Quantitative problems on the size of $G$-operators

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HAL Id: hal-02521687

https://hal.archives-ouvertes.fr/hal-02521687v2

Preprint submitted on 30 Mar 2020

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Gabriel Lepetit

To cite this version:

Gabriel Lepetit. Quantitative problems on the size ofG-operators. 2020. �hal-02521687v2�

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Gabriel Lepetit March 30, 2020

Abstract

G-operators, a class of differential operators containing the differential operators of minimal order annihilating Siegel’sG-functions, satisfy a condition of moderate growth calledGalochkin condition, encoded by ap-adic quantity, the size. Previous works of Chudnovsky, André and Dwork have provided inequalities between the size of aG - operator and certain computable constants depending among others on its solutions.

First, we recall André’s idea to attach a notion of size to differential modules and detail his results on the behavior of the size relatively to the standard algebraic operations on the modules. This is the corner stone to prove a quantitative version of André’s generalization of Chudnovsky’s Theorem: for f(z)=P

α,k,`c_α,k,`z^αlog(z)^kf_α,k,`(z), where f_α,k,`(z) areG-functions, we can determine an upper bound on the size of the minimal operatorLoverQ(z) of f(z) in terms of quantities depending on the f_α,k,`(z) and the rationalsα. We give two applications of this result: we estimate the size of a product of twoG-operators in function of the size of each operator; we also compute a constant appearing in a Diophantine problem encountered by the author.

Remarks and notations

• In this paper, we will call "minimal operator of f overQ(z)", or "minimal operator" where there is no possible ambuigity, any nonzero operatorL∈Q(z) [d/dz] such thatL(f(z))=0, and whose order is minimal for f.

• For u₁, . . . ,u_n∈Q, we denote byd(u₁, . . . ,u_n) thedenominatorofu₁, . . . ,u_n, that is to say the smallestd∈N^∗such thatd u₁, . . . ,d u_nare algebraic integers.

• IfKis a subfield ofQ, we denote byO_Kthe set of algebraic integers ofK.

• Givenα∈Q, thehouseofαis α := max

σ∈Gal(Q/Q)|σ(α)|.

1 Introduction

AG-functionis a power seriesf(z)= ^∞P

n=0

a_nzⁿ∈Qzsatisfying the following hypotheses:

a) f is solution of a nonzero linear differential equation with coefficients inQ(z);

b) There existsC₁>0 such that∀n∈N, a_n ÉC₁ⁿ⁺¹;

c) There existsC₂>0 such that∀n∈N,d(a₀, . . . ,a_n)ÉC₂ⁿ⁺¹.

This family of special functions has been studied together with the family ofE-functions, which are the functionsf(z)= P^∞

n=0

(a_n/n!)zⁿsatisfyinga)and such that thea_nsatisfy the con- ditionsb)andc). Siegel defined both classes in [13]. The most basic example ofG-function,

1

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which gives it its name, is the geometric series f(z)= P^∞

n=0

zⁿ=1/(1−z). Other examples in- cludes the polylogarithms functions Lis(z) := ^∞P

n=1

zⁿ/n^s, or some hypergeometric series with rational parameters.

The theory ofG-functions was largely developed from the 1970s with the works of Ga- lochkin ([7]), Chudnovsky ([4]) and then André’s book ([1]), which is the first extensive and systematic review of this theory, later enhanced by Dwork ([5]).

The corner stone of this theory is the study of the nonzero minimal operators ofG- functions over Q(z). It turns out that they have specific properties, including the fact that they are Fuchsian differential operators with rational exponents. They belong to the class of G-operators, which are believed to "come from geometry" (see [1, chapter II]).

Chudnovsky’s Theorem states that the nonzero minimal operatorLof a nonzeroG-function f satisfies an arithmetic condition of geometric growth on some denominators, calledGa- lochkin condition. This condition can be equivalently expressed by the fact than a quantity σ(L), which is defined inp-adic terms (see Definition 7 in Section 2 below), is finite. This quantity is called thesizeofL. More generally, it is possible to associate a sizeσ(G) with a differential systemy⁰=G y,G∈Mn(Q(z)).

Similarly, we can associate with aG-function its size, encoding the conditionc)above.

Chudnovsky’s Theorem was proved in such a way that there is an explicit relation between the size ofG and the size of f (see [5, chapter VII]). This is particularly useful for Diophantine applications, as shown in [11].

