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, FÉLIX 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 K is 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 K be 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 R is any ring and n ∈ N≥1, Mn(R) denotes the ring of square matrices of size n with entries in R. A matrix A ∈Mn(LK) is identified with the formal meromorphic linear system of ODEs [A] Y ′ = AY, where Y = (Y1, . . . , Yn)t is a column vector of n variables. 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 i and A0 6= 0. Also, if N is a non-negative integer, the truncated system up to degree N is defined by JNA := xν(A)(A0 + xA1 + · · ·+ xNAN ). The system A is called singular (at x = 0) if ν(A) < 0. The Poincaré 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 A as the system of formal linear ODEs xq+1Y ′ = ÃY, where à = A0 + xA1 + . . . . A singular system with Poincaré rank q = 0 (resp. q > 0) is usually referred to be of first kind (resp. of second kind). We denote by In the identity matrix of size n. We define the radiality index of A as the non-negative integer k = k(A) := min({j : Aj 6∈ 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] : Y ′ = AY into the system [B] : Z ′ = BZ where B = P−1AP − P−1P ′. The map ΨP : Mn(LK) → Mn(LK) sending A to ΨP [A] := P−1AP − P−1P ′ is bijective. It is called the gauge transformation associated with P . In particular, if P ∈ GLn(K) is a constant matrix, then ΨP is the conjugation by P . (ii) Given r ∈ N≥1, the change of the independent variable x = zr transforms the system dY dx = A(x)Y into 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 ΨP will be called: regular, if ν(P ) = 0 and detP (0) 6= 0; polynomial, if each entry of P belongs to K[x] (in which case the degree of ΨP is defined as the maximum of the degrees of the entries of P ); diagonal monomial, if P = diag(xk1 , . . . , xkn) with kj ∈ N≥0 for each j. Notice that if P,Q ∈ GLn(LK) then we have ΨPQ = ΨQ ◦ΨP . Also, if r, s ∈ N≥1 then 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, K algebraically 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-Pflügel [6]. The dif- ferent 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 K is 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 A was 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 state- ments 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 Poincaré 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 trans- formations 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 exponen- tial 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 Poincaré rank q = q(A) and let µ ∈ N≥0. We say that A is in Turrittin- Ramis-Sibuya form of degree µ (and of rank q), or in (TRS)qµ-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 A is singular of first kind (that is q = 0) then A is already in (TRS)0 0-form, with exponential part equal to D(x) = 0 and residual matrix C = A0. On the other hand, if A is in (TRS)qµ-form for some µ and 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 C is non-resonant if for any pair of distinct eigenvalues λ, λ′ ∈ K of C we have λ − λ′ 6∈ Z. Other authors, for instance Balser in his book [2], use the terminology “C has good spectrum”. Now, Turrittin’s results for the case where K = K can be stated in the two following theorems. Theorem 1.3 (Complex polynomial normal form). Suppose that K is an algebrai- cally closed field and let A ∈ Mn(LK) be a singular system with Poincaré rank equal to q = q(A). (i) There exist some r ∈ N≥1 and finitely many polynomial gauge transforma- tions ϕ1, . . . , ϕm, either regular or diagonal monomial, such that, denoting ψ = ϕm ◦ · · · ◦ ϕ1 ◦Rr, the transformed system à = ψ[A] is in (TRS)q̃0-form, where q̃ = q(Ã). Moreover, if B ∈ Mn(LK) is another singular system with q(B) = q and JnqA = JnqB then B̃ = ψ[B] is also in (TRS)q̃0-form with the same prin- cipal part as Ã; i.e, q(B̃) = q̃ and Jq̃ψ[A] = Jq̃ψ[B]. (ii) Assume that A is in (TRS)q0-form and that its residual matrix is non- resonant. Then, for any given µ ≥ 0, there exists a regular polynomial gauge transformation φµ = ΨPµ , where Pµ(0) = In, such that φµ[Ã] is in (TRS)q̃µ-form with the same principal part as the original system A. Moreover, the family {Pµ}µ can be chosen such that Pµ is of degree at most q + µ and satisfying that Jq+µP µ′ = Jq+µP µ for any µ′ > µ. (iii) Assume that A is in (TRS)q0-form. Then there exists a gauge transfor- mation φ, given by a finite composition of regular polynomial or diagonal monomial transformations, such that φ[A] is in (TRS)q0-form with non- resonant 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 K is an algebraically closed field and let A ∈ Mn(LK) be a singular system with Poincaré rank equal to q = q(A). There exists a formal gauge transformation ΨP and a ramification Rr such that the transformed system F = (ΨP ◦ Rr)[A] has Poinaré rank equal to q̃ = q(F ) and can be written in a Formal Normal Form [F ] Y ′ = x−(q̃+1)(D(x) + xq̃C)Y, where D(x) and C satisfy the requirements in Definition 1.1; i.e., F is in (TRS)q̃0- form and Jq̃F = F . Moreover, the transformation ΨP can be chosen to be equal to ΨP = ΨQ ◦ψ, where Q ∈Mn(K[[x]]) with Q(0) = In and ψ is 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-Schäfke [18]. Below, we propose a proof with the slightly improved order N := n(q − k) + k, where k is the radial index of the initial system A. (b) To obtain the formal normal form [F ] for the system A in Theorem 1.4 is equivalent to say that there exists a matrix P (t) ∈ Mn(K[[t]]) and some r ∈ N≥1 such that Z(x) = P (x1/r) exp (∫ D(x1/r) x(q̃+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 q̃/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 A and B such that B = ΨT [A] with T ∈ GLn(LK) and we obtain (TRS)-normal forms of A and B as in item (i) of Theorem 1.3 with resulting Poincaré ranks q̃A and q̃B and exponential parts DA(x) and DB(x), respectively, then there are integers r1, r2 such that q̃A/r1 = q̃B/r2 and DA(x1/r1) = DB(x1/r2). In particular, q̃ = 0 iff the system A is 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 Poincaré 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 Poincaré 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 φ in item (iii). 1.2. Real case. Suppose that K is 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 Λλ has roots a ± bi and is irreducible if and only if λ 6∈ K, i.e., b 6= 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 Θm will be called a complex matrix over K, or a C-matrix, for short. We extend Θm to a monomorphism of K-algebras, denoted with the same letter, from Mm(LK) into M2m(LK); that is, from formal meromorphic linear systems over K to formal meromorphic linear systems over K of 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 ⊕ V the square matrix of size k + l given in blocks U ⊕ V := ( U 0 0 V ) . Definition 1.6 (Real Turrittin-Ramis-Sibuya form). Suppose that K is a real and closed field. Let A ∈ Mn(LK) be a system with Poincaré rank q = q(A) and let µ ∈ N≥0. We say that A is in Real Turrittin-Ramis-Sibuya form of degree µ (and of rank q), or in (RTRS)qµ-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). • C1 and C2 are constant matrices with entries in K of sizes n1 and 2n2, respectively, and C2 is 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 K is a real closed field and let A ∈Mn(LK) be a singular system with Poincaré rank equal to q = q(A). (i) There exists r ∈ N≥1 and there exists finitely many polynomial gauge trans- formations ϕ1, . . . , ϕm (with coefficients in K), either regular or diagonal monomial, such that, denoting ψ = ϕm ◦ · · · ◦ ϕ1 ◦Rr, the transformed system à = ψ[A] is in (RTRS)q̃0-form. Moreover, if B ∈ Mn(LK) is another singular system with q(B) = q and JnqA = JnqB then, with the same transformation ψ, the transformed system B̃ = ψ[B] is also in (RTRS)q̃0 with the same principal part as Ã, i.e., Jq̃ψ[A] = Jq̃ψ[B]. (ii) Assume that A is in (RTRS)q0-form and that its residual matrix is non- resonant. Then, for any µ ≥ 0 there exists a regular polynomial gauge trans- formation φµ = ΨPµ where Pµ(0) = In such that φµ[A] is in (RTRS)qµ- form and with the same principal part as the system A. Moreover, the family {Pµ}µ can be chosen such that Pµ is of degree at most q + µ and satisfying that JµP µ′ = JµP µ for any µ′ > µ. (iii) Assume that A is in (RTRS)q0-form. Then there exists a polynomial gauge transformation φ, given by a finite composition of regular or diagonal mono- mial transformations, such that φ[A] is in (RTRS)q0-form with non-resonant residual matrix (and the same exponential part as A). EJDE-2023/79 TURRITTIN’S THEOREM 7 Theorem 1.8. Let K be 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 Rr such that the transformed system FR = (ψP ◦Rr)[A] has Poincaré rank equal to q̃ and is written as: [FR] Y ′ = x−(q̃+1)(D1(x)⊕D2(x) + xq̃(C1 ⊕ C2))Y, where D1, D2, C1, C2 satisfy the conditions in Definition 1.6; that is FR is in (RTRS)-form and FR = Jq̃F R. Moreover, the transformation ψP can be chosen to be equal to ΨP = ΨQ ◦ ψ, where Q ∈Mn(K[[x]]) satisfies Q(0) = In and ψ is a fi- nite 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 Poincaré 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−1 is preserved by any gauge transformation, the coefficient Ak is 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 Ak becomes the new leading coefficient (and q drops 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) + k as 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 transforma- tions 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 lΨP be the associated polynomial gauge transformation. Then there exists another polynomial gauge transformation ΨP̃ satisfying Rr ◦ ΨP = ΨP̃ ◦