Multi-resurgence of formal solutions of linear meromorphic differential systems

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meromorphic differential systems

Pascal Remy

To cite this version:

Pascal Remy. Multi-resurgence of formal solutions of linear meromorphic differential systems. 2015.

�hal-01140091�

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meromorphic di¤erential systems

Pascal Remy

6 rue Chantal Mauduit, F-78 420 Carrières-sur-Seine email : [email protected]

Abstract

In this paper, we consider a linear meromorphic di¤erential system at the origin. For any of its levels , we prove with the factorization theorem that the Borel transforms of its -reduced formal solutions are resurgent and we give a complete description of all their singularities.

Then, restricting ourselves to some special geometric con…gurations of the singular points of these Borel transforms, we make explicit formulæ relating the Stokes multipliers of level of the given system to some connection constants in the Borel plane. So, we generalize the results already obtained by M. Loday-Richaud and the author for systems with a unique level and for the lowest and highest levels of systems with multi-levels. As an illustration, we develop one example.

Keywords. Linear di¤erential system, multisummability, Stokes phe- nomenon, Stokes multipliers, resurgence, singularities, connection constants

AMS subject classi…cation. 34M03, 34M30, 34M35, 34M40

1 Introduction

All along the article, we consider a linear meromorphic di¤erential system (in short, a di¤erential system or a system) of dimensionn 2at the origin 02C of the form

x^r+1dY

dx =A(x)Y (A)

where r 1 is a positive integer and where A(x) 2 M_n(Cfxg) is a n n- analytic matrix at 0 such thatA(0) 6= 0. Using a …nite algebraic extension x7 !x with 2N and a meromorphic gauge transformationY 7 !T(x)Y

1

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with a suitable polynomial matrixT(x)inxand1=xif needed, we can always assume (see [5]) that system (A) admits as formal fundamental solution at0 a matrix of the form Ye(x) =Fe(x)x^Le^Q(1=x) with

(N₁) Fe(x) 2 M_n(C[[x]]) a formal power series in x satisfying Fe(x) = I_n+ O(x^r), whereI_n denotes the identity matrix of size n,

(N₂) L = MJ

j=1

( _jI_n_j +J_n_j), where J is an integer 2, the eigenvalues j

satisfy 0 Re( _j)<1and where

J_n_j = 8>

>>

><

>>

:

0 if n_j = 1

2 66 66 4

0 1 0

... . .. ... ...

... . .. 1

0 0

3 77 77

5 if n_j 2 is an irreductible Jordan block of size n_j,

(N₃) Q(1=x) a diagonal matrix of the form

Q 1

x =

MJ j=1

q_j 1 x I_n_j

where the q_j(1=x) are polynomials in 1=x of degree r and without constant terms.

Recall that normalizations (N₁) and (N₂) guarantee the unicity of Fe(x) as formal series solution of the homological system (AH) associated with system (A) (see [5]).

Under the hypothesis that system (A) has the unique level r 1 (see de…nition 2.1 below for the exact de…nition of levels), M. Loday–Richaud and the author investigated in [12] (case r = 1) and [23] (case r 2) the resurgence of the Borel transforms of the r-reduced series (= sub-series of terms r by r) of Fe(x) and displayed a complete description ofall their singularities. Then, as an application, they stated some Stokes-to-connection formulæ making explicit the Stokes multipliers of system (A) in terms of some connection constants in the Borel plane, providing thus an e¢ cient tool for the e¤ective calculation of the Stokes-Ramis matrices of system (A).

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When system (A) has multi-levelsr₁ < ::: < r_p, these results were general- ized later to thelowest [22] andhighest[21] levels by respectively considering, on one hand, the r₁-reduced series andr_p-reduced series ofFe(x)and, on the other hand, the lowest and highest levels’Stokes-Ramis matrices.

In the present paper, we propose to extend the results above to any level r_k of system (A). To do that, we shall proceed similarly as the approach developed in [22] for the lowest level by …rst showing that the study of level r_k can always be reduced to the study of the highest level of a convenient system. This point, which is central in our present approach, is based on the factorization theorem of Fe(x) [9, 19, 20] (see section 2, theorem 2.7 below) and on a block-diagonalisation theorem allowing to write system (A) on a convenient block-diagonal form (section 3.2, theorem 3.6). Using that and the results of [21], we then prove that the Borel transforms Fb^[k;u]( ), u = 0; :::; r_k 1, of the r_k-reduced series of Fe(x) areresurgent (section 3.3, theorem 3.10) and we give a complete description of all their singularities (section 3.4, theorem 3.15). In next section 4, we restrict our study to some special geometric con…gurations of singular points of the Fb^[k;u]’s; then, for such con…gurations, we display connection-to-Stokes formulæ of level r_k relating the Stokes multipliers of level r_k of Fe(x) to the connection constants of the Fb^[k;u]’s in the Borel plane (theorem 4.11). As an illustration of these formulæ, we develop one example (section 4.3).

2 Preliminaries

Split the matrix Fe(x) into J column-blocks …tting to the Jordan block- structure of matrix L(for `= 1; :::; J, the matrix Fe ^;`(x) hasn_` columns):

Fe(x) =h

Fe ^;1(x) Fe ^;2(x) Fe ^;J(x) i

:

The aim of this section is to brie‡y recall some basic de…nitions/results about the summation theory and to introduce some notations we are needed in the sequel.

2.1 Some de…nitions and notations

Given a pair (q_j; q_`)such that q_j 6 q_`, we denote (q_j q_`) 1

x = ^j;`

x^r^j;` +o 1

x^r^j;` ; _j;`6= 0:

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De…nition 2.1 (Levels, Stokes values and anti-Stokes directions of Fe ^;`(x)) Let j; `2 f1; :::; Jg such thatq_j 6 q_`.

