Turrittin's normal forms for linear systems of meromorphic ODEs over the real field

Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 79, pp. 1–23. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.79 TURRITTIN’S NORMAL FORMS FOR LINEAR SYSTEMS OF MEROMORPHIC ODES OVER THE REAL FIELD MOULAY BARKATOU, F´ ELIX A. CARNICERO, FERNANDO SANZS Abstract. We establish a version of Turrittin’s result on normal forms of linear systems of meromorphic ODEs when the base field Kis real and closed. Both the proposed normal forms and the transformations used have coefficients in K. Our motivation comes from applications to the study of trajectories of real analytic vector fields (already treated in the literature in dimension three). For the sake of clarity and completeness, we first review Turrittin’s theorem in the case of an algebraically closed base field. 1. Preliminaries Let Kbe a field of characteristic zero and let LK=K[[x]][x−1] be the field of formal meromorphic series with coefficients in K, endowed with the usual derivation with respect to x(denoted only by a prime), and the usual valuation ν:LK→ Z∪{∞} defined as the minimum of the support of the series, also called the order. As a matter of notation, if Ris any ring and n∈N≥1,Mn(R) denotes the ring of square matrices of size nwith entries in R. A matrix A∈ Mn(LK) is identified with the formal meromorphic linear system of ODEs [A]Y0=AY, where Y= (Y1, . . . , Yn)tis a column vector of nvariables. Define the order of A to be ν(A) := min{ν(aij):1≤i, j ≤n}, where A= (aij). Sometimes we use the notation A=A(x) to make explicit that we are dealing with meromorphic series in the variable x. Correspondingly, we will usually write the system as a series of matrices in the form A=xν(A)(A0+xA1+. . . ),(1.1) where Ai∈ Mn(K) for all iand A06= 0. Also, if Nis a non-negative integer, the truncated system up to degree Nis defined by JNA:= xν(A)(A0+xA1+···+xNAN). The system Ais called singular (at x= 0) if ν(A)<0. The Poincar´e rank of the system is defined as the non-negative integer q=q(A) := max{−ν(A)−1,0}. We 2020 Mathematics Subject Classification. 34C20, 34C08, 34M03, 34M25. Key words and phrases. Linear systems of meromorphic ODE; formal normal form; Turrittin’s theorem. ©2023. This work is licensed under a CC BY 4.0 license. Submitted July 4, 2023. Published November 27, 2023. 1 2 M. BARKATOU, F. A. CARNICERO, F. SANZS EJDE-2023/79 usually rewrite Aas the system of formal linear ODEs xq+1Y0=e AY, where e A=A0+xA1+. . . . A singular system with Poincar´e rank q= 0 (resp. q > 0) is usually referred to be of first kind (resp. of second kind). We denote by Inthe identity matrix of size n. We define the radiality index of Aas the non-negative integer k=k(A) := min({j:Aj6∈ KIn}∪{q}). The truncation Jk−1A=a(x)In, where a(x) is a polynomial with degree at most k−1, is called the radial part of A. We are interested in the problem of getting formal normal forms of a given singular system under transformations of one of the following types: (i) Given P∈GLn(LK), the linear change of variables Y=PZ transforms the system [A] : Y0=AY into the system [B] : Z0=BZ where B=P−1AP −P−1P0. The map ΨP:Mn(LK)→ Mn(LK) sending Ato ΨP[A] := P−1AP − P−1P0is bijective. It is called the gauge transformation associated with P. In particular, if P∈GLn(K) is a constant matrix, then ΨPis the conjugation by P. (ii) Given r∈N≥1, the change of the independent variable x=zrtransforms the system dY dx =A(x)Yinto a system dY dz =B(z)Y, where B(z) = rzr−1A(zr). Re-written with the same letter x, we define the map Rr:Mn(LK)→ Mn(LK) given by Rr[A] := rxr−1A(xr), called the ramification of order r. It is an injective map but not bijective for r > 1. A gauge transformation ΨPwill be called: regular, if ν(P) = 0 and det P(0) 6= 0; polynomial, if each entry of Pbelongs to K[x] (in which case the degree of ΨPis defined as the maximum of the degrees of the entries of P); diagonal monomial, if P= diag(xk1, . . . , xkn) with kj∈N≥0for each j. Notice that if P, Q ∈GLn(LK) then we have ΨP Q = ΨQ◦ΨP. Also, if r, s ∈N≥1then Rrs =Rr◦Rs. In addition, we are interested in polynomial (truncated) normal forms obtained by means of polynomial gauge transformations and ramifications (so that, if the initial system is polynomial or convergent, we preserve this character). The case K=C(or more generally, Kalgebraically closed) is classical and treated with different approaches in the literature (Birkhoff [7], Hukuhara [13, 14], Turrittin [21], Wasow [23], Moser [19], Levelt [15], Balser-Jurkart-Lutz [3], Babbitt- Varadarajan [1], Hsieh-Sibuya [12], Barkatou [4, 5], Barkatou-Pfl¨ugel [6]. The different avatars of the algorithms for obtaining normal forms are commonly referred (as we will do here) by the generic expression Turrittin’s Theorem. In this article, we extend Turrittin’s Theorem to the real case K=R, or more generally to the case where Kis a real closed field (see [8, Ch1] for the definition and basic facts about real closed fields). As far as we know, this case has not been treated yet (except, of course, in the situation of a constant system A∈ Mn(K) for which the usual well known real Jordan canonical form of Awas proposed by Turrittin himself in [22]). We present versions of real (formal and polynomial) normal forms for any system, in such a way EJDE-2023/79 TURRITTIN’S THEOREM 3 that they can be obtained by transformations written in the base field K, without passing through the algebraic closure K=K(√−1). Certain cases of our real Turrittin’s result have already been considered in the study of trajectories of real analytic vector fields around a formal invariant curve, mainly in dimension three [9, 10]. Our motivation was to establish general statements to serve to this study in any dimension, as well as other possible applications where systems of ODEs with real coefficients are involved. 