In [3] and [2], André studied the properties of theNilsson-Gevrey series of arithmetic type of order 0,i.e.functions than can be expressed as a finite sum

f(z)= X

(α,k,`)∈S

c_α,k,`z^αlog(z)^kf_α,k,`(z) (1) whereS⊂Q×N×N,c_α,k,`∈C^∗, and the f_α,k,`(z) are nonzeroG-functions. André proved in particular an analogue of Chudnovsky’s Theorem for these functions:

Theorem 1 ([3], p. 720)

LetS⊂Q×N×Nbe a finite set,(c_α_,k,_`)₍_α_,k,_`₎_∈_S∈(C^∗)^S and a family(f_α_,k,_`(z))₍_α_,k,_`₎_∈_S of nonzeroG-functions. We consider

f(z)= X

(α,k,`)∈S

c_α,k,`z^αlog(z)^kf_α,k,`(z)

a Nilsson-Gevrey series of arithmetic type of order0. Then f(z)is solution of a nonzero linear differential equation with coefficients inQ(z)and the minimal operatorL of f(z) overQ(z)is aG-operator.

In this paper, we consider only Nilsson-Gevrey series of order 0, so that we shall not write

"order 0" anymore.

We aim to provide a more precise result: is it possible to find a quantitative relation between the sizeσ(L) of the minimal operator off(z) overQ(z) and the sizes of theG-functions

f_α_,k,_`(z)?

This question will be studied and given a positive answer in Section 3. The main result of this paper – and the answer to the problem above – is the following theorem:

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Theorem 2

We keep the notations of Theorem 1. For every(α,k,`)∈S, letL_α_,k,_`6=0denote a minimal operator of theG-function f_α,k,`(z). We setκthe maximum of the integersk such that(α,k,`)∈Sfor some(α,`)∈Q×NandA={α∈Q:∃(k,`)∈N², (α,k,`)∈S}. Then, we have

σ(L)Émax µ

1+log(κ+2), 2¡

1+log(κ+2)¢ log¡

maxα∈Ad(α)¢ ,

(αmax,k,`)∈S

¡(1+log(k+2))σ(L_α_,k,_`)¢

¶

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It turns out that the size of a differential system is invariant by equivalence of differential systems. Thus, André was able to generalize the notion of size to differential modules and to study the behavior of the size with respect to the usual algebraic operations on differential modules: submodule, quotient, direct sum, tensor product, etc.

We will follow this point of view to solve the problem above and devote Section 2 to some useful reminders on the differential modules and to a synthesis of the results of André on the size of differential modules.

In Section 4, we will finally give two applications of these results. The first one consists, given twoG-operatorsL₁andL₂, in finding an explicit inequality between the size ofL₁L₂ and the sizes ofL₁andL₂. Using the first application and Theorem 2, we then give an application consisting in the evaluation of a constant appearing in a Diophantine problem, which is studied in another paper [11].

Acknowledgements: I thank T. Rivoal for carefully reading this paper and for his useful comments and remarks that improved it substantially.

2 Size of a differential module

2.1 Reminder on differential modules

In this part, we will essentially synthetise the presentation of the theory of differential modules contained in [14, chapter 2], that will be useful in the rest of this paper. More details and proofs about differential modules can be found in the above-quoted book.

Definition 1

Let(k,δ) be a differential field. The ring of differential operators overk is denoted by k[∂]and defined as the set of noncommutative polynomials in∂with the compatibility condition

∀a∈k, ∂a=a∂+δ(a).

This is a left (and right) euclidean ring for the euclidean function ord :L=

n

X

k=0

a_k∂^k7→n (where a_n6=0).

Definition 2

Let(k,δ)be a differential field. A differential moduleM over k is ak-vector space of finite dimension which is a leftk[∂]-module, or equivalently ak-vector space of finite dimension endowed with a map∂:M →M such that

∀λ∈k,∀a,b∈M,∂(a+b)=∂(a)+∂(b) and ∂(λa)=λ∂(a)+δ(λ)a.

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LetM be ak-differential module of dimension n endowed with a k-basis (e₁, . . . ,e_n).

Then there exists a matrix A =(a_i_,j)_i_,j ∈M_n(k) such that∀1Éi Én, ∂ei = − Pⁿ

j=1

a_j,ie_j. A direct computation shows that ifu=

n

P

i=1

uiei∈M, then∂u=0 ⇐⇒ u⁰=Au,u=^t(u1, . . . ,un).

Remark.If we choose anotherk-basis (f₁, . . . ,f_n) ofM such that∂fi = − Pn j=1

b_j_,if_j, then there existsP∈GL_n(k) such thatB=P[A]=P⁻¹P⁰+P⁻¹AP.

The differential systems defined byAandBare then said to beequivalentoverk. This is equivalent to the fact thaty7→P yis a one-to-one correspondance between the solution sets ofy⁰=Ayandy⁰=B y.

Therefore, the matrixAobtained by the construction above doesn’t depend on the choice of basis we make, up to equivalence of differential systems.