The degree r_j;` is called a level of Fe ^;`(x).

The coe¢ cient j;` is called a Stokes value of level r_j;` of Fe ^;`(x).

The directions of maximal decay of e^(q^j ^q^`^)(1=x), i.e., ther_j;` directions arg( _j;`)=r_j;` mod (2 =r_j;`) along which j;`=x^r^j;` is real negative, are called anti-Stokes directions of level r_j;` of Fe ^;`(x).

Note that a Stokes value (resp. an anti-Stokes direction) ofFe ^;`(x)may be with several levels. Note also that the denomination “anti-Stokes directions”

is not universal: sometimes, one calls such directions “Stokes directions”.

Notation 2.2 The set R^(`) :=fr₁^(`)< ::: < rp^(`)`g with p_` 1 denotes the set of all levels of Fe ^;`(x).

Note that, according to normalization(N₃), all the levelsr_k^(`) are integer;

one refers sometimes this case as the unrami…ed case.

Note also that, for all`, we haver_p^(`)_` rthe rank of system (A). Actually, if there exists ` such that r^(`)p` < r, then rp^(`)` < r for all ` 2 f1; :::; Jg and polynomialsq_j have the same degreerand the same terms of highest degree.

One then reduces to the caser^(`)p` =rby means of a change of unknown vector of the form Y = Ze^q(1=x) with a convenient polynomial q(1=x)2 x ¹C[x ¹].

Recall that such a change does not a¤ect levels or Stokes-Ramis matrices of system (A).

De…nition 2.3 (Levels, Stokes values and anti-Stokes directions of Fe(x)) We call

level of Fe(x) (or of system (A)) any level of theFe ^;`(x)’s,

Stokes value ofFe(x)(or of system (A))any Stokes value of theFe ^;`(x)’s, anti-Stokes direction ofFe(x) (or of system (A)) any anti-Stokes direction of the Fe ^;`(x)’s.

Notation 2.4 The set R :=fr₁ < ::: < r_pg with p 1 denotes the set of all levels of Fe(x) (or of system (A)).

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We clearly haveR = [J

`=1

R^(`) and r_p =r.

When p = 1, system (A) is said to be with the unique level r. Recall that such a system was already investigated in great details in [12] (case r = 1) and [23] (case r 2). Henceforth, we suppose from now on p 2, i.e., system (A) has at least two levels. Note however that some column- blocks Fe ^;`(x) may have the unique levelr, i.e.,p_` = 1 and R^(`) =frg.

2.2 Multisummability

/ Multisummability of Fe(x). The multisummability of formal power series inC[[x]]was investigated by many authors and several multisummation process based on various methods such as asymptotic, cohomology, integral operators, etc... were built [2–4, 6, 9, 16, 18]. Of course, all these process provide a same and unique multisum (see [11] for instance). In this article, we shall use either of these process depending on our needs.

Notation 2.5 Given a direction 2 R=2 Z and k := (k₁ < ::: < k_s) a s-tuple of positive numbers, we denote by

Cfxg^k; the set of k-summable formal series in direction , s_k; (eh)(x)thek-sum of eh(x)2Cfxg^k; in direction .

Recall thats_k; (eh)(x)de…nes an analytic function1=k₁-Gevrey asymptotic to eh(x)on a germ of sector with vertex0, bisected by and opening larger than

=k_s ¹. In particular, eh(x) is a 1=k₁-Gevrey formal series (denoted below by h(x)e 2 C[[x]]_1=k₁), i.e., its formal Borel transform Be^k1(eh) of level k₁ is analytic at the origin 0 2C. Recall also that, for k := (k), the set Cfxg^k;

coincides with the set Cfxg^k; of classicalk-Borel-Laplace-summable formal series in direction [17]. We also denote by

Cfxg^k the set of k-summable formal series, i.e., the set ofk-summable formal series in all directions but …nitely many.

Note that Cfxg Cfxg^k for any k.

Back to Fe(x), one has the following classical theorem:

1When opening is<2 , the sector can be seen as a sector ofCnf0g; otherwise, it must be considered as a sector of the Riemann surfaceCe:=Cnf^0g of the logarithm.

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Theorem 2.6 ([4, 6, 9, 16, 18]) 1. Multisummability of Fe(x).

Let 2R=2 Z be a non anti-Stokes direction of Fe(x).

Let r := (r1 < ::: < rp) be the p-tuple of all the levels of Fe(x).

Then, Fe(x)2Cfxg^r; . 2. Multisummability of Fe ^;`(x).

Let ^(`) 2R=2 Z a non anti-Stokes direction of Fe ^;`(x).

Let r^(`) := (r^(`)₁ < ::: < rp^(`)_`) the p_`-tuple of all the levels of Fe ^;`(x).

Then, Fe ^;`(x)2Cfxgr^(`); ^(`).

/ Factorization theorem. The factorization theorem 2.7 below tells us that Fe(x) can be written essentially uniquely as a product of r_k-summable formal series Fe_r_k(x)for the di¤erent levelsr_k ofFe(x). It was …rst proved by J.–P. Ramis in [19,20] by using a technical way based on Gevrey estimates. A quite di¤erent proof based on Stokes cocycles and mainly algebraic was given later by M. Loday–Richaud in [9]. Both proofs are nonconstructive. However, as we shall see in section 3, this theorem provides su¢ cient informations to allow us to investigate the resurgence and the singularities of the Borel transforms of the r_k-reduced series of Fe(x).

Theorem 2.7 (Factorization theorem, [9, 19, 20])

Let R=fr₁ < r₂ < ::: < r_p =rg denote the set of levels of Fe(x) ².