1.1. Complex case. To make precise statements and expose some of the steps that are useful to treat the real case, we propose first a brief revision of the complex case. Despite of its prevalence in the literature, we are led ourselves to sketch the different steps of the corresponding proofs (in section 2), instead of simply addressing the reader to the references. There are additional reasons to justify this revision: - Although there are other proofs (even better ones from the point of view of computational effectiveness, see [5]), maybe the most commonly used reference for the complex case is Wasow’s book [23]. We decided to follow also this last reference here. However, in that proof, the final arguments concerning the induction on the Poincar´e rank is perhaps not sufficiently clarified: it drops as long as we do not need to make a ramification, but it increases after ramifications, an operation which is unavoidable in general. The required modification, even its simplicity, is worth to be made, in any case. - In existing proofs of Turrittin’s Theorem, it is frequently allowed the use of exponential shiftings when the leading matrix has a single eigenvalue. Such transformations have not an algebraic or formal nature and are “strange” to the initial setting of the systems. Although they commute with the whole matrix of the system so that the resulting system has also formal meromorphic coefficients, the exponential shiftings may behave very badly with respect to non-linear terms in general systems. Thus, for applications, it is better to avoid these operations. - The search of precise statements for polynomial normal forms of Turrittin’s result make necessary to enter in some details of the proofs; such statements are not exactly pursued in the common references, mostly devoted to obtain expressions of a fundamental matrix of solutions (cf. Remark 1.5, (b) below). We start by defining the normal forms that we expect to obtain. Definition 1.1 (Turrittin-Ramis-Sibuya form).Let A∈ Mn(LK) be a system with Poincar´e rank q=q(A) and let µ∈N≥0. We say that Ais in Turrittin- Ramis-Sibuya form of degree µ(and of rank q), or in (TRS)q µ-form for short, if it is written as A(x) = x−(q+1) D(x) + xqC+O(xq+µ+1), where D(x) = diag(d1(x), . . . , dn(x)) is a diagonal matrix with polynomial entries dj(x)∈K[x] of degree at most q−1 (equal zero if and only if q= 0) and C∈ Mn(K) is a constant matrix commuting with D(x). In this case, the truncated system Jq(A) = x−(q+1)(D(x) + xqC) is called the principal part of A, while D(x), resp. C, is called the exponential part, resp. the residual matrix. Notice that if the system Ais singular of first kind (that is q= 0) then Ais already in (TRS)0 0-form, with exponential part equal to D(x) = 0 and residual matrix C=A0. On the other hand, if Ais in (TRS)q µ-form for some µand q > 0 then its exponential part D(x) is not zero; in fact, it satisfies D(0) 6= 0. 4 M. BARKATOU, F. A. CARNICERO, F. SANZS EJDE-2023/79 The names Ramis and Sibuya in the definition above come from those authors paper [20], devoted to summability properties of formal solutions of systems of holomorphic ODEs where the linear part is in (TRS)-form of some degree. We have added the name Turrittin by obvious reasons. It is worth to notice that analogous expressions as the (TRS)-forms appear also in the context of germs of biholomorphisms in [16, 17] (where the name of “Ramis-Sibuya form” is used). Definition 1.2. Let C∈ Mn(K) be a constant square matrix with entries in K. We say that Cis non-resonant if for any pair of distinct eigenvalues λ, λ0∈Kof Cwe have λ−λ06∈ Z. Other authors, for instance Balser in his book [2], use the terminology “Chas good spectrum”. Now, Turrittin’s results for the case where K=Kcan be stated in the two following theorems. Theorem 1.3 (Complex polynomial normal form).Suppose that Kis an algebraically closed field and let A∈ Mn(LK)be a singular system with Poincar´e rank equal to q=q(A). (i) There exist some r∈N≥1and finitely many polynomial gauge transformations ϕ1, . . . , ϕm, either regular or diagonal monomial, such that, denoting ψ=ϕm◦···◦ϕ1◦Rr, the transformed system e A=ψ[A]is in (TRS)˜q 0-form, where ˜q=q(e A). Moreover, if B∈ Mn(LK)is another singular system with q(B) = qand JnqA=JnqBthen e B=ψ[B]is also in (TRS)˜q 0-form with the same principal part as e A; i.e, q(e B) = ˜qand J˜qψ[A] = J˜qψ[B]. (ii) Assume that Ais in (TRS)q 0-form and that its residual matrix is nonresonant. Then, for any given µ≥0, there exists a regular polynomial gauge transformation φµ= ΨPµ, where Pµ(0) = In, such that φµ[e A]is in (TRS)˜q µ-form with the same principal part as the original system A. Moreover, the family {Pµ}µcan be chosen such that Pµis of degree at most q+µand satisfying that Jq+µPµ0=Jq+µPµfor any µ0> µ. (iii) Assume that Ais in (TRS)q 0-form. Then there exists a gauge transformation φ, given by a finite composition of regular polynomial or diagonal monomial transformations, such that φ[A]is in (TRS)q 0-form with nonresonant residual matrix (and the same exponential part as A). As a consequence of the theorem above, one obtains the following version of Turrittin’s formal normal forms of complex meromorphic linear ODEs. Theorem 1.4 (Complex formal normal form).Suppose that Kis an algebraically closed field and let A∈ Mn(LK)be a singular system with Poincar´e rank equal to q=q(A). There exists a formal gauge transformation ΨPand a ramification Rrsuch that the transformed system F= (ΨP◦Rr)[A]has Poinar´e rank equal to ˜q=q(F)and can be written in a Formal Normal Form [F]Y0=x−(˜q+1)(D(x) + x˜qC)Y, where D(x)and Csatisfy the requirements in Definition 1.1; i.e., Fis in (TRS)˜q 0- form and J˜qF=F. Moreover, the transformation ΨPcan be chosen to be equal to ΨP= ΨQ◦ψ, where Q∈ Mn(K[[x]]) with Q(0) = Inand ψis a finite composition of regular polynomial or diagonal monomial gauge