. Case M is given by [DHR18, Proposition 3.1]), combined with some to C. Such argument is similar to the one given indescent argument to go from C
the proof of [HS08, Corollary 3.2].

5.2. Connected and irreducible Galois groups. As a second step, we consider
the situation where the diﬀerence Galois group is both connected and irreducible.

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HYPERTRANSCENDENCE AND LINEAR DIFFERENCE EQUATIONS

495

Proposition 5.3. Let us assume that n ≥ 2. If f ∈ F0 is a nonzero solution to
Equation (1.2) and if the corresponding diﬀerence Galois group G over K is bothconnected and irreducible, then f is ∂-transcendental over K.
Proposition 5.3 is obtained as a direct consequence of the following more general
result.
Proposition 5.4. Let us consider a linear system of the form (4.1) with n ≥ 2.
Let us assume that the diﬀerence Galois group G for (4.1) over K is connected and be a (ρ, δ)-Picard-Vessiot ﬁeld extension for (4.1) over K, withirreducible. Let Q
fundamental matrix of solutions U . Then every column of U contains at least one
element that is δ-transcendental over K.
Proof of Proposition 5.3. We argue by contradiction, assuming that f is ∂-algebraic
over K. Then all coordinates of the vector (f, . . . , ρn−1 (f )) are also ∂-algebraic
over K. This follows from the fact that ρ and ∂ almost commute, that is, ∂ρ = cρ∂with c = 1 in Cases S0 , S∞ , and Q, and c = p in Case M. We deduce fromthe construction of δ with respect to ∂ given in Section 3 (and the fact that log
is δ-algebraic for Case M) that all coordinates of the vector (f, . . . , ρn−1 (f )) are
also δ-algebraic over K.
Now, let A denote the companion matrix associated with Equation (1.2). Since(f, . . . , ρn−1 (f )) ∈ F0n is nonzero, Corollary 4.13 ensures the existence of a positive
 for ρr (Y ) = A[r] Y over Kinteger r and a (ρr , δ)-Picard-Vessiot ﬁeld extension Qn−1(f )) is the ﬁrst column of a fundamental matrix
such that the vector (f, . . . , ρU . Furthermore, Remark 4.11 ensures that the Galois group of ρr (Y ) = A[r] (Y )over K is equal to G, for the latter is connected. Thus, Proposition 5.4 applies with
ρ replaced by ρr , providing a contradiction.
It remains to prove Proposition 5.4. As a key argument, we will use the following
result due to Arreche and Singer [AS17, Lemma 5.1]. It says that the parametrized
Galois group must be as big as possible when the diﬀerence Galois group has an
identity component that is semisimple.
Proposition 5.5. Let us consider a system ρ(Y ) = AY , where A ∈ GLn (C(x)) in
Cases S0 and S∞ , and where A ∈ GLn (C(x1/ )) for some positive integer  in Cases denote the diﬀerence
Q and M. In Cases S0 , S∞ , and M, we let G ⊂ GLn (C) In
Galois group over K, and H ⊂ GLn (C) denote the (ρ, δ)-Galois group over K.
1/ denote the diﬀerence Galois group over C(x), andCase Q, we let G ⊂ GLn (C)1/H ⊂ GLn (C) denote the (ρ, δ)-Galois group over C(x ). As in Proposition 2.5,
we see H as a subgroup of G. If the identity component of G is semisimple, then
H = G.
Proof. Cases S0 and S∞ are explicitly proved in [AS17, Lemma 5.1], since, by is C. Cases Q and M are not
Remark 3.1, the ﬁeld of diﬀerential constants of C
explicitly proved in [AS17, Lemma 5.1] for we consider the slightly more generalsituation where A ∈ GLn (C(x1/ )). However, the result follows easily from theargument given in the proof of [AS17, Lemma 5.1].

We are now ready to prove Proposition 5.4.

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
Proof of Proposition 5.4. We let H denote the (ρ, δ)-Galois group of (4.1) over K,
that is, K)
 ⊂ GLn (C) .
H = Galδ (Q/ denote the Galois group of ρ(Y ) = AY over K.
 Let us argue by
We also let G
contradiction, assuming the existence of one column of U whose coordinates are allδ-algebraic over K.
 Since G is irreducible,
We ﬁrst show that all entries of U are δ-algebraic over K.
 is irreducible too. By Proposition 2.5, H is Zariski dense
Lemma 4.6 implies that G This implies that H is irreducible, since otherwise, H
in the irreducible group G.
would be conjugated to a group formed by block upper triangular matrices, as well
 Now, let S denote theas its Zariski closure, contradicting the irreducibility of G.
C-vector space of solution vectors over Q whose entries are all δ-algebraic over K.
By assumption, S is not reduced to {0}. Furthermore, S is invariant under theaction of any σ ∈ H. Since H is irreducible, we deduce that S contains n linearly
independent solutions to (4.1). In other words, all columns of U belong to S, which as claimed.
implies that all entries of U are δ-algebraic over K,
We observe that the determinant det(U ) is solution to the equation(5.2)