Then, Fe(x) can be factored in Fe(x) = Fe_r_p(x):::Fe_r₂(x)Fe_r₁(x) where, for all k = 1; :::; p,Fe_r_k(x)2M_n(C[[x]])is ar_k-summable formal series with singular directions the anti-Stokes directions of level r_k of Fe(x).

This factorization is essentially unique: let Fe(x) =Ge_r_p(x):::Ge_r₂(x)Ge_r₁(x) be another decomposition of Fe(x); then, there exist p 1 invertible matrices P_r₁(x); :::; P_r_p ₁(x) 2 GL_n(Cfxg[x ¹]) with meromorphic entries at 0 such thatGe_r₁ =P_r₁Fe_r₁, Ge_r_k =P_r_kFe_r_kP_r ¹

k 1 for k= 2; :::; p 1and Ge_r_p =Fe_r_pP_r_p¹₁. In particular, we can always choose Fe_r_k so that Fe_r_k(x) = I_n+O(x^r) for all k = 1; :::; p ³.

Notation 2.8 Given a level 2 R of Fe(x), we denote by

2Recall that we supposep 2 in this paper.

3Actually, such conditions, like the initial conditionFe(x) =In+O(x^r), allow us to have

“good” normalizations for the rk-reduced series and thus to simplify future calculations (see sections 3.3 and 3.4 below).

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:= (r₁ < ::: < ) the tuple of levels of R which are ,

+1 the level of R immediately greater than when < r,

+ := ( ⁺¹ < ::: < r) the tuple of levels of R which are > , with the convention ⁺ = +1 when =r,

Fe (x) the sub-product ofFe(x)de…ned by Fe (x) :=Fe(x):::Fe_r₁(x), e

F ⁺(x) the sub-product of Fe(x) de…ned by Fe⁺(x) := Fe_r(x):::Fe⁺¹(x) with the convention Fe⁺(x) =Fe₊₁(x) =I_n when =r.

Note that, following [18, Lem. 7], Fe (x)2Cfxg ^; and Fe⁺(x)2Cfxg ⁺^; for any non anti-Stokes direction of Fe(x).

Let us now consider the matrix

A (x) :=Fe⁺(x) ¹A(x)Fe⁺(x) x^r+1Fe⁺(x) ¹dFe⁺ dx (x)

of the system obtained from system (A) by the formal gauge transformation Y 7! Fe⁺(x)Y. Then [9], A (x) is analytic at 0 and the matrix Ye (x) :=

Fe (x)x^Le^Q(1=x) is a formal fundamental solution of system x^r+1dY

dx =A (x)Y: (A )

Note that system (A ) and matrix Ye (x) coincide with system (A) and matrix Ye(x) when = r. Note also that all systems (A ) have same levels r₁ < r₂ < ::: < r_p as system (A) and that all matrices Ye (x) have same normalizations as Ye(x).

When < r, the structure of A (x) will be precised in theorem 3.6 below. In particular, we shall show that A (x) (and, consequently, Fe (x)) can always be chosen with a convenient “block-diagonal form”.

3 Main results

Since any of the J column-blocks Fe ^;`(x) can be positionned at the …rst place by means of a convenient permutation P on the columns of Ye(x) and since this same permutation acting on the rows of Ye(x)allows to keep initial normalizations ofYe(x)⁴, we can restrict ourselves, without loss of generality,

4The new formal fundamental solution reads PYe(x)P =PFe(x)P x^P ¹^LPe^P ¹^Q(1=x)P withPFe(x)P =I_n+O(x^r).

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to the study of the …rst column-blockFe ^;1(x)which we denote below byfe(x).

Note that the size of f(x)e is n n₁. Note also that f(x) =e I_n;n₁ +O(x^r), where I_n;n₁ denotes the …rstn₁ columns of the identity matrixI_n.

3.1 Setting the problem

In addition to normalizations (N₁) (N₃)of Ye(x), we suppose that (N₄) ₁ = 0 and q₁ 0,

conditions that can always be ful…lled by means of the change of unknown vector Y =x ¹e^q¹^(1=x)Z. Doing that, the levels r⁽¹⁾₁ < ::: < r⁽¹⁾p1 of fe(x) (see de…nition 2.1 and notation 2.2) are the degrees of nonzero polynomials q_j of Q. To simplify notations, we denote them below by 1 < ::: < p1. Recall that p1 =r the highest level ofFe(x).

Notation 3.1 To simplify calculations below, we suppose from now on that matrix Q reads on the form

Q=Q₁ ::: Q_p₁

where

Q1 is a diagonal matrix whose entries are all the polynomials qj of degree 1, i.e., all the polynomials q_j 0 (in particular,q₁) and all the polynomials q_j of degree 1,

for all k 2,Q_k is a diagonal matrix whose entries are all the polynomials q_j of degree k and whose the leading termQ^k :=x ^kQ_kj^x=0 has a block-decomposition of the form

sk

M

`=1

Q_k;`I_m_k_;` ; Q_k;` 2Cnf0g and Q_k;`6=Q_k;`0 if ` 6=`⁰:

Note that decomposition of Q can always be ful…lled by means of a convenient permutation acting both on the rows and columns with indices n₁+ 1ofYe(x). In particular, such a permutation does not a¤ect normalizations (N₁) (N₄)of Ye(x)or the …rst place of f(x). Note also thate Q=Q₁ when p₁ = 1.