We can also do the converse operation:

Definition 3

ForA=(a_{i j})_i,_j∈M_n(k), we define the differential moduleMAassociated with the differential systemy⁰=Ay byMA=kⁿendowed with the derivation∂such that∀i ∈{1, . . . ,n},

∂e_i= −

n

P

j=1

a_{j i}e_j, where(e₁, . . . ,e_n)is the canonical basis ofkⁿ. We associate with a differential operatorL=∂ⁿ+

n

P

k=0

a_k∂^k∈k[∂] the differential module ML:=MAL, whereA_Lis the companion matrix ofL:

A_L=







0 1 (0)

. .. ...

(0) 0 1

−a₀ . . . −a_n−1





 .

Definition 4

A morphism of differential modulesis ak-linear mapϕ: (M,∂_M)→(N ,∂_N)such that

∂_N ◦ϕ=ϕ◦∂_M.

Proposition 1

Let A,B ∈M_n(k). Then there is an isomorphism of differential modules betweeenMA

andMBif and only ifAandB are equivalent overk.

Thus, the classes of equivalences of differential systems overkclassify thek-differential modules up to isomorphism.

We can perform usual algebraic constructions with the differential modules:

• Adifferential submoduleofM is a leftk[∂]-submodule ofM.

• IfN is a differential submodule ofM, then thequotient differential moduleM/N is endowed with the quotient derivation∂:m7→∂(m).

• The cartesian product (or direct sum) of (M1,∂1) and (M2,∂2) can be endowed with the derivation∂: (m₁,m₂)7→(∂1(m₁),∂2(m₂)), which makes it a differential module.

• Thetensor productM1⊗kM2is a differential module when endowed with∂:m₁⊗m₂7→

m₁⊗∂2(m₂)+∂1(m₁)⊗m₂(extended on the whole vector space with the Leibniz rule). Note that the tensor product cannot be defined over the noncommutative ringk[∂], sinceM1and M2are only leftk[∂]-modules. We will therefore often denoteM1⊗M2without precising the base field.

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• The derivation ∂^∗ : ϕ7→¡

m7→ϕ(∂(m))−δ(ϕ(m))¢

makes the dual M^∗ =Homk(M,k) a differential module

• More generally, the preceding constructions enable us to define a structure of differential module on thespace of morphismsHomk(M1,M2)'M1⊗kM₂^∗.

Remark.Actually, for any A∈M_n(k), we haveM_A^∗'MA^∗, whereA^∗= −^tA. IfX is a fundamental matrix of solutions of the systemy⁰=Ay, thenY :=^tX⁻¹satisfiesY⁰= −^tAY.

Lemma 1

There is an isomorphism of differential modulesML'

³Q(z)[∂]/Q(z)[∂]L´_∗ .

Definition 5

Theadjoint operatorofL∈Q(z)[∂]is the operatorL^∗such thatML^∗'Q(z)[∂]/Q(z)[∂]L.

We can compute directlyL^∗: if L =∂ⁿ+

n−1

P

k=0

a_k∂^k, then L^∗= (−∂)ⁿ+

n−1

P

k=0(−∂)^ka_k. The adjoint operator has useful properties, for instance thatL^∗∗=Land (L₁L₂)^∗=L^∗₂L^∗₁.

We finally mention that thecyclic vector Theorem, implies that any differential system y⁰=Ay is equivalent to some systemy⁰=A_Ly, L∈k[∂]. This yieldsin finean equivalence between differential systems and differential operators (see [14, pp. 42–43]).

2.2 Size and operations on differential modules

LetKbe a number field, andG∈M_n(K(z)). In this subsection, following André’s notations, we introduce a quantity σ(G) encoding an arithmetic condition of moderate growth on a differential system, calledGalochkin condition.

Fors ∈N, we defineG_s as the matrix such that, if y is a vector satisfying y⁰=G y, then y^(s)=Gsy. In particular,G₀ is the identity matrix. The matricesGs satisfy the recurrence relation

∀s∈N, G_s+1=G_sG+G⁰_s, whereG⁰_sis the derivative of the matrixG_s.

Definition 6 (Galochkin, [7])

LetT(z)∈K(z)be such that T(z)G(z)∈M_n(K[z]). For s ∈N, we consider q_s Ê1the least common denominator of all coefficients of the entries of the matricesT(z)^mG_m(z)

m! , whenm∈{1, . . . ,s}. The systemy⁰=G y is said tosatisfy the Galochkin conditionif

∃C>0 : ∀s∈N, q_sÉC^s⁺¹.

Chudnovsky’s Theorem (cf [4]) states that ifG is the companion matrix of the minimal non-zero differential operatorLassociated to aG-function, then the systemy⁰=G ysatisfies the Galochkin condition. That is why we say thatL is aG-operator (see [3, pp. 717–719] for a review of the properties ofG-operators). Following [5, chapter VII], we are now going to rephrase this condition inp-adic terms.