transformations. EJDE-2023/79 TURRITTIN’S THEOREM 5 Remark 1.5. Concerning the statements in Theorems 1.3 and 1.4, we have the following statements. (a) The sufficient truncation order nq in the second sentence of item (i) is already obtained for instance by Babbit-Varadarajan [1] or Lutz-Sch¨afke [18]. Below, we propose a proof with the slightly improved order N:= n(q−k) + k, where kis the radial index of the initial system A. (b) To obtain the formal normal form [F] for the system Ain Theorem 1.4 is equivalent to say that there exists a matrix P(t)∈ Mn(K[[t]]) and some r∈N≥1such that Z(x) = P(x1/r) exp ZD(x1/r) x(˜q+1))/r xC/r (1.2) is a fundamental matrix of formal solutions of the system A. (c) Another consequence of the expression (1.2) is that the ratio ˜q/r and the exponential part D(x), modulo ramification of x, are both invariant under formal meromorphic gauge transformations. More precisely, if we have two systems Aand Bsuch that B= ΨT[A] with T∈GLn(LK) and we obtain (TRS)-normal forms of Aand Bas in item (i) of Theorem 1.3 with resulting Poincar´e ranks ˜qAand ˜qBand exponential parts DA(x) and DB(x), respectively, then there are integers r1, r2such that ˜qA/r1= ˜qB/r2 and DA(x1/r1) = DB(x1/r2). In particular, ˜q= 0 iff the system Ais equivalent to a system of first kind under a formal gauge transformation. (d) In the proof proposed below, one could see that the sequence of gauge transformations used in items (i) or (iii) can be chosen so that any one of them, individually, do not increase the Poincar´e rank of the system it applies to in the process. As we know, this observation only concerns the diagonal monomial gauge transformations, since a regular gauge transformations always preserves the Poincar´e rank. (e) Below, we propose a bound, in terms of the eigenvalues of the residual matrix C, for the degree of the polynomial gauge transformation φin item (iii). 1.2. Real case. Suppose that Kis a real closed field, i.e., K K(i) = K, where i=√−1. Given λ=a+bi ∈K, denote by Λλ=a−b b a . Recall that the characteristic polynomial of Λλhas roots a±bi and is irreducible if and only if λ6∈ K, i.e., b6= 0. For any m∈N≥1, define the monomorphism of K-algebras Θm:Mm(K)→ M2m(K), sending a matrix C= (cuv)∈ Mm(K) to the (2 ×2)-block matrix (Λcuv )∈ M2m(K). A square matrix in the image of Θmwill be called a complex matrix over K, or a C-matrix, for short. We extend Θmto a monomorphism of K-algebras, denoted with the same letter, from Mm(LK) into M2m(LK); that is, from formal meromorphic linear systems over Kto formal meromorphic linear systems over Kof double dimension. A system in the image of this map will be called a complex system (over K)or a C-system. 6 M. BARKATOU, F. A. CARNICERO, F. SANZS EJDE-2023/79 In what follows, if U, V are two square matrices of sizes k, l, respectively, we denote by U⊕Vthe square matrix of size k+lgiven in blocks U⊕V:= U0 0V. Definition 1.6 (Real Turrittin-Ramis-Sibuya form).Suppose that Kis a real and closed field. Let A∈ Mn(LK) be a system with Poincar´e rank q=q(A) and let µ∈N≥0. We say that Ais in Real Turrittin-Ramis-Sibuya form of degree µ(and of rank q), or in (RTRS)q µ-form for short, if it can be written in the form A(x) = x−(q+1) D1(x)⊕D2(x) + xq(C1⊕C2) + O(xq+µ+1), where •D1(x) = diag(e1(x), . . . , en1(x)) is diagonal polynomial with entries ej(x)∈ K[x] of degree at most q−1 (equal to zero if q= 0). •D2(x)=Θn2(diag(d1(x), . . . , dn2(x))) is a diagonal 2 ×2-block complex matrix such that the entries dj(x) belong to K[x]\K[x] and are of degree at most q−1 (equal to zero if q= 0). •C1and C2are constant matrices with entries in Kof sizes n1and 2n2, respectively, and C2is a C-matrix. •[D1(x), C1] = 0 and [D2(x), C2] = 0. In this case, the truncated system Jq(A) = x−(q+1)(D1(x)⊕D2(x) + xq(C1⊕C2)) is called the principal part of A, while D1(x)⊕D2(x), resp. C1⊕C2, is called the exponential part, resp. the residual matrix. Our main result in the paper is the following real version of Theorems 1.3 and 1.4. Theorem 1.7 (Real polynomial normal forms).Suppose that Kis a real closed field and let A∈ Mn(LK)be a singular system with Poincar´e rank equal to q=q(A). (i) There exists r∈N≥1and there exists finitely many polynomial gauge transformations ϕ1, . . . , ϕm(with coefficients in K), either regular or diagonal monomial, such that, denoting ψ=ϕm◦···◦ϕ1◦Rr, the transformed system e A=ψ[A]is in (RTRS)˜q 0-form. Moreover, if B∈ Mn(LK)is another singular system with q(B) = qand JnqA=JnqBthen, with the same transformation ψ, the transformed system e B=ψ[B]is also in (RTRS)˜q 0with the same principal part as e A, i.e., J˜qψ[A] = J˜qψ[B]. (ii) Assume that Ais in (RTRS)q 0-form and that its residual matrix is nonresonant. Then, for any µ≥0there exists a regular polynomial gauge transformation φµ= ΨPµwhere Pµ(0) = Insuch that φµ[A]is in (RTRS)q µform and with the same principal part as the system A. Moreover, the family {Pµ}µcan be chosen such that Pµis of degree at most q+µand satisfying that JµPµ0=JµPµfor any µ0> µ. (iii) Assume that Ais in (RTRS)q 0-form. Then there exists a polynomial gauge transformation φ, given by a finite composition of regular or diagonal monomial transformations, such that φ[A]is in (RTRS)q 0-form with non-resonant residual matrix (and the same exponential part as A). EJDE-2023/79 TURRITTIN’S THEOREM 7 Theorem 1.8. Let Kbe a real closed field and let A∈ Mn(LK)be a singular system. Then there exists a formal gauge transformation ψP, with P∈LK, and a ramification Rrsuch that the transformed system FR= (ψP◦Rr)[A]has Poincar´e rank equal to ˜qand is written as: [FR]Y0=x−(˜q+1)(D1(x)⊕D2(x) + x˜q(C1⊕C2))Y, where D1, D2, C1, C2satisfy the conditions in Definition 1.6; that is FRis in (RTRS)-form and FR=J˜qFR. Moreover, the transformation ψPcan be chosen to be equal to ΨP= ΨQ◦ψ, where Q∈ Mn(K[[x]]) satisfies Q(0) = Inand ψis a finite composition of regular polynomial or diagonal monomial gauge transformations (with coefficients in K). 