Notation 3.2 Following decomposition ofQ,

we denote by N_k the size of the square matrix Q_k, k = 1; :::; p₁,

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we split matrix L of exponents of formal monodromy like Q:

L=L₁ ::: L_p₁ with L_k 2M_N_k(C):

For k 2 f1; :::; p₁g, we denote by fe^[k;u](t), with u = 0; :::; _k 1, the k- reduced series of fe(x), i.e., the sub-series of terms k by k of f(x). Recalle that these series are uniquely determined by relation

fe(x) = fe^[k;0](x ^k) +xef^[k;1](x ^k) +:::+x ^k ¹fe^[k; ^k ^1](x ^k):

Following proposition 3.3 gives us a …rst property of the formal Borel transforms fb^[k;u]( ) :=Be¹(ef^[k;u])( ) of level 1.

Proposition 3.3 Let 2R=2 Z be a non anti-Stokes direction of fe(x).

Let k2 f1; :::; p₁g and ^[k]:= _k .

Case k = 1. Then, fb^[1;u]( ) is analytic at 0:

fb^[1;u]( )2Cf g.

Case k 2. Then, fb^[k;u]( ) is summable in direction ^[k]: fb^[k;u]( ) 2Cf g ^[k]; ^[k], where ^[k] := ¹

k 1

; :::; ^k ¹

k k 1

.

We denote by fb^[k;u]_[k]( ) the sum thus de…ned by fb^[k;u]( ) in direction ^[k] and by V₀(bf^[k;u]_[k] ) the domain of de…nition of fb^[k;u]_[k]( ) ⁵.

Proof. Since fe(x) is ( ₁; :::; _p₁)-summable in direction with 1 1 (see theorem 2.6), [1] tells us that f(x)e can be split into the form

fe(x) =eg₁(x) +:::+eg_p₁(x) with eg_j(x)2Cfxg j; for all j = 1; :::; p₁. Thereby, denoting byeg^[k;u]_j (t),u= 0; :::; _k 1, the k-reduced series ofeg_j(x), the formal series fe^[k;u](t)reads as

fe^[k;u](t) =eg^[k;u]₁ (t) +:::+eg^[k;u]_p₁ (t) with eg^[k;u]_j (t)2Cftg j= k; ^[k]:

5Precisely,V0(fb^[k;u][k])is a disc centered at0ifk= 1and a sector with vertex0, bisected by ^[k] and opening larger than ( _k _k ₁)= _k ₁ ifk 2 [18].

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Hence, [3, pp. 81 and 101] implies identity

fb^[k;u]( ) =bg^[k;u]₁ ( ) +:::+bg^[k;u]_p₁ ( ) where

b

g^[k;u]_j ( )2 Cf g j=( k j); ^[k] if j= _k<1 Cf g if j= _k 1

In particular, we havebg^[k;u]_k ( )+:::+bg^[k;u]p1 ( )2Cf gand [18, Lem. 7] implies b

g^[k;u]₁ ( ) +:::+bg_k^[k;u]₁( )2Cf g ^[k]; ^[k]. This ends the proof.

The aim of section 3 is to investigate the resurgent character of functions fb^[k;u][k] ( ) and to give a complete description of all their singularities. Note that, since p1 =r is the highest level of Fe(x), the case k =p₁ was already treated in [21]. For other casesk 2 f1; :::; p₁ 1g, we shall see in sections 3.3 and 3.4 that their study can actually be reduced to this case of “highest level”.

To do that, we shall use an approach based on factorization theorem 2.7 and on block-diagonalisation theorem 3.6 below which will allow us, on one hand, to isolate levels k of fe(x) by means of the relation Fe(x) =Fe⁺

k(x)Fe

k(x) and, on the other hand, to write the corresponding matrix A _k(x) into a convenient block-diagonal form. Recall that such an approach was already used in [22] for lowest level 1.

3.2 Block-diagonalisation theorem

In this section, we …x k 2 f1; :::; p₁ 1g. Our aim is to prove that system (A _k) can be written as a convenient direct sum of sub-systems allowing to isolate the levels k of f(x).e

Notation 3.4 Using notations 3.1 and 3.2, we denote by

Q_<d =Q₁ ::: Q_d ₁, Q _d =Q_<d Q_d and Q_>d =Q_d+1 ::: Q_p₁, N_<d =N₁+:::+N_d ₁,N _d =N_<d+N_d and N_>d =N_d+1+:::+N_p₁, L<d =L1 ::: Ld 1, L d =L<d Ld and L>d =Ld+1 ::: Lp1

when sums make sense.

According to notation 3.4 above, matrix Qreads as

Q=Q _k Q_>k (3.1)

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with Q _k (resp. Q_>k) of size N _k (resp. N_>k). Block-diagonalisation theorem 3.6 below, which is an improved version of the one stated in [22, Thm.

3.3], tells us that, up to analytic gauge transformation, system (A _k) can be split into a direct sum of two sub-systems …tting to the block-decomposition (3.1) of Q. In particular, it shows that matrix A _k(x) can be reduced into a block-diagonal form A _k(x) = A⁰

k(x) A⁰⁰

k(x).

Theorem 3.6 stems from following technical lemma 3.5 which is based on the results of B. Malgrange proved in [13] and on Tauberian theorems due to J. Martinet and J.-P. Ramis [18].

Before stating this lemma, let us recall that a (formal) meromorphic gauge transformation Z =T(x)W transforms any system

x^r+1dW

dx =A(x)W into the system

x^r+1dZ

dx = ^TA(x)Z where ^TA(x) = TA(x)T ¹+x^r+1dT dxT ¹: Lemma 3.5 Let d2 fk+ 1; :::; p1g. Let a system

x ^d⁺¹dW

dx =A(x)W ; A(x)2M_N _d(Cfxg) (3.2) together with a formal fundamental solution at 0 of the form

fW(x) =H(x)xe ^L ^de^Q ^d^(1=x)

where H(x)e 2M_N _d(C[[x]]) satis…es H(x) =e I_N _d+O(x^r).