ρ(y) = det(A)yand that the diﬀerence Galois group of this equation over K is the group det(G).
 we obtain that det(U ) is also δ-algebraic
Since all entries of U are δ-algebraic over K,
 We ﬁrst consider the particular case where det(U ) ∈ K, and then we moveover K.
to the general case.
Let us ﬁrst assume that det(U ) belongs to K. In that case, det(G) = {1} andtherefore G ⊂ SLn (C). We recall that G is assumed to be connected and irreducible.
According to Lemma 4.2, we have that G is primitive. By [SU93, Proposition 2.3],
 = G(C) is semisimple
we ﬁnally obtain that G is semisimple. By Lemma 4.5, Gtoo.
 Since
In Cases S0 , S∞ , and M, we infer from Proposition 5.5 that H = G.
all entries of U are δ-algebraic over K, the δ-dimension of the (ρ, δ)-Picard-Vessiot
extension is zero6 , and therefore the δ-dimension of H is zero by [HS08, Proposition 6.26]. By a result of Kolchin [Kol73, Chap. IV, Proposition 10], the dimension
 is the same as the δ-dimension of G viewed as aof an algebraic group G over C This proves that the dimension of H equals the dimendiﬀerential group over C.
 . Since C is algebraically closed, the dimension of G(C) over C equalssion of G(C)
the dimension of G over C. Thus, the above equality H = G = G(C) implies thatthe dimension of the algebraic group G over C is zero. We recall that an algebraic
group of dimension zero is just a ﬁnite group, and that a ﬁnite connected grouphas cardinality one. Thus, we deduce that G = {In }, where we let In denote theidentity matrix of size n. Since by assumption n ≥ 2, this provides a contradiction
with the assumption that G is irreducible. This ends the proof in these cases.
In Case Q, the system ρ(Y ) = AY has coeﬃcients in C(x1/ ) and the entries 1/ ), for some positive integer . Let G  denoteof U are δ-algebraic over C(xK, is an algebraic
 1/ ). Since K
the diﬀerence Galois group of ρ(Y ) = AY over C(x6 We refer the reader to [HS08, P. 374] for a deﬁnition of the notion of δ-dimension of a (ρ, δ)Picard-Vessiot ring and of a (ρ, δ)-Galois group.

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497

 1/ ), we deduce from [Roq18, Theorem 7] that G  and G  haveextension of C(xKK,
the same identity component, which is GK since the latter is connected. Henceis semisimple. By Proposition 5.5, it follows thatthe identity component of GK,
 1/ ). Furthermore,H = G  , where H denote the (ρ, δ)-Galois group over C(xK, 1/ ) which implies that the δ-dimension of
the entries of U are δ-algebraic over C(xH is zero. Since H = GK,
 , we get as previously that the algebraic group GK,
has dimension zero. Since GK ⊂ GK,
 , this proves that the dimension of GK is also
zero, and we can argue as previously to get a contradiction.
Now, let us consider the general case. We remind that det(U ) is δ-algebraic
 and ρ(det(U )) = det(A)det(U ). By Lemma 5.2, there exist some rational
over K
number α and nonzero elements c ∈ C and g ∈ K such that det(A) = cxα ρ(g)/g.
Furthermore, α = 0 in Cases S0 , S∞ , and M.
Let us consider the rank one linear diﬀerence system
ρ(y) = c−1/n x−α/n y .

(5.3)