For every prime idealpofOK, we define| · |pas thep-adic absolute value onK, with the choice of normalisations given in [5, p.223].

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We recall that theGauss absolute valueassociated with| · |pis the non-archimedean absolute value

| · |p,Gauss: K(z) −→ R PN

i=0

a_izⁱ

M

P

j=0

b_jz^j 7−→

0ÉmaxiÉN|ai|p 0maxÉjÉM|b_j|p

.

The absolute value| · |p,Gaussnaturally induces a norm onMµ,ν(K(z)), defined for allH= (h_i,_j)_i,_j∈M_µ,ν(K(z)) askHkp,Gauss=max_i,_j|hi,j|p,Gauss. It is called theGauss norm. Ifµ=ν,

| · |p,Gaussis a norm of algebra onM_µ(K(z)). We now use this notation to define the notion of size of a matrix:

Definition 7 ([5], p. 227)

LetG∈M_n(K(z)). ThesizeofGis

σ(G) :=lim sup

s→+∞

1 s

X

p∈Spec (O_K)

h(s,p)

where

∀s∈N, h(s,p)=sup

mÉs

log⁺

°

° G_m

m!

°

°p,Gauss

, withlog⁺:x7→log (max(1,x)). ThesizeofY = P^∞

m=0

Y_mz^m,Y_m∈Mp,q(K)is σ(Y) :=lim sup

s→+∞

1 s

X

p∈Spec (O_K)

sup

mÉs

log⁺kY_mkp.

The relation between Galochkin condition and size is given by the following result.

Proposition 2

a) With the notations of Definitions 6 and 7, we have

σ(G)+h⁻(T)

[K:Q]Élim sup

s→+∞

1

s log(q_s)É[K:Q]σ(G)+h⁺(T) , where

h⁻(T)= X

p∈Spec (O_K)

log¡ min¡

1,|T|p,Gauss

¢¢ and h⁺(T)= X

p∈Spec (O_K)

log⁺|T|p,Gauss

([11, Proposition 5, p. 16]).

b) In particular, the differential systemy⁰=G ysatisfies the Galochkin condition if and only ifσ(G)< +∞([5, p. 228]).

WhenG∈M_n(Q(z)) andT(z)∈Z[z] has at least one coefficient equal to 1, we have the equality

σ(G)=lim sup

s→+∞

1

slog(q_s).

The size does not depend on the number fieldKwe consider, and it is invariant under equivalence of differential systems, as this lemma shows. It is due to André ([1, p. 71]). We give a more detailed proof than in [1] for the sake of completeness.

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Lemma 2 ([1], p. 71)

LetKa number field,G∈M_n(K(z))andP∈GL_n(K(z)), letH=P[G]=PGP⁻¹+P⁰P⁻¹be a matrix defining a differential systemy⁰=H y which is equivalent toy⁰=G y overQ(z).

Thenσ(G)=σ(H).

Proof. It follows from [10, p. 12] that

∀s∈N, Hs=

s

X

m=0

Ãs m

!

P^(s⁻^m)GmP⁻¹. Hence forp_∈_{Spec (O}_K),

¯

¯ Hs

s!

¯

¯p,Gauss

É max

0ÉmÉs

¯

P^(s⁻^m) (s−m)!

¯

¯p,Gauss

· |P⁻¹|p,Gauss·

¯

¯ Gm

m!

¯

¯p,Gauss

É |P|p,Gauss· |P⁻¹|p,Gauss· max

0ÉmÉs

¯

¯ Gm

m!

¯

¯p,Gauss

because

¯

P^(s⁻^m) (s−m)!

¯

¯p,Gauss

É |P|p,Gauss(cf [5, p. 118]). DenotingC(p)= |P|p,Gauss· |P⁻¹|p,Gauss, we then have, for all integerss,

0maxÉmÉs

¯

¯ H_s

s!

¯

¯p,Gauss

ÉC(p) max

0ÉmÉs

¯

¯ G_m

m!

¯

¯p,Gauss

. Hence

σ(H)Éσ(G)+lim sup

s→+∞

1 s

X

p∈Spec (O_K)

log⁺C(p)=σ(G).

Symmetrically, we haveσ(G)Éσ(H), asG=P⁻¹[H].

Lemma 2 and Proposition 1 imply that we can define without ambiguity thesizeofM by σ(M) :=σ(A), because every differential moduleM can be associated with a unique class of equivalence of differential systems [A].

Definition 8 ([1], pp. 74–76)

LetL ∈Q(z)[∂]. We setσ(L)=σ(ML). Ifσ(L)< +∞, the operatorL is said to be a G- operator.

This terminology is justified by the fact that every solution of the equationL(y(z))=0 around a non singular point of theG-operatorLis aG-function.

The following result is key to this paper. It is due to André. It proves that the notion of size is compatible with most of the usual operations on differential modules.