2. Proof of the complex Turrittin’s theorem Let A(x)∈ Mn(LK) be a system with Poincar´e rank q=q(A), written as in (1.1). Let k=k(A) be the radiality index. Since the radial part A0+xA1+···+ xk−1Ak−1is preserved by any gauge transformation, the coefficient Akis considered as the first significant matrix of the system. This must be compared with the usual proofs of Turrittin’s theorem, where the radial part is ruled out by an exponential shifting so that Akbecomes the new leading coefficient (and qdrops to q−k). In our approach, where we stress the finitely determined nature of the transformations, we do not allow the use of exponential shifting, so that the radial part is carried all along the procedure. We denote N(n, q, k) = n(q−k) + kas in Remark 1.5, (a) and consider the statement (i)’ to be the same as item (i) in Theorem 1.3 but substituting in the second part the truncation order nq by N(n, q, k). For the proof of items (i)’-(iii) of Theorem 1.3, we perform, a priori in arbitrary ordering, several ramifications, or regular polynomial or diagonal monomial gauge transformations. The desired expression of the composition of those transformations required in the different items of the statement will be a consequence of the following lemma, whose proof is straightforward. Lemma 2.1. Fix a field κ. Let r∈N≥1,P(x)∈ Mn(κ[x]), and lΨPbe the associated polynomial gauge transformation. Then there exists another polynomial gauge transformation Ψ˜ Psatisfying Rr◦ΨP= Ψ ˜ P◦Rr. In fact, we can take ˜ P(x) := P(xr). In particular, ΨPis regular or diagonal monomial if and only if Ψ˜ Pis so. Another important tool is the following result, known with the name of Splitting Lemma, which is valid for any given base field, algebraically closed or not. It permits to reduce the dimension of the system when the first significant matrix has two disjoint subsets of eigenvalues. It is usually stated in the formal setting (see for instance [23, 2, 5]), but it has a finitely determined nature in terms of truncations of the system. Lemma 2.2 (Splitting lemma).Let Kbe a field. With the same notations as above, if kis the radiality index of the system A, assume that k < q and that Akis conjugated to A11 k⊕A22 k, where the characteristic polynomials χA11 k(λ)and χA22 k(λ) are coprime, both of positive degrees, say n1and n2, respectively. Then there exists a formal regular gauge transformation ΨT, where T∈ Mn(K[[x]]) satisfies T(0) = In, such that the transformed system B= ΨT[A]writes as B=B11 ⊕B22, where 8 M. BARKATOU, F. A. CARNICERO, F. SANZS EJDE-2023/79 Bii ∈ Mni(LK)is a system of dimension nifor i= 1,2. Moreover, q(B) = q, k(B) = kand, writing B(x) = x−(q+1)(Pj≥0xjBj), we have Aj=Bjfor j= 0,1, . . . , k and for any m>k, the truncation JmBonly depends on Jm−kTand JmA. In other words, if e Ais another system with the same Poincar´e rank q(e A) = q and satisfying Jme A=JmAthen JmΨT(m−k)[e A] = JmB, where T(m−k)=Jm−kT. 2.1. Proof of Theorem 1.3, (i)’. Getting a (TRS)-form of degree 0 and some rank ˜q. First, notice that the cases q= 0 and q=k(the former being a particular case of the later, by definition) are trivial: in these cases, the system is already in (TRS)q 0-form. We proceed by induction on the dimension nof the system. The starting case n= 1 is also trivial. Assume then that n > 1. For readability, let us summarize the proof that follows. We consider first the case where Akhas more than one eigenvalue, in which case, Splitting Lemma permits to decompose a truncation of the system into two systems of lower dimension and apply induction. Then we consider the case where Akhas a single eigenvalue. In this case, we define a tuple I(A) = (γ(Ak), q −k)∈Nn+1 associated with the system, where γ(Ak) only depends on the conjugation class of Ak. The aim is to define a series of transformations so that the resulting system Bis in the precedent cases or else verifies I(B)< I(A) in the lexicographical order. To choose the transformations, we define an “exponent” g=g(A), a positive rational number that depends only on the truncation JNAof the system, where N=N(n, q, k), after a finer preparation of Aby a regular polynomial gauge transformation. If gis an integer, it determines a special type of monomial diagonal transformation to be done to win (a shearing transformation). If g=h/r is the irreducible expression for gand r > 1, we perform first the ramification Rrand then the shearing associated with the numerator hdoes the work. 2.1.1. Case with different eigenvalues. Suppose that we are in the case where Ak has at least two different eigenvalues. Then we can reduce to a smaller dimension as follows. First, up to a constant regular gauge transformation we can assume that Ak=A11 k⊕A22 kwhere A11 kand A22 kare matrices of respective sizes n1, n2, both smaller than n, and having no common eigenvalue. Using Lemma 2.2 for N=N(n, q, k), there exists a regular polynomial gauge transformation ΨPsuch that the N-truncation of B= ΨP(A) is written as JNB=B11 ⊕B22, where, for i= 1,2, Bii is a system of dimension ni. Moreover, JNBhas the same Poincar´e rank, the same radiality index and the same k-truncation than A. In particular, if qi=q(Bii) and ki=k(Bii) then qi≤qand qi−ki≤q−k. Taking into account that n1and n2are both positive and hence strictly smaller than n, we obtain for i= 1,2 that N(n, q, k)≥ni(q−k) + q≥ni(qi−ki) + qi≥ni(qi−ki) + ki=N(ni, qi, ki). Using the induction hypothesis to system Bii for i= 1,2, there is a finite composition ψii of transformations in dimension ni(as in statement (i)) such that e Bii =ψii(Bii) is in (TRS)qi 0-form and such that the second part of statement (i)’ holds for the truncation order N(ni, qi, ki) in the place of niqi. Moreover, in the composition ψii there is but a single ramification Rriwith ri∈N≥1(including the case