 is a (ρ, δ)-ﬁeld with Qρ = C, there exists a (ρ, δ)-Picard-Vessiot extension
Since Q1 ) be a fundamental matrix associated with
Q1 for (5.3) over Q. Let (λ) ∈ GL1 (Qthis system. Thenρ(λ) = c−1/n x−α/n λ1 . Using the commutativity of δ and ρ, we obtain thatand λ is invertible in Qδ(λ)
ρ = C,∈Qδ1λ In particular, it is δ-algebraic over K.
which shows that λ is δ-algebraic over C.
1 ) are δ-algebraic over K.
Thus, all entries of the matrix λU ∈ GLn (QOn the other hand, we set
B = c−1/n x−α/n A ∈ Gln (K) and UB := λU.
Note that ρ(UB ) = BUB . Let GB be the diﬀerence Galois group of the system
ρ(Y ) = BY over K. Our choice of B ensures that det(B) = ρ(g)/g. Thus, theequation
ρ(y) = det(B)y
has a solution in K. It follows that its diﬀerence Galois group det(GB ) is reducedto {1}. Hence, GB ⊂ SLn (C). For k ≥ 1, we let GB[k] denote the diﬀerence Galois
group of ρk (Y ) = B[k] Y over K. By Proposition 4.10, there exists r ≥ 1 such that
GB[r] is connected. Furthermore, ρr (g)/g = det(B[r] ) and the diﬀerence Galois
group of ρr (Y ) = B[r] Y over K remains included in SLn (C). Let us notice that
A[r] = αr B[r]×for some αr ∈ K .
We claim that GB[r] is irreducible. Let us argue by contradiction, assuming that
GB[r] is reducible. By Lemma 4.4, there exists T ∈ GLn (K) such that ρr (T )B[r] T −1
is a block upper triangular matrix, and we deduce thatαr ρr (T )B[r] T −1 = ρr (T )A[r] T −1 .
By Lemma 4.4, we obtain that Gr , the diﬀerence Galois group of the system
ρr (Y ) = A[r] Y over K, is conjugated to a group of block upper triangular matrices, which implies that Gr is reducible. By Remark 4.11, Gr = G, providing a

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498

BORIS ADAMCZEWSKI ET AL.

contradiction with the assumption that G is irreducible. This proves that GB[r] isirreducible.
Finally by Lemma 4.2, the irreducible connected group GB[r] is primitive. Again,
since GB[r] ⊂ SLn (C), we infer from [SU93, Proposition 2.3] that GB[r] is semisim and all coordinates of UB , we have
1 s a pseudo δ-ﬁeld that contains K
ple. Since Q B ⊂ Q
1 (see Section 2.2.2 for the notation). Furthermore, since Qρ = C,
 the
KU
1
r
r

pseudo δ-ﬁeld KUB  is a (ρ , δ)-Picard-Vessiot extension for ρ (Y ) = B[r] Y overK.
 we can apply Proposition 5.5 and,Since all entries of UB are δ-algebraic over K,
arguing as in the ﬁrst part, we deduce that GB[r] = {In }. Since n ≥ 2, we obtain acontradiction with the fact that GB[r] is irreducible.

5.3. The general case. We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2. We argue by induction on n. More precisely, our induction
assumption reads as follows.
(Hn ) For all positive integers k and all f ∈ F0 that is solution to a ρk -linear
equation of order at most n with coeﬃcients in K, we have either f is∂-transcendental over K or f ∈ K.
Proposition 5.1 implies that (H1 ) hold true. Let n ≥ 2 and let us assume (Hn−1 ).
Let f ∈ F0 be solution to a ρk -linear equation of order n with coeﬃcients in K.
Without any loss of generality, we can assume that f = 0 and k = 1. Considering
the companion matrix associated with this equation, Corollary 4.13 ensures theexistence of a positive integer r such that the following properties hold.
(a) The vector (f, . . . , ρn−1 (f )) is solution to the system ρr (Y ) = AY forsome A ∈ GLn (K).
 for ρr (Y ) = AY(b) There exists a (ρr , δ)-Picard-Vessiot ﬁeld extension Q such that the vector (f, . . . , ρn−1 (f )) is the ﬁrst column of aover K,
fundamental matrix U ∈ GLn (Q).
(c) The diﬀerence Galois group G of the system ρr (Y ) = AY over K is connected.
If G is irreducible, Proposition 5.3 shows that f is ∂-transcendental. Hence (Hn )holds. From now on, we assume that G is reducible. Furthermore, we assume thatf is ∂-algebraic over K. Thus, it remains to prove that f ∈ K. Without loss ofgenerality, we can assume that r = 1.
By Lemma 4.4, there exists a gauge transformation T = (ti,j ) ∈ GLn (K) suchthat

A1 A1,2
ρ(T )AT −1 =,0
A2where Ai ∈ GLni (K), n1 + n2 = n, and 0 < n2 < n. Furthermore, let us assume
that n1 is minimal with respect to this property.
Set
i−1
n
(f ))(gi )
i≤n := T (ρi≤n ∈ F0 .

(5.4)


The vector (gi )n1 +1≤i≤n ∈ F0n2 is solution to the system ρ(Y ) = A2 Y . Furthermore,since f is ∂-algebraic over K, the gi are also ∂-algebraic over K. By (Hn2 ) and

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HYPERTRANSCENDENCE AND LINEAR DIFFERENCE EQUATIONS

499

Remark 2.1, we obtain that(gi )n1 +1≤i≤n ∈ K n2 .