Recall that forn∈N^∗, then-th symmetric power of the differential moduleM, Symⁿ

Q(z)(M), is the quotient ofM^⊗ⁿby the submodule generated overQ(z) by them₁⊗ · · · ⊗mn−m_σ(1)⊗

· · · ⊗m_σ_(n), whenσis any permutation of {1, . . . ,n}.

Proposition 3 ([1], p. 72)

LetM1,M2,M3be differential modules overQ(z). Then

a) IfM2 is a differential submodule ofM1, then σ(M2)Éσ(M1)andσ(M1

±M2)É σ(M1).

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b) σ(M1×M2)=max (σ(M1),σ(M2))Éσ(M1)+σ(M2).

c) σ³ Sym^N

Q(z)(M1)

´ É¡

1+log(N)¢

σ(M1).

d) σ(M₁^∗)Éσ(M1)¡

1+log(µ1−1)¢

, whereµ1=dim_Q_(z)M1.

e) If there is an exact sequence0→M2→M1→M3→0, then we have

σ(M1)É1+2σ(M2×M3×(M3)^∗)É1+2 max (σ(M2),σ(M3)+µ3−1) (3) withµ3=dim_Q_(z)M3.

Ine), we can actually deduce from André’s proof the following slightly more precise result:

σ(M1)É1+11 6 σ¡

M2×M3×(M3)^∗¢

. (4)

Alternatively, we have

σ(M1)Éµ3+σ(M2)+σ(M3). (5) Note that (4) is not always a better estimate than (5), as the example withσ(eL_β) in Sub- section 4.2 shows.

The proof ofa)ande)relies essentially on the following lemma, mentioned in [1, p. 72], of which we give a proof for the reader’s convenience:

Lemma 3 ([1], p. 72)

Let M1,M2,M3 be differential modules overQ(z). If there is an exact sequence0→ M2→M1→M3→0, then, with a suitable choice of basis,M1is associated with a differential system of the form∂y=G y, whereG=

µG⁽²⁾ G⁽⁰⁾ 0 G⁽³⁾

¶

and the systems∂y=G⁽²⁾yand

∂y=G⁽³⁾yare respectively associated withM2andM3. Proof. We read on the exact sequence thatM3'M1

±M2.

Linear algebra convinces us that, if (e₁, . . . ,e_n) is ak-basis ofM2and (f₁, . . . ,f_m) is ak- basis ofM1/M2, thenB=(e₁, . . . ,e_n,f₁, . . . ,f_m) is a basis ofM1.

Thus, if∀1Éi Ém, ∂f_i = −

m

P

j=1

g⁽³⁾_j,if_j, we have∂f_i+

n

P

j=1

g⁽³⁾_j,if_j∈M2so that there exists a matrixG⁽⁰⁾=(g_i,⁽⁰⁾_j)∈Mm,n(k) such that

∀1ÉiÉm, ∂f_i= −

m

X

j=1

g⁽³⁾_j,if_j−

n

X

j=1

g⁽⁰⁾_j,ie_j.

Furthermore, we can find a matrixG⁽²⁾∈M_n(k) associated withM2such that

∀1Éi Én, ∂e_i= −

m

X

j=1

g⁽²⁾_j,ie_j.

Hence, the matrixGsuch that∂^tB= −^tG^tBis of the formG=

µG⁽²⁾ G⁽⁰⁾ 0 G⁽³⁾

¶ .

Proof of Proposition 3. a)is a direct consequence of Lemma 3, andb)follows straight- forwardly from the definition. For the pointc), see [1, p. 72] (and [1, pp. 17–18] for the key argument). The pointd)is a consequence of Katz’s Theorem (see [1, p. 80]).

Fore), André gave a proof of which we provide the details here.

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By Lemma 3, we can find suitable bases ofM1,M2, M3such that, in these bases, M1

(resp.M2,M3) represents the system∂y=G y (resp.∂y=G⁽²⁾yand∂y=G⁽³⁾y) where G=

µG⁽²⁾ G⁽⁰⁾ 0 G⁽³⁾

¶ .

LetKbe a number field such thatG∈M_n(K(z)). We fixp_∈Spec (O_K). We considert_pa free variable onKcalled thegeneric point. We can then build a complete and algebraically closed extension of¡

K(tp),| · |p,Gauss

¢. Concretely,Ωpis the completion of the algebraic closure of the completion ofK(t_p) endowed with the Gauss valuation| · |p,Gauss(see [5, p. 93]).