R1=id). Now, for {i, j}={1,2}write, using Lemma 2.1, Rrj◦ψii =eϕii ◦Rr1r2, EJDE-2023/79 TURRITTIN’S THEOREM 9 where eϕii is a composition of transformations in dimension ni, either regular polynomial or diagonal monomial (that is, no ramification). Notice that the composition Rrj◦ψii satisfies the requirements of Theorem 1.3, (i)’ for the system Rrj(Bii), for which the Poincar´e rank and radiality index are equal to rjqiand rjki, respectively. Writting eϕii = ΨQi, where Qi∈ Mni(K[x]), we put r=r1r2and define ψ= ΨQ1⊕Q2◦Rr◦ΨP=ϕ◦Rr, where ϕ= ΨQ1⊕Q2◦ΨP(xr)is a composition of gauge transformations, either regular polynomial or diagonal monomial. We check that ψsatisfies the requirements of statement (i)’ for the initial system A. To be convinced, we need to observe two facts. On one hand, for {i, j}={1,2}, the composition Rrj◦ψii satisfies all the requirements of (i)’ for the system Bii, since so does ψii by construction (notice that if Bis any system then we have q(Rr(B)) = rq(B), k(Rr(B)) = rk(B) and for any M≥0, the truncation JrM Rr(B) is univocally determined by JMB). On the other hand, use the second part of Lemma 2.2 to conclude that JNΨP(A) only depends on JNAfor N=N(n, q, k) and the inequality N(ni, qi, ki)≤Nfor i= 1,2 proved above. 2.1.2. Case with a single eigenvalue. Suppose now that Akhas a single eigenvalue λk∈K. Recall that we are assuming that K=K. So, up to a constant gauge transformation, we may suppose that Akis in Jordan normal form. Explicitly, there exists a (unique) sequence 1 ≤n1≤n2≤ ··· ≤ n`≤nwith n=n1+···+n`such that Ak= (λkIn1+H(n1))⊕···⊕(λkIn`+H(n`)),(2.1) where H(nj)=        0 1 . . . 0 0 0 1 . . . 0 . . . 0 0 . . . 0 1 0 0 . . . 0 0       nj×nj (each such matrix will be called in the sequel a shifting matrix). Notice also that, since Akis not a radial matrix, we have nj>1 for at least one index j. We divide the proof in different steps. Step 1. The tuple I(A). In the situation above, for i= 1, . . . , n, denote by γi(Ak)∈ N≥0the degree, as a polynomial in λ, of the g.c.d. of the family of all i×iminors of the characteristic matrix Ak−λIn. I n particular γn(Ak) = n. For such a system A(when Akhas a single eigenvalue), we define the following tuple of non-negative integer numbers I(A) := (γ1(Ak), . . . , γn(Ak), q −k). We remark that each γ(Ak) depends only on the conjugation class of Ak. We need to recall also the following result on Linear Algebra (see in Wasow [23, Lemma 19.4] for a proof) concerning the behaviour of the values γifor a perturbation of the matrix Akin the case where Akhas at least two blocks. Lemma 2.3. Consider Akas a block-diagonal Jordan matrix Ak=A(ij) k, where the diagonal blocks are given by A(ii) k=λkIni+H(ni). Assume that `≥2. With the same block structure, let G=G(ij)∈ Mn(K)be a block-lower-triangular matrix 16 M. BARKATOU, F. A. CARNICERO, F. SANZS EJDE-2023/79 shows that Buu l=Auu l,if l6=q, Buv l=Auv l,if u6= (11) and v6= (11), B(11)v l=A(11)v l+1 ,if v6= (11), Bu(11) l=Au(11) l−1,if u6= (11) B(11)(11) q=A(11)(11) q−It. (2.16) In particular, we obtain q(B) = q,Bis in (TRS)q 0-form with the same exponential part D(x) and the residual matrix C0=Bqis upper triangular with respect to the block structure (Buv q) and with diagonal equal to diag(C0) = ((C11 −It)⊕···⊕C1s1)⊕···⊕(Cr1⊕···⊕Crsr). We deduce that m(C0) = m(C)−1 and we are done. General case. Notice that, in preceding case, the degree of a polynomial gauge transformation needed to obtain a non-resonant residual matrix can be bounded by m(C) (cf. Remark 1.5, (e)). Moreover, such transformation depends only on the truncation Jm(C)+q(A). We use this remark and Step 1 in paragraph 2.2 to reduce the general case to the precedent case. More precisely, consider the decomposition of the exponential part in radial matrices D(x) = D11(x)⊕ ··· ⊕ D``(x), as in equation (2.8). Consider also C=C11 ⊕···⊕C``, decomposed into the same block structure. Put m:= max{m(Cii), i = 1, . . . , `}. Taking into account Step 1 in the proof of (ii), there is regular formal gauge transformation ψsuch that ψ[A] decomposes as ψ[A] = A11(x)⊕···⊕A``(x), where the principal part of Ajj (x) is x−ν(Djj )(Djj(x) + xqCjj). If ψ= ΨPand we put P:= Jm+qP,ψ:= ΨP, we have that ψis a polynomial regular transformation of degree at most m+qthat satisfies Jm+q(ψ[A]) = Jm+q(A11(x)) ⊕···⊕Jm+q(A``(x)) The system Ajj(x) is in the radial case treated above, so that there is a finite composition of constant regular and monomial diagonal transformations ϕjsuch that ϕj[Ajj ] has the same exponential part as Ajj(x) and a non-resonant residual matrix. Moreover, as already noticed, the degree of each ϕjis bounded by m(Cjj) and the principal part of ϕj[Ajj] only depends on Jm(Cjj )+q(Ajj). We conclude that the composition φ= (ϕ1⊕··· ⊕ ϕ`)◦ψis a polynomial gauge transformation so that φ[A] is a system in (TRS)q 0-form with non-resonant residual matrix. Moreover, the degree of φcan be bounded by 2m+q. This ends the proof of item (iii) of Theorem 1.3 and completes the statement in Remark 1.5, (e).  One final comment on how to conclude Remark 1.5, (d) in what it concerns for this item (iii). We need to take into account that the diagonal monomial transformations used in the process are only those of the form ΨS, where Sis as in equation (2.15). As we have already observed from equations (2.16), such transformations preserve the Poincar´e rank. EJDE-2023/79 TURRITTIN’S THEOREM 17 3. Proof of the real Turrittin’s theorem In this section we fix a real closed field Kand we prove Theorem 1.7, the real version of Turrittin’s theorem on polynomial normal forms. As mentioned, the formal statement Theorem 1.8 will be a consequence of it. We use the monomorphism of K-algebras defined in paragraph 1.2. That is, for any m∈N, we consider Θm:Mm(K)→ M2m(K),(cuv)7→ (Λcuv ). and (with the same name), its extension to a morphism of K-algebras from Mm(LK) to M2m(LK) sending an m-dimensional system B=x−(q+1) Pj≥0xjBjwith coefficients in Kto the system Θm(B) = x−(q+1) Pj≥0xjΘm(Bj). Notice that Θm preserves the Poincar´e rank but not necessarily the radiality index of the system. On the other hand, one can check easily that Θmcommutes with the gauge transformations and with ramifications. To be precise, if B, P ∈ Mm(LK) with det(P)6= 0, we have Θm(ΨP[B]) = ΨΘm(P)[Θm(B)],(3.1) and, if ris a natural non-zero number, then Θm◦Rr=Rr◦Θm.