Suppose that H(x)e is summable of levels k.

Then, there exists an invertible matrix T(x) 2 GL_N _d(Cfxg) with analytic entries at 0 such that

1. T(x) =I_N _d+O(x^r),

2. the gauge transformation Z = T(x)W transforms system (3.2) into a system

x ^d⁺¹dZ

dx = A⁰(x) 0

0 A⁰⁰(x) Z (3.3)

with A⁰(x)2M_N_<d(Cfxg) and A⁰⁰(x)2M_N_d(Cfxg),

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3. the formal fundamental solutionZ(x) =e T(x)fW(x)of system (3.3) has a block-diagonal decomposition

Ze(x) = He⁰(x)x^L^<de^Q^<d^(1=x) He⁰⁰(x)x^L^de^Q^d^(1=x)

where

(a) He⁰(x) and He⁰⁰(x) satisfy He⁰(x) =He⁰⁰(x) =I +O(x^r),

(b) He⁰(x)x^L^<de^Q^<d^(1=x) is a formal fundamental solution of system x ^d ¹⁺¹dZ

dx =A⁰(x)Z; (3.4)

(c) He⁰⁰(x)x^L^de^Q^d^(1=x) is a formal fundamental solution of system x ^d⁺¹dZ

dx =A⁰⁰(x)Z:

Moreover, both formal series He⁰(x) and He⁰⁰(x) are summable of levels k. Proof. /Since H(0) =e I_N _d, the matrix A(x) of system (3.2) reads

A(x) =x ^d⁺¹dQ _d

dx +x ^dB(x)

with B(x) analytic at 0. Hence, according to the block-decomposition of matrix Q (see notation 3.1), the heading term A(0) = 0_N_<d ( _dQ^d) of A(x) reads

A(0) = 0_N_<d

sd

M

`=1

dQ_d;`I_m_d;`

!

with Q_d;` 6= 0 and Q_d;` 6= Q_d;`0 if ` 6= `⁰. Thereby, applying [13, Thm. 1.5], there exists an invertible matrix T1(x) 2 GLN _d(C[[x]]₁₌ _d[x ¹]) with meromorphic 1= _d-Gevrey entries at 0 such that the matrix ^T¹A(x) has a block- decomposition like A(0). Note that the entries of ^T¹A(x) are generally meromorphic 1= d-Gevrey and not convergent. Denote by A^(`)(x), ` = 0; :::; sd, the blocks of ^T¹A(x). By construction, all the sub-systems

x ^d⁺¹dW

dx =A^(`)(x)W ; `= 0; :::; s_d;

have levels < _d. Then, [13, Thm. 1.4] applies and, consequently, there exists, for all ` = 0; :::; s_d, an invertible matrix T₂^(`)(x) with meromorphic

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1= _d-Gevrey entries at 0 such that the matrix ^T²^(`)A^(`)(x) has meromorphic entries at 0. Finally, normalizing the formal fundamental solutions of these last systems by means of convenient polynomial gauge transformations in x and 1=x if needed, calculations above tell us that there exists a matrix T(x) 2 GL_N _d(C[[x]]₁₌ _d[x ¹]) satisfying points 2 3. of lemma 3.5. Note that point 1 results from equalities

T(x)H(x) =e He⁰(x) He⁰⁰(x) = IN _d +O(x^r) (3.5) and from assumption H(x) =e I_N _d +O(x^r).

/ We are left to prove that T(x) is analytic at 0 and that formal series e

H⁰(x) and He⁰⁰(x) are both summable of levels k. According to construction above, we already known that He⁰(x) and He⁰⁰(x) are both summable of levels< _d. Then, the …rst equality of (3.5) and hypothesis “H(x)e summable of levels k”tell us that T(x) is actually both1= _d-Gevrey and summable of levels < _d (indeed, k < _d for all d = k + 1; :::; p₁). Hence, applying Tauberian theorem [18, Prop. 7, p. 349], T(x) is analytic at 0. As a result, T(x)H(x)e is still summable of levels k and, consequently,He⁰(x) and

e

H⁰⁰(x) are both summable of levels k too. This ends the proof of lemma 3.5.

Note that the hypothesis “H(x)e is summable of levels k” is crucial in the proof of lemma 3.5: without it, we can not prove the analyticity of T(x). Note also that lemma 3.5 can be again applied to sub-system (3.4) when d k+ 2... and so on as long as d6=k+ 1.

In the case of system (A _k), an iterative application of lemma 3.5 starting with d=p₁ allows us to state the following result:

Theorem 3.6 (Block-diagonalisation theorem) There exists an invertible matrix T_k(x)2GL_n(Cfxg) with analytic entries at 0 such that

1. T_k(x) =I_n+O(x^r),

2. the gauge transformation Z =T_k(x)Y transforms system (A _k) into a system

x^r+1dZ

dx = ^T^kA _k(x)Z (^T^kA _k) where the matrix ^T^kA _k(x)2M_n(Cfxg) has a block-diagonal decomposition like block-decomposition (3.1) of Q:

TkA _k(x) =A⁰

k(x) A⁰⁰

k(x)