We set

X_t⁽²⁾

p (z)= X∞ s=0

G⁽²⁾_s (t_p)

s! (z−t_p)^s∈GLn(Ωpz−t_p)

(resp. X_t⁽³⁾_p ∈GLm(Ωpz−tp)) a fundamental matrix of solutions of∂y =G⁽²⁾y (resp. ∂y = G⁽³⁾y) at the generic pointt_p such that X_t⁽²⁾_p (t_p)=I_n (resp. X_t⁽²⁾_p (t_p)=I_m). Consider X_t⁽⁰⁾_p ∈ Mn,m(Ωpz−tp) a solution of

∂X_t⁽⁰⁾_p =G⁽³⁾X_t⁽⁰⁾_p +G⁽⁰⁾X_t⁽²⁾_p (6) such thatX_t⁽⁰⁾_p (tp)=0.

ThenXtp=

ÃX_t⁽²⁾_p X_t⁽⁰⁾_p 0 X_t⁽³⁾_p

!

is a fundamental matrix of solutions of∂y=G ysuch thatXtp(tp)= I_n+m. Thus we have∀s ∈N, ∂^s(X_t_p)(t_p)=G_s(t_p). Since we know that∂^s(X_t⁽²⁾_p )(t_p)=G⁽²⁾_s (t_p) (and likewise forX_t⁽³⁾_p ), it suffices to estimate the Gauss norm of∂^s(X_t⁽⁰⁾_p )(t_p) to obtain an estimate on the size ofG.

For simplicity, we will omit the indext_pin what follows.

It follows from (6) that

∂(X⁽³⁾⁻¹X⁽⁰⁾)=X⁽³⁾⁻¹∂(X⁽⁰⁾)−X⁽³⁾⁻¹∂(X⁽³⁾)X⁽³⁾⁻¹X⁽⁰⁾

=X⁽³⁾⁻¹(G⁽³⁾X⁽⁰⁾+G⁽⁰⁾X⁽²⁾)−X⁽³⁾⁻¹G⁽³⁾X⁽⁰⁾=X⁽³⁾⁻¹G⁽⁰⁾X⁽²⁾ so that, for`∈N, by Leibniz’ formula,

∂^`X⁽⁰⁾

`! = X`

i=0

∂^`−ⁱ(X⁽³⁾) (`−i)!

∂ⁱ(X⁽³⁾⁻¹X⁽⁰⁾)

i! =∂^`(X⁽³⁾)X⁽³⁾⁻¹X⁽⁰⁾+ X`

i=1

∂^`−ⁱ(X⁽³⁾)

(`−i)! ∂ⁱ⁻¹(X⁽³⁾⁻¹G⁽⁰⁾X⁽²⁾)

=∂^`(X⁽³⁾)X⁽³⁾⁻¹X⁽⁰⁾+ X`

i=0

∂^`−ⁱ(X⁽³⁾) (`−i)!

1 i

X

i1+i2+i3=i−1

∂ⁱ¹(X⁽³⁾⁻¹) i₁!

∂ⁱ²(G⁽⁰⁾) i₂!

∂ⁱ³(X⁽²⁾) i₃! . We now evaluate this equality at the generic pointtp. SinceX⁽⁰⁾(tp)=0, we get

∂^`X⁽⁰⁾

`! (tp)= X`

i=0

∂^`−i(X⁽³⁾)(t_p) (`−i)!

1 i

X

i1+i2+i3=i−1

∂ⁱ¹(X⁽³⁾⁻¹)(t_p) i₁!

∂ⁱ²(G⁽⁰⁾)(t_p) i₂!

∂ⁱ³(X⁽²⁾)(t_p) i₃! . We make the following observations:

• SinceG⁰(tp)∈K(tp), we have

°

∂ⁱ²(G⁽⁰⁾) i₂!

°

°p,Gauss

É kG⁰kp,Gaussby [5, p. 94].

• Setδ_`=lcm(1, 2, . . . ,`). Then X

p_∈Spec (OK)

sup

1ÉiÉ`

1

|i|p

= 1

[K:Q]log(δ_`)É `

[K:Q](1+o(1)), by [5, Lemma 1.1 p. 225].

(11)

Hence, sincek · kp,Gaussis non-archimedean, we have log⁺

°

∂^`X⁽⁰⁾

`!

°

°p,Gauss

É max

0ÉiÉn i1+i2+i3=i−1

µ log⁺

°

∂^`−ⁱ(X⁽³⁾) (`−i)!

°

°p,Gauss

+log⁺

°

° 1 i

°

°p,Gauss

+

log⁺

°

∂ⁱ¹(X⁽³⁾⁻¹) i₁!

°

°p,Gauss

+log⁺

°

∂ⁱ²(G⁽⁰⁾) i₂!

°

°p,Gauss

+log⁺

°

∂ⁱ³(X⁽²⁾) i₃!

°

°p,Gauss

! (7) But

°

∂^`−i(X⁽³⁾⁻¹) (`−i)!

°

°p,Gauss

=

°

¡G⁽³⁾¢

`−i

(`−i)!