(5.5)

Let G1 denote the diﬀerence Galois group of the system ρ(Y ) = A1 Y over K. We
claim that G1 is connected and irreducible. Indeed, if G1 were reducible, then by
Lemma 4.4, there would exist a gauge transformation changing A1 into a block
upper triangular matrix, contradicting the minimality of n1 . Furthermore, theGalois group of ρ(Y ) = A1 Y over K is a quotient of the connected group G andthereby a connected group.
The main step of the proof consists in showing the following result.
Claim. One has n1 = 1.
Proof of the claim. By assumption, f is ∂-algebraic over K. Arguing as in the
proof of Proposition 5.3, we get that all coordinates of the vector (f, . . . , ρn−1 (f )) Then, (5.4) implies that all the gi ’s are also δ-algebraic overare δ-algebraic over K.
 We have to distinguish two cases.
K.
 By (5.4) and Lemma 4.7, weLet us ﬁrst assume that all the gi ’s belong to K.
obtain that
 =K∀i ∈ {1, . . . , n}, gi ∈ F0 ∩ K
n−1

−1

(f ))
= T (g1 , . . . , gn )∈ K n . Thus the vector
and hence (f, . . . , ρn−1
(f, . . . , ρ(f )) is ﬁxed by the diﬀerence Galois group G. By Lemma 4.4, there
exists P ∈ GLn (K) such that
b1 B1,2−1
ρ(P )AP =,0
B2
for some matrices B1,2 and B2 with coeﬃcients in K and a nonzero b1 ∈ K. By
minimality of n1 , we obtain that n1 = 1, as wanted.
 Note that, byNow, let us assume that at least one the gi ’s does not belong to K.
 too. We let H
assumption, the fi ’s all belong to Q and thus the gi ’s all belong to Qdenote the (ρ, δ)-Galois group of ρ(Y ) = AY over K. By the parametrized Galois
correspondence [HS08, Theorem 6.20], we deduce the existence of some σ ∈ H, suchthat
(σ(gi ))i≤n = (gi )i≤n ,while (5.5) implies that σ(gi ) = gi for every i, n1 + 1 ≤ i ≤ n. Setu1 := (gi )
i≤n1 , u2 := (gi )n1 +1≤i≤n and v1 := (σ(gi ))i≤n1 .

Hence w := u1 − v1 is a nonzero vector. Since the coordinates of u1 are δ-algebraic
 and σ belongs to the (ρ, δ)-Galois group H, the coordinates of v1 are also δover K
 Hence the coordinates of w are δ-algebraic over K.
 Furthermore,
algebraic over K.
u1 and v1 are both solution to the system
ρ(Y ) = A1 Y + A1,2 u2 .
It follows that
ρ(w) = A1 w .
Since we haveρ=K
 ρ ⊂ K(w)ρ = C,C⊂Q
Corollary 4.13 ensures the existence of a positive integer s and a (ρs , δ)-Picard such that w is
1 for the system ρs (Y ) = (A1 )[s] Y over K
Vessiot ﬁeld extension Q

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BORIS ADAMCZEWSKI ET AL.

the ﬁrst column of a fundamental matrix. Furthermore, the diﬀerence Galois groupof ρs (Y ) = (A1 )[s] Y is equal to G1 for the latter is connected. The coordinates of w
 and G1 being connected and irreducible, Proposition 5.4
being δ-algebraic over Kimplies that n1 = 1.
Now, let us prove that (g1 , . . . , gn ) ∈ K n . Set
u2 := (gi )
2≤i≤n .
By (5.5), we have u2 ∈ K n−1 and thus A1,2 u2 ∈ K. Furthermore, g1 ∈ F0 is
solution to the inhomogeneous order one equation
ρ(Y ) = A1 Y + A1,2 u2 .
Since g1 is ∂-algebraic over K, Proposition 5.1 implies that g1 ∈ K, and hence(g1 , . . . , gn ) ∈ K n . By (5.4), we obtain that(f, . . . , ρn−1 (f )) = T −1 (g1 , . . . , gn ) .
Since the coeﬃcients of T belong to K, it follows that f ∈ K, as wanted.



Acknowledgments
We warmly thank Michael F. Singer for his careful reading of an earlier version ofthis paper and for making many useful suggestions. We are also grateful to Daniel
Bertrand for his comments.
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Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille
Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France
Email address: boris.adamczewski@math.cnrs.fr
Institut de Recherche Mathématique Avancée, UMR 7501 Université de Strasbourget CNRS, 7 rue René Descartes, 67084 Strasbourg, France
Email address: dreyfus@math.unistra.fr
Institut de Mathématiques de Toulouse; UMR 5219, Université de Toulouse; CNRS,
UPS, F-31062 Toulouse Cedex 9, France
Email address: hardouin@math.univ-toulouse.fr

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