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with A⁰

k(x)2M_N _k(Cfxg) and A⁰⁰

k(x)2M_N_>k(Cfxg), 3. the formal fundamental solutionZe

k(x) = T_k(x)Ye

k(x)of system(^T^kA _k) has a block-diagonal decomposition

e Z

k(x) =Fe⁰

k

(x)x^L ^ke^Q ^k^(1=x) Fe⁰⁰

k

(x)x^L^>ke^Q^>k^(1=x)

where (a) Fe⁰

k

(x) and Fe⁰⁰

k

(x) satisfyFe⁰

k

(x) = Fe⁰⁰

k

(x) = I +O(x^r), (b) the matrixYe⁰

k

(x):= Fe⁰

k

(x)x^L ^ke^Q ^k^(1=x) is a formal fundamental solution of system

x ^k⁺¹dZ dx =A⁰

k(x)Z; (A⁰_k)

(c) the matrixYe⁰⁰

k

(x):= Fe⁰⁰

k

(x)x^L^>ke^Q^>k^(1=x) is a formal fundamental solution of system

x^r+1dZ dx =A⁰⁰

k(x)Z: (A⁰⁰_k) In particular, the matrix T_k(x)Fe

k(x) has the block-decomposition T_k(x)Fe

k(x) = Fe⁰

k

(x) Fe⁰⁰

k

(x)

where Fe⁰

k

(x) and Fe⁰⁰

k

(x) are both _k-summable.

Remark 3.7 According to the analyticity of T_k(x)and the “unicity”of factorization theorem 2.7, block-diagonalisation theorem 3.6 tells us that we can always choose as matrix Fe

k(x)the matrixT_k(x)Fe

k(x) and as system(A _k) the system (^T^kA _k). This we do from now on.

Note that one of the interests of the choice of system (^T^kA _k) for system (A _k) is that its sub-system (A⁰_k) “contains” all the levels k of f(x)e and has k as highest level.

Note also that block-diagonalisation theorem 3.6 and remark 3.7 above can be extended to the highest level p1 =r offe(x)by settingN_>p₁ = 0 and T_p₁(x) =I_n. Doing that, we clearly have

Fe(x) =Fe_r (x) =T_p₁(x)Fe_r (x) =Fe_r⁰ (x) and systems (A), (Ar), (^T^p¹A_r) and (A⁰_r) coincide.

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3.3 Resurgence

In this section, we shall investigate the resurgent character of functions fb^[k;u]_[k] ( ) given in proposition 3.3. In particular, we shall prove a resurgence theorem which generalizes resurgence theorems stated by M. Loday-Richaud and the author in [12, 23] for systems with single-level and in [21, 22] for lowest and highest levels of systems with multi-levels.

3.3.1 Resurgence theorem

Recall that a resurgent function is an analytic function near the origin which can be analytically continued on all a convenient Riemann surface. More precisely, one has the following.

De…nition 3.8 (Resurgent function) Let C be a …nite subset of C containing 0. A function de…ned and analytic near 0 is said to be

resurgent with singular support ;0 when it can be analytically continued on all the Riemann surfaceR de…ned as (the terminal end of) all homotopy classes in Cn of paths issuing from0 and bypassing all points of (only homotopically trivial paths are allowed to turn back to 0); in particular, such a function is analytic at 0 in the …rst sheet, resurgent with singular support ;e0when it can be analytically continued on all the Riemann surface Re :=the universal cover of Cn . We denote byRes _;0 andRes _;e₀ the sets of resurgent functions with singular support ;0and of resurgent functions with singular support ;e0.

Recall that the di¤erence betweenR andRe just lies in the fact thatR has no branch point at 0in the …rst sheet. In particular, we have a natural injection Res _;0 ,! Res _;e₀. Recall also that the choice of the Riemann surface Re orR only depends on the fact that the function we consider has a singular point at 0or not.

De…nition 3.9 (Resurgent function with exponential growth) Given

>0, a resurgent function of Res _;0 (resp. Res _;e₀) is said to be with exponential growth of order if it grows at most exponentially with an order

on any bounded sector of in…nity of R (resp. Re ).

We denote by Res _;0 (resp. Res _;e

0) the set of resurgent functions of Res _;0 (resp. Res _;e₀) with exponential growth of order . As before, we have a natural injection Res _;0 ,! Res _;e

0.

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When = 1, any function ofRes _;0(resp. Res _;e

0) is said to besummable- resurgent with singular support ;0 (resp. ;e0). Following notations of [12, 21–23], we denote Res^sum_;0 (resp. Res^sum_;e₀ ) for Res ¹_;0 (resp. Res ¹_;e

0).

We are now able to state the main result of this section.

Theorem 3.10 (Resurgence theorem) Let k2 f1; :::; p₁g.

Let 2R=2 Z be a non anti-Stokes direction of f(x)e and ^[k] := _k . Let _k be the set of Stokes values of level k of f(x)e (see de…nition 2.1) and

k :=

k[ f0g.

Let 1; :::; _p₁ >0 be the positive numbers de…ned by

j :=

+ j +

j j

for j = 1; :::; p₁ 1 and p1 := 1.

Case k = 1. Then, for all u= 0; :::; ₁ 1, fb^[1;u][1]( ) 2 Res ¹

1;0. Case k 2. Then, for all u= 0; :::; _k 1,

fb^[k;u]_[k] ( ) 2 Res ^k

k;e0.

Remark 3.11 When fe(x) has the unique level r (i.e., p1 = 1 and so 1 =

p1 =r), we …nd again, of course, the resurgence theorem already stated by M. Loday-Richaud and the author in [12] and [23], namely

fb^[1;u][1]( ) 2 Res^sum

1;0 for all u= 0; :::; ₁ 1.

The proof of theorem 3.10 is developed in section 3.3.2 below. It is based on factorization theorem 2.7, block-diagonalisation theorem 3.6 and on the results of [21].