(3.2) 3.1. Propagating a C-matrix to higher order coefficients. The key result for the proof of Theorem 1.7 is the following proposition. Proposition 3.1. Consider a system A∈ Mn(LK)with Poincar´e rank equal to q and written as A=x−(q+1) (A0+xA1+. . . ). Let kbe the radiality index of Aand assume that k < q and that the spectrum of Akin Kconsists in a pair of conjugated values a±ib with a, b ∈Kand b6= 0 (thus in particular n= 2mis even). Then there exists a formal regular gauge transformation ΨT, where T∈ Mn(K[[x]]), such that the transformed system B= ΨT[A]is a C-system. Moreover, writing B=x−(q+1)(Pj≥0xjBj), we have Aj=Bj∈KInfor j= 0,1, . . . , k −1and for any µ≥k, the truncation JµBonly depends on Jµ−kTand JµA. In other words, if e Ais another system with q(e A) = qand satisfying Jµe A=JµA, then Jµ(ΨJµ−kT[e A]) = JµB. The proof of Proposition 3.1 is similar to the one of the Splitting Lemma (cf. Lemma 2.2) or of Theorem 1.3, (ii). This time, it is based on the following result for C-matrices of size two. Lemma 3.2. Let λ∈K\Kand let Λλ= Θ1(λ)be the corresponding C-matrix of size 2. Given S∈ M2(K)an arbitrary matrix with coefficients in K, there exists a matrix X∈ M2(K)such that ΛλX−XΛλ+Sis a C-matrix. Proof. Put λ=a+ib with a, b ∈Kand b6= 0 and write S= (sij) and X= (xij) with 1 ≤i, j ≤2. Computing we have ΛλX−XΛλ+S=−u+s11 v+s12 v+s21 u+s22,(3.3) where u=b(x12 +x21) and v=b(x11 −x22). The matrix in (3.3) is a C-matrix iff we have −u+s11 =u+s22 and v+s12 =−(v+s21). These two last equations have solutions in u, v once we are given the entries sij of Sand, taking into account that b6= 0, we conclude the lemma.  18 M. BARKATOU, F. A. CARNICERO, F. SANZS EJDE-2023/79 Proof of Proposition 3.1. First, using a real canonical form of Ak, there is a nonsingular matrix T0with entries in Ksuch that T−1 0AkT0= Λ + Hwhere Λ=Λλ⊕Λλ⊕···⊕Λλ, H =        01I2 02I2 ... m−1I2 0        , with j∈ {0,1}. Up to replacing Aby ΨT0[A], we may assume that the original coefficient Akhas already the form above Ak= Λ + H, a C-matrix. We look for a regular formal gauge transformation ΨTwith T=In+xT1+x2T2+. . . satisfying the required property. For that, we compute the coefficients of the transformed system B:= ΨT[A] in terms of the coefficients of Aand of T. With similar computations as already done in the preceding section, if we write B=x−(q+1)(B0+xB1+. . . ) then we obtain: - The radial part does not change; i.e., B`=A`for `= 0,1, . . . , k −1. -Bk=Ak= Λ + H. - For j≥k+ 1, we obtain Bj= [Ak, Tj−k] + Aj+Qj,(3.4) where Qjis a matrix which depends polynomially only on the matrices of the family {Ak+s, Ts}s<j−k. Let us show that we can choose recursively T1, T2, . . . such that each Bjin equation (3.4) is a C-matrix for any j≥k. This will complete the proof of Proposition 3.1. The starting case j=kis done since Bk=Akis already a C-matrix. Suppose that for j > k we have already constructed T1, . . . , Tj−k−1such that B`is a C- matrix for `<j. For each value of the letter Y∈ {A, T, B, Q}and for each `≤j, we write Y`= (Yuv `)1≤u,v≤min a block structure of 2 ×2 matrices. We construct the different blocks Tuv j−kin the following order. We start by the bottom of the first column: the block Tm1 j−ksatisfies, after equation (3.4), [Λλ, Tm1 j−k] + Am1 j+Qm1 j=Bm1 j. Using Lemma 3.2, we choose Tm1 j−kin such a way that Bm1 jis a C-matrix. Then we continue with the block Tm−1,1 j−kwhich satisfies [Λλ, Tm−1,1 j−k] + m−1Tm1 j−k+Am−1,1 j+Qm−1,1 j=Bm−1,1 j. Taking into account that Tm1 j−khas already been chosen and using Lemma 3.2, we choose Tm−1,1 j−ksuch that Bm−1,1 jis a C-matrix. The process can be repeated in this way until we construct all blocks in the first column, that is, those of the form Tu1 j−kin inverse order for ufrom u=mto u= 1. After that, we construct the blocks in the second column Tu2 j−k, again from u=mto u= 1: by (3.4) we obtain [Λλ, Tu,2 j−k] + uTu+1,2 j−k−1Tu,1 j−k+Au2 j+Qu2 j=Bu2 j (with m= 0), and we choose Tu2 j−ksuch that Bu2 jis a C-matrix, once the blocks Tu+1,2 j−k,Tu1 j−kin the above equation are already known. We continue in this way in EJDE-2023/79 TURRITTIN’S THEOREM 19 order to complete the construction of all blocks Tuv j−kso that any block Buv j(and hence the whole matrix Bj) is a C-matrix.  3.2. Proof of Theorem 1.7, (i). Getting a (RT RS)-form of degree 0. Fix a singular system A∈ Mn(LK) with Poincar´e rank q=q(A) and write A= x−(q+1) (A0+xA1+. . . ) as in (1.1) with A06= 0. Denote by k=k(A) the radiality index of A. As in the complex case, we prove a slightly improvement of item (i) in Theorem 1.7 (called item (i’) in what follows), where the sufficient jet order nq to obtain the same real Turrittin-Ramis-Sibuya form is replaced by the order N=N(n, q, k) := n(q−k) + k. We start with the trivial case where q=k(this includes the case q= 0). Using the real Jordan canonical form of Ak=Aq, there is a non-singular matrix T0∈ Mn(K) such that T−1 0A0T0=C1⊕C2, where C1is a matrix with eigenvalues in K and C2is a C-matrix. The radial part A0+xA1+···+xq−1Aq−1is preserved by ΨT0 and can be written in the form D1(x)⊕Θn2(D2(x)) where D1(x)∈ Mn1(K[X]) and D2(x)∈ Mn2(K[x]) are both diagonal polynomial. Hence ΨT0[A] is in (RTRS)q 0- form and (i’) follows (notice that N=qin this case). Assume that 0 ≤k < q. We proceed by induction with respect to the size n of the system. The case n= 1 is also trivial: Ais already in (RT RS)q 0-form with n1=n= 1 and n2= 0 and