°

°p,Gauss

and

°

∂ⁱ³(X⁽²⁾) i₃!

°

°p,Gauss

=

°

¡G⁽²⁾¢

i3

i₃!

°

°p,Gauss

. Moreover we have

°

∂ⁱ¹(X⁽³⁾⁻¹) i₁!

°

°p,Gauss

=

°

¡−^tG⁽³⁾¢

i1

i₁!

°

°p,Gauss

,

because^tX⁽³⁾⁻¹ is a fundamental matrix of solutions of the system y⁰= −^tG⁽³⁾y, associated with the dual moduleM₃^∗. This yields for every`∈N^∗andNÊ`

log⁺

°

∂^`(X⁽⁰⁾)(t_p)

`!

°

°p,Gauss

Éh(N,p,G⁽³⁾)+h(N,p,−^tG⁽³⁾)+h(N,p,G⁽²⁾)

+log⁺kG⁽⁰⁾kp,Gauss+log max

0ÉiÉN

1

|i|p

. We therefore obtain

lim sup

N→+∞

1 N

X

p∈Spec (O_K)

sup

nÉN

log⁺

°

∂^`(X⁽⁰⁾)(t_p)

`!

°

°p,Gauss

É 1

[K:Q]+σ(M3)+σ(M₃^∗)+σ(M2) sincekG⁽⁰⁾kp,Gauss=1 for all but a finite number of primes. Furthermore,

°

° G_`

`!

°

°p,Gauss

=

°

∂^`(X)(t_p)

`!

°

°_p

,Gauss

=max Ã°

°

∂^`(X⁽⁰⁾)(t_p)

`!

°

°p,Gauss

,

°

∂^`(X⁽²⁾)(t_p)

`!

°

°p,Gauss

,

°

∂^`(X⁽³⁾)(t_p)

`!

°

°p,Gauss

!

so finally

σ(M1)É1+σ(M3)+σ(M₃^∗)+σ(M2).

We can actually obtain a more refined inequality by using the following argument. Let

` ∈N andi1+i2+i3É`−1. We assume for examplei1Êi2Êi3, the other cases can be treated likewise and lead to the same final conclusion. By considering separately the cases i₁É`−1

2 andi₁>`−1

2 , we prove thati₂É`−1

2 andi₃É`−1 3 . Thus we obtain in inequality (7)

max Ã

log⁺

°

∂ⁱ¹(X⁽³⁾⁻¹) i₁!

°

°p,Gauss

+log⁺

°

∂ⁱ²(G⁽⁰⁾) i₂!

°

°p,Gauss

+log⁺

°

∂ⁱ³(X⁽²⁾) i₃!

°

°p,Gauss

!

Éh(`,p,G⁽³⁾)+h µ`

2,p,−^tG⁽³⁾

¶ +h

µ`

3,p,G⁽²⁾

¶

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so that 1

` X

p∈Spec (O_K)

h(`,p,G)Élog(δ`)

` +1

` X

p∈Spec (O_K)

log⁺kG⁽⁰⁾kp,Gauss+v_`+1

`

¹` 2 º

v^j_`

2

k+1

`

¹` 3

º v^j_`

3

k

with v_`= 1

` P

p∈Spec (O_K)

max¡

h(`,p,G⁽³⁾),h(`,p,−^tG⁽³⁾),h(`,p,G⁽²⁾)¢

. Taking the superior limit on both sides, we get

σ(M1)É1+11 6 max¡

σ(M₃^∗),σ(M3),σ(M2)¢ , bounded by 2 max¡

σ(M₃^∗),σ(M3),σ(M2)¢

in [1]. This is the desired conclusion.

Remark.The inequality d) is a consequence of Katz’s theorem on the exponents of a G- operator. We now explain how to obtain another bound onσ(M₁^∗).

Following the notations of [5, p. 226], we denote byρ(M1) the global inverse radius of M1. Then André-Bombieri Theorem ([1, p. 74]) yields

ρ(M1)Éσ(M1)Éρ(M1)+µ−1,

whereµ=dim_Q(z)M1. Moreover, [1, Lemma 2 p. 72] states thatρ(M₁^∗)=ρ(M1). Thus, the application of André-Bombieri Theorem to the adjoint moduleM₁^∗yieldsρ(M1)Éσ(M₁^∗)É ρ(M1)+µ−1, hence

σ(M1)−µ+1Éσ(M₁^∗)Éσ(M1)+µ−1.

Pointsa) c) of Proposition 3 actually imply the following result, which was not stated explicitly by André and is important for our application to Theorem 1.

Proposition 4

Let(Mi)_1Éi_ÉN be a family of differential modules overQ(z). Then σ(M1⊗_Q_(z)· · · ⊗_Q_(z)MN)É¡

1+log(N)¢ max¡

σ(M1), . . . ,σ(MN)¢ . In particular, we haveσ¡

M₁^⊗N¢

É(1+log(N))σ(M1).