3.3.2 Proof of theorem 3.10

/A fundamental identity. Letk 2 f1; :::; p₁g. According to factorization theorem 2.7, block-diagonalisation theorem 3.6 and remark 3.7, the formal series fe(x)can be written on the form

f(x) =e Fe⁺

k(x)fe

k(x) with fe

k(x) =

"

fe⁰

k

(x) 0_N_>k _n₁

#

(3.6) where

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e F ⁺

k(x)2M_n(C[[x]])is a ⁺_k-summable formal series satisfyingFe⁺

k(x) = I_n+O(x^r)when k < p₁ and Fe⁺

k(x) =I_n when k =p₁, fe⁰

k

(x) denotes the …rst n₁ columns of Fe⁰

k

(x)2M_N _k(C[[x]]), 0_N_>k _n₁ denotes the null-matrix of size N_>k n₁.

Note that fe

k(x) =fe⁰

k

(x) when k =p₁. As before, we denote by Fe^[k;u]+

k

(t) and fe⁰^[k;u]

k

(t) with u 2 f0; :::; _k 1g the k-reduced series of Fe⁺

k(x) and of fe

k(x). We also denote by fe^[k](t) =

2 64

fe^[k;0](t) ... fe^[k; ^k ^1](t)

3

752 M _kn;n1(C[[t]]) the matrix formed by the k-

reduced series of f(x),e

fe^[k;u]

k

(t) =

"

fe⁰^[k;u]

k

(t) 0_N_>k _n₁

#

and fe^[k]

k

(t) = 2 66 4

fe^[k;0]

k

(t) ... fe^[k; ^k ^1]

k

(t) 3 77 5.

Then, relation (3.6) above implies relation fe^[k](t) = Fe^[k]+ k

(t)ef^[k]

k

(t) where

Fe^[k]+ k

(t) :=

2 66 66 66 66 64

Fe^[k;0]+ k

(t) tFe^[k;+ ^k ^1]

k

(t) tFe^[k;1]+

k

(t) Fe^[k;1]+

k

(t) Fe^[k;0]+ k

(t) . .. ...

... . .. . .. . .. ...

... . .. Fe^[k;0]+

k

(t) tFe^[k;+ ^k ^1]

k

(t) Fe^[k;+ ^k ^1]

k

(t) Fe^[k;1]+

k

(t) Fe^[k;0]+ k

(t) 3 77 77 77 77 75

is a

+1 k

k

; :::; r

k

-summable formal series satisfying Fe^[k]+ k

(t) = I _k_n +O(t)

when k < p₁ and where Fe^[k]+ k

(t) = I_rn when k = p₁. In particular, applying [3, p. 81], its formal Borel transform Fb^[k]+

k

( ) reads as Fb^[k]+

k

( ) = I _k_n+Gb_k( ) whenk < p₁ I_rn whenk =p₁ ;

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where Gbk( ) de…nes an entire function on all C with exponential growth of order k = ⁺¹_k =( ⁺¹_k _k) at in…nity. Indeed, ⁺¹_k = _k > 1. This brings then us to the following lemma:

Lemma 3.12 Letk 2 f1; :::; p₁g. Then, the formal Borel transformsfb^[k;u]( ) of fe^[k;u](t) and the formal Borel transforms fb⁰^[k;u]

k

( ) of fe⁰^[k;u]

k

(t) are related, for all u= 0; :::; _k 1, by relation

fb^[k;u]=

"

fb⁰^[k;u]

k

0_N_>k _n₁

#

+E_k;u

"

fb⁰^[k;u]

k

0_N_>k _n₁

#

(3.7) where Ek;u is a convenient entire function on all C with exponential growth of order k at in…nity when k < p₁ and where E_k;u 0 whenk =p₁.

/ Resurgence of fb⁰^[k;u]

k

( ). By construction (see block-diagonalisation theorem 3.6 and remark 3.7), the matrix Fe⁰

k

(x)x^L ^ke^Q ^k^(1=x) is a formal fundamental solution of a system of the form

x ^k⁺¹dY dx =A⁰

k(x)Y (A⁰_k)

with a convenient matrix A⁰

k(x) 2 M_N _k(Cfxg) satisfying A⁰

k(0) 6= 0. In particular, one can easily check the following points:

system (A⁰_k) has k as highest level, the levels of fe⁰

k

(x) are the levels k of fe(x), namely 1 < ::: < _k, the Stokes values (hence, the anti-Stokes directions) of level` 2 f ¹; :::; _kg of fe⁰

k

(x) and fe(x)coincide.

Thereby, choosing a direction 2R=2 Z as in theorem 3.10, it is clear that 1. is not an anti-Stokes direction offe⁰

k

(x), 2. the fb⁰^[k;u]

k

( )’s are, as the fb^[k;u]( )’s, analytic at 0 if k = 1 and ^[k]- summable in direction ^[k]= _k if k 2(see proposition 3.3).

Hence, denoting as before by fb⁰^[k;u]

k; ^[k]( ) the sum thus de…ned and applying [21], we have the following.

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Proposition 3.13 ([21, Thm. 4.9]) Let k2 f1; :::; p₁g. Let 2R=2 Z and _k as in theorem 3.10.

Case p₁ = 1. Then, for all u= 0; :::; ₁ 1:

fb⁰^[1;u]

1; ^[1]( )2 Res^sum

1;0:

Case p₁ 2. Then, for all u= 0; :::; _k 1:

fb⁰^[k;u]

k; ^[k]( )2 Res^sum

k;e0:

We are now able to end the proof of theorem 3.10.