N=qin this case. Suppose then that n > 1. Suppose first that the first non-radial term Akhas at least two non-conjugated eigenvalues in K. In this case, after a constant regular gauge transformation ΨT0, with T0∈GLn(K), we can assume that Ak=A11 k⊕A22 k, where each Aii kis a square matrix with positive size niwith entries in Kand Spec(A11 k)∩Spec(A22 k) = ∅. Apply the Splitting Lemma to Aup to order N=N(n, q, k). That is, there exists a regular polynomial gauge transformation ΨTwhere T=I+xT1+···+xN−kTN−ksuch that JN(ΨT[A]) = B11 ⊕B22, where Bii is a (polynomial) system of size ni< n with coefficients in K. By induction on the size, item (i’) holds for both systems Bii k. In a way completely analogous as we did for the complex case in paragraph 2.1, we use this to conclude item (i’) for the original system A. Suppose now that Spec(Ak) = {λk,λk}for some λk∈K. We consider the two possible situations: Case 1: λk6=λk. Notice that n= 2mis even in this case. We apply Proposition 3.1 to the system A. Notably, let ΨTbe a formal regular gauge transformation with T∈ Mn(K[[x]]) such that B= ΨT[A] is a C-system. Let Bbe the system with coefficients in Ksatisfying B= Θm(B). Apply Theorem 1.3, (i) to B: we obtain a natural number r≥1 and gauge transformations ϕ1, . . . , ϕs, either regular polynomial or monomial diagonal (with coefficients in K) such that, putting ψ=ϕs◦. . . ◦ϕ1◦Rr, we have (a) The system ψ[B] is in (T RS)˜q 0-form for some ˜q≥0. (b) Being N=N(n, q, k), if Eis another system with q(E) = qand JNE= JNB, the system ψ[E] is also in (TRS)˜q 0-form with the same principal part as ψ[B]. Now, for any i= 1, . . . , s, if ϕi= ΨPiwith Pi∈ Mn(K[[x]]), we put eϕi:= ΨΘm(Pi). Notice that eϕiis a gauge transformation with coefficients in K, either regular polynomial or diagonal monomial. Denote by e ψ=eϕs◦···◦ eϕ1◦Rrand let 20 M. BARKATOU, F. A. CARNICERO, F. SANZS EJDE-2023/79 us see that the composition e ψ◦ΨJNT=eϕs◦···◦ eϕ1◦ΨJNT(xr)◦Rr satisfies the requirements of Theorem 1.7, (i’). Using the property (b) above and equations (3.1) and (3.2) we have Θm(ψ[JN(B)]) = e ψ[JN(Θm(B))] = e ψ[JN(B)] is in (RTRS)˜q 0-form. On the other hand, we have the following Claim. If E∈ Mn(LK) is any system with Poincar´e rank equal to qand JNE= JNBthen J˜q(e ψ[E]) = J˜q(e ψ[JNB]). Applying this claim to E= ΨJNT[A], taking into account that JN(ΨJNT[A]) = JNB, we conclude Theorem 1.7, (i’) in this Case 1. It remains to show the Claim. It is a consequence of the property (b) above (using (3.1) and (3.2)) in the case that Eis a C-system). To be convinced that it is true for any system Ein the hypothesis of the statement, we notice, using the description of a general gauge transformation or a ramification, that there exists some integer M > 0 such that J˜q(e ψ[E]) = J˜q(e ψ[JME]) for such systems and that the map HM:JME7→ J˜q(e ψ[JME]) is a polynomial map in the entries of the coefficient matrices E0, . . . , EMof JME. Necessarily the minimum Mwith this property must be greater or equal than N. But, as we have just said, property (b) implies that the value of HMdoes not depend on the entries of EN+1, . . . , EMif Eis a C-system. Since the set of JM-jets of C-systems with a fixed Poincar´e rank qhas non-empty interior in the space of JM-jets of all systems (with that fixed Poincar´e rank), we conclude that M=Nsatisfies the property above. The Claim follows. Case 2: λk=λk. In this case, Akhas a single eigenvalue λk∈K. After a constant gauge transformation with entries in K, we can write Akin its Jordan canonical form as in equation (2.1). We can define in this case the tuple I(A)=(γ1(Ak), . . . , γn(Ak), q −k) and proceed exactly as in the proof of the complex Turrittin theorem in paragraph 2.1 from the step in which Akhas the Jordan form (2.1). Notably, the terms in the truncation JNA, with N=N(n, q, k), determine a shearing order g=h/r ∈Q(cf. Definition 2.4). Then, we consider the transformed system B= ΨSh◦Rr[A], where Rris the ramification of index rand Sh= diag(1, xh, . . . , x(n−1)h). Denoting by q0and k0the Poincar´e rank and the radiality index of Brespectively, one of the following situations occurs: •k0=q(including the case q0= 0): we finish since this the trivial case. •0≤k0< q0and Bk0has at least two non-conjugated eigenvalues in K: we finish using splitting lemma and induction on nas above. •0≤k0< q0and Bk0has a unique pair of conjugated eigenvalues that do not belong to K: we finish since we are in Case 1 above. •0≤k0< q0and Bk0has a unique eigenvalue that belongs to K: in this case, the arguments in Steps 3 and 4 in paragraph 2.1 are valid for the real closed field Kand they permit to conclude I(B)< I(A) (in lexicographical order). We finish again since this tuple of non-negative integers number cannot decrease indefinitely. This completes the proof of Theorem 1.7, (i’).  EJDE-2023/79 TURRITTIN’S THEOREM 21 3.3. Proof of Theorem 1.7, (ii): getting (RTRS)-form of higher degree. The proof can be done similarly to the case where Kis algebraically closed in paragraph 2.2, with only some minor changes. Let us indicated them. Suppose that the system Ais in (RTRS)q 0-form with exponential part equal to D(x) = D1(x)⊕D2(x) and residual matrix C=C1⊕C2in the conditions of Definition 1.6. In particular, D2(x) = Θn2(E2(x)) and C2= Θn2(G2), where G2∈ Mn2(K) and E2(x) = diag(d1(x), . . . , dn2(x)) with dj(x)∈K[x]q−1\K[x]. We also assume that Cis non-resonant, which is equivalent to say that both C1 and G2are non-resonant matrices (of sizes n1and n2, respectively). We consider a block structure to write our system, similar to the one given in (2.8), but compatible with the fact that D2(x) is a C-matrix. Notably, we write D(x) = D11(x)⊕···⊕Dss(x)⊕Φk1(E11(x)) ⊕···⊕Φkt(Ett(x)),(3.5) where •Each Djj (x) = fj(x)Imjis a radial matrix in Mmj(K[x]) (i.e., the coefficients of fj(x) are in K). •Each Ejj (x) = gj(x)Ikjis a radial matrix in