Proof. We denote byαthe class of an elementα∈(M1× · · · ×MN)^⊗ⁿin the quotient space Symⁿ(M1× · · · ×MN). There is an injective morphism of differential modules

ϕ: M1⊗ · · · ⊗MN −→ Symⁿ(M1× · · · ×MN)

m₁⊗ · · · ⊗m_N 7−→ (m₁, 0, . . . , 0)⊗ · · · ⊗(0, . . . , 0,m_N).

Indeed, for alli, letBi =(e⁽ⁱ₁⁾, . . . ,e⁽ⁱ⁾_n_i) be aQ(z)-basis ofMi. AQ(z)-basis ofM1⊗ · · · ⊗MN is the familyFwhose elements are thee⁽¹⁾_i

1 ⊗e⁽²⁾_i

2 ⊗ · · · ⊗e_i^(N⁾

N when 1Éi_kÉn_k.

DenoteB=(f₁, . . . ,f_n) the basis ofM1×. . .MN obtained by concatenation of the bases Bi. Then the familyF⁰ whose elements are the fi₁⊗fi₂⊗ · · · ⊗fi_N when 1Éi₁Éi₂É · · · É i_NÉnis a basis of Symⁿ(cf [8, p. 218]). Moreover,

ϕ(e_i⁽¹⁾₁ ⊗e_i⁽²⁾

2 ⊗ · · · ⊗e^(N)_i

N )=fi₁⊗fn₁+i₂⊗ · · · ⊗fn₁+···+n_N−1+i_N

soFis sent byϕon a free family of Symⁿ(M), which proves thatϕis injective.

Hence the combination ofa)andc)in Proposition 3 yields σ(M1⊗ · · · ⊗MN)É¡

1+log(N)¢ max¡

σ(M1), . . . ,σ(MN)¢ .

(13)

3 A generalization of Chudnovsky’s Theorem for Nilsson-Ge- vrey series of arithmetic type

The goal of this section is to prove Theorem 2, a quantitative version of André’s result on a Chudnovsky type Theorem for Nilsson-Gevrey series of arithmetic type ([3]).

In what follows, we will denote the standard derivation d/dzonQ(z) by∂.

3.1 Product and sum of solutions ofG-operators

The following result is proved in André in [3, p. 720]:

Proposition 5 ([3], p. 720)

Ify₁(resp. y₂) is solution of aG-operatorL₁(resp.L₂), then

a) y1+y2is solution of aG-operatorL3; b) y1y2is solution of aG-operatorL4.

Note thaty₁andy₂need not beG-functions, they are in general Nilsson-Gevrey series of arithmetic type. This proposition is obvious ify₁andy₂areG-functions, because the set of G-functions is a ring.

We are going to use the results of Section 2 (Proposition 3) in order to bound the sizes of L₃andL₄.

Proposition 6

LetL₁andL₂be the respective minimal operators ofy₁andy₂, which areG-operators.

We can find nonzeroG-operatorsL₃andL₄as in Proposition 5 such that σ(L3)Émax¡

σ(L1),σ(L2)¢

and σ(L4)É¡

1+log(2)¢ max¡

σ(L1),σ(L2)¢ .

Proof of Propositions 5 and 6. In [1, p. 720], André gave a sketch of proof of Proposition 5, which we make explicit here.

IfL∈Q(z)[∂] is the minimal operator of a function y, we notice that there is a natural isomorphism of differential modules betweenQ(z)[∂]/Q(z)[∂]Land

Q(z)[∂](y) :=©

M(y)|M∈Q(z)[∂]ª given byM modL7→M(y). Henceσ(L)=σ³³

Q(z)[∂](y)´_∗´ .

We also define foru,v solutions ofG-operators,Q(z)[∂](u,v) :=

³Q(z)[∂](u)´

[∂](v) which is the set of linear combinations with coefficients inQ(z) of the∂^k(u)∂^`(v), fork,`∈N. The existence of differential equations overQ(z) satisfied byuandvensures that this is indeed a finite dimensionalQ(z)-vector space, so that it can be endowed with a structure of differential module.

• LetL3be the minimal operator ofy1+y2overQ(z). Thenσ(L3)=σ³³

Q(z)[∂](y1+y2)

´_∗´ . LetM be the image of the morphism of leftQ[z][∂]-modules

ι: Q(z)[∂] −→ Q(z)[∂](y1)×Q(z)[∂](y₂) L 7−→ (L(y₁),L(y₂)).

Thus, M is a differential submodule of Q(z)[∂](y1)×Q(z)[∂](y2) because it is a sub-left- Q(z)[∂]-module ofQ(z)[∂](y1)×Q(z)[∂](y₂) which is of finite dimension overQ(z).