/ Conclusion. According to lemma 3.12, functions fb^[k;u]_[k] and fb⁰^[k;u]

k; ^[k] are de…ned on the same domain V₀(bf^[k;u]_[k]) (see proposition 3.3) and are related by relation

fb^[k;u]_[k] =

"

fb⁰^[k;u]

k; ^[k]

0_N_>k _n₁

#

+E_k;u

"

fb⁰^[k;u]

k; ^[k]

0_N_>k _n₁

#

: (3.8)

Theorem 3.10 follows then from proposition 3.13 and from the fact that the exponential growth k of Eu at in…nity is greater than 1when k < p₁. This ends the proof.

3.4 Singularities

Resurgence theorem 3.10 above tells us in particular that the only possible singular points of fb^[k;u]_[k] ( ) are 0 and the Stokes values ! 2 k of level k

of fe(x). In this section, we propose to give a complete description of all the singularities of the fb^[k;u][k] ( ) at the various Stokes values of _k. Before starting the calculations, let us recall some de…nitions and notations about singularities. For more precise details, we refer to [7, 14, 24].

3.4.1 Some spaces of singularities

Denote by O the space of holomorphic germs at 0 2 C and by Oe the space of holomorphic germs at 0 on the Riemann surface Ce of the logarithm. One calls any element of the quotient space C :=Oe=O a singularity at 0. Recall that C is also denoted by SING₀ by J. Écalle and al. (cf. [24] for instance).

Recall also that the elements ofC are calledmicro-functions by B. Malgrange

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[14, 15] by analogy with hyper- and micro-functions de…ned by Sato, Kawai and Kashiwara in higher dimensions.

The elements of C are usually denoted with a nabla, like ', for a singu-^r larity of the function '. A representative of '^r in Oe is often denoted by ' b and is called a major of '.

It is worth to consider the two natural maps

can :O ! Ce =Oe=O the canonical map and var :C !Oe the variation map, action of a positive turn around 0 de…ned by var'^r = '

b

( ) ' b

( e ²ⁱ ), where '

b

( e ²ⁱ ) is the analytic continuation of ' b

( ) along a path turning once clockwise around 0 and close enough to 0 for '

b

to be de…ned all along (the result is independent of the choice of the major '

b

). The germ var'^r is called the minor of '.^r

One can not multiply two elements of C, but an element of C and an element of O: '^r := can( '

b

) = ^r'for all 2 O and '^r 2 C.

On the other hand, one can de…ned a convolution product ~ on C by setting'^r₁~'^r₂ := can('

b

1 u'

b

2), where' b

1 u'

b

2 is the truncated convolution product

(' b

1 u'

b

2)( ) :=

Z u

u

' b

1( )'

b

2( )d 2Oe

with u arbitrarily close to0 satisfying 2]0; u[and arg( u) = arg( ) . Note that '^r1 ~'^r2 makes sense since it does not depend on u, nor on the choice of the majors '

b

1 and '

b

2. The convolution product ~ is commutative and associative on C with unit := can _2i¹ .

In the sequel of this article, we shall use especially the following sub- spaces of C :

/ The subspace C ¹ of singularities for which the variation de…nes an entire function on allCe with exponential growth of order 1on any bounded sector of in…nity. Recall that this space is isomorphic, via the Borel-Laplace transformation, to the space of analytic functions with subexponential growth at 0 2 Ce [7, pp. 46-48]; in particular, any power t with 2 C and any exponential e^P^(t^1=p⁾ with p 2andP(t) polynomial int of degree< p de…ne singularities in C ¹.

/ The subspace N^ril^res;_;0 (resp. D^ret^res;_;e

0 ) of resurgent singularities of Nilsson class (resp. of …nite determination)with singular support ;0(resp.

;e0) and exponential growth of order at in…nity ( denotes a positive

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number and a …nite subset of C containing0). Recall that these singularities are the singularities of C for which the variation reads on the form

X

…nite

' _;p( ) (ln )^p

with 2C,p2Nand' _;p( )2 Res _;0 (resp. ' _;p( ) 2 Res _;e

0 holomorphic on a punctured disc at0). When = 1, such singularities are saidsummable- resurgent and we simply denoteN^ril^{s res}_;0 (resp. D^ret^{s res}_;e

0 ) for N^ril^{res; 1}_;0 (resp.

r

Det^{res; 1}_;e

0 ).

For any! 2C , we denote byCj! the space of singularities at !,i.e., the space C translated from 0to!. A function'

b

is then a major of a singularity at ! if '

b

(!+ ) is a major of a singularity at 0. In the same way, we de…ne the translated space C ¹j!, etc...

3.4.2 Description of singularities

Letk 2 f1; :::; p₁gand u2 f0; :::; _k 1g. The behavior offb^[k;u]_[k] ( )at any of its singular points ! 2 k depends, of course, on the sheet of the Riemann surface where we are, i.e., it depends on the “homotopic class” of the path of analytic continuation followed from any pointa 6= 0 of V₀(bf^[k;u]_[k] ) ⁶ to a neighborhood of !. Note in particular that “homotopic class” implies that the behavior of fb^[k;u][k] ( ) does not depend on the choice of a.

We denote below by f^r^[k;u]_[k]

;!; the singularity of fb^[k;u]_[k] ( ) de…ned by the analytic continuation offb^[k;u]_[k] ( )along the path . Before starting the calculations, let us …rst introduce the key notion offront of a singularity [21–23].

/ Front of a singularity. Let !2 k. We callfront of level k of ! the set of all the polynomials q_j(1=x) of Q(1=x) with leading term !=x ^k. We denote it by F r _k(!) and we have

F r _k(!) := !

x ^k +q_!;k;` 1

x ; `= 1; :::; s_k

where s_k is an integer 1 and where all the q_!;k;`(1=x) are polynomials in 1=x with degree< _k and without constant term.

6See proposition 3.3 for the exact de…nition ofV₀(fb^[k;u][k] ).