Mkj(K[x]) (i.e., the coefficients of gj(x) are in K). •D1(x) = D11(x)⊕···⊕Dss(x) and D2(x)=Φn2(E11(x)⊕···⊕Ett(x)). In accordance with the notation of equation (2.8), we denote `:= s+tand Djj(x) := Φkj−s(Ej−s,j−s(x)) for j=s+ 1, . . . , `. We write also A=x−(q+1) PlxlAlas in (1.1) and each coefficient Al= (Auv l)1≤u,v≤`in the block structure of (3.5). Notice that the residual matrix C=C1⊕C2is block-diagonal in this structure, since it commutes with D(x) = D1(x)⊕D2(x) and C2is a C-matrix. Thus, Auv q+1 =Cuv = 0 if u6=v. We want to eliminate all coefficients Aq+1, Aq+2, . . . by means of a formal regular gauge transformation ΨPwith P∈ Mn(K[[x]]) and P(0) = In. We proceed as in paragraph 2.2 in two steps: first we eliminate the non-diagonal blocks Auv l, for u6=vand l≥q+1, and then the diagonal blocks Auu l, for u= 1, . . . , ` and l≥q+1. The first step is proved by induction on q, as in the mentioned paragraph. The case q= 0 is trivial. If q > 0, we consider a coarser block structure than the one given by (2.8) More precisely, we consider a similar block structure as the one in (2.9), where each block Djj(x) with j∈ {1, . . . , `1}is of maximal size such that its value Djj 0:= Djj (0) at zero is: (a) Either a radial matrix (i.e., Djj 0=ajIhjwith some aj∈K). (b) Or a radial C-matrix (i.e., Djj 0= Φhj((aj+ibj)Ihj) for some aj+ibj∈ K\K). Notice that a block Djj(x) in the case (a) may contain several of the blocks in the decomposition (3.5), even of the two different types {Dll(x)}l≤sand {Dll(x)}l>s. In any case, using the same equations (2.10) and Lemma 2.5, we can construct a formal matrix T=In+xq+1Tq+1 +··· ∈ K[[x]] such that the system B= ΨT(A) has zero non-diagonal blocks with respect to this last structure D(x) = ⊕1≤j≤`1Djj(x). Hence, B=⊕1≤j≤`1Bjj , where Bjj is of size hjwhen Djj 0is in the case (a), or Bjj is of size 2hjwhen Djj 0is in the case (b). In this last case, using Proposition 3.1, 22 M. BARKATOU, F. A. CARNICERO, F. SANZS EJDE-2023/79 we can assume that Bjj is a C-system (notice that bj6= 0 in this case, so that k(Bjj ) = 0 and q(Bjj) = q > 0). Put e Bjj := Bjj −x−(q+1)Djj 0for j= 1, . . . , `1, a system with Poincar´e rank strictly smaller than q. At this point, the proof continues as the one in Step 1 of paragraph 2.2 by constructing, using the induction hypothesis, a regular transformation ΨTjthat applies and transform the subsystem e Bjj into a block-diagonal one with respect to the structure induced on the block Bjj by (3.5). We only have to take care about the following: for any index j∈ {1, . . . , `1}such that Djj 0in the case (b) above, the matrix Tjmust be chosen to be a C-matrix, so that ΨTjpreserves the system x−(q+1)Djj 0and thus this transformation applied to Bjj produces the same result. Finally, the second step (eliminating the diagonal blocks Auu lfor l≥q+ 1) is obtained exactly in the same way as in Step 2 of the proof of Theorem 1.3, (ii) in paragraph 2.2: we have to solve recursively the same equations (2.12) for the blocks Ujj , and this can be done independently of the base field K, since we only need Lemma 2.5 (valid for any field) and the hypothesis that Cis non-resonant. 3.4. Proof of Theorem 1.7, (iii). Getting a non-resonant matrix. The proof of this item is made entirely equal to the corresponding complex case (cf. Theorem 1.3, (iii)) in paragraph 2.3. The only difference is that the first case treated there, called the “radial case”, must be treated here in two different cases: either we are in the similar “radial case” with coefficients in K(that is D(x) = f(x)In with some polynomial f(x)∈K[x]q−1), or we are in the C-radial case (that is D(x) = Θn/2(g(x)In/2), where g(x)∈K[x]q−1\K[x]). In the second of these two cases, we have that D(x)6= 0 and 0 ≤k(A)< q(A), so that we can apply Proposition 3.1 and assume, after a regular polynomial gauge transformation of degree m=m(C) (cf. equation (2.13)), that the truncation Jq+m(A) is a C- system, image by Θn/2of some system e A∈ Mn/2(LK) with exponential part equal to e D(x) = g(x)In/2. By Theorem 1.3, (iii), we transform e Ainto another one with the same exponential part and non-resonant residual matrix by a regular gauge transformation ΨS, where S=In/2+xS1+··· ∈ Mn/2(K[x]m). In this case, the regular transformation ΨΘn/2(S)proves item (iii) for the real system A. The general case is done as in paragraph 2.3 by means of the decomposition (3.5) of D(x) and using step 1 of the proof of Theorem 1.7, (ii), already discussed in the previous paragraph 3.3. This completes the proof of Theorem 1.7.  Acknowledgments. This work was supported by the Agencia Estatal de Investigaci´on, Ministerio de Ciencia e Innovaci´on, Spain (Projects MTM2016-77642-C2-1- P and PID2019-105621GB-I00). The first author thanks UVa for the support during several research stays at the Departamento de ´ Algebra, An´alisis Matem´atico, Geometr´ıa y Topolog´ıa. Authors thank the anonymous referees for their carefully reading of the article and for their comments. References [1] Babbitt, D. G.; Varadarajan, V. S.; Formal reduction of meromorphic differential equations: a group theoretic view, Pacific Journal of Mathematics, 109(1) (1983), 1–80. [2] Balser, W.; Formal Power Series, Linear Systems of Meromorphic Ordinary Differential Equations, Universitext, Springer-Verlag, 2000. 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Thomas, 87060 Limoges cedex, France Email address:[email protected] F´ elix A. Carnicero Departamento de ´ Algebra, An´ alisis Matem´ atico, Geometr´ ıa y Topolog´ ıa. Universidad de Valladolid, Facultad de Ciencias, Paseo de Bel´ en, 7, E-47011, Valladolid, Spain Email address:carnicero [email protected] Fernando Sanzs Departamento de ´ Algebra, An´ alisis Matem´ atico, Geometr´ ıa y Topolog´ ıa, Universidad de Valladolid, Facultad de Ciencias, Paseo de Bel´ en, 7, E-47011. Valladolid, Spain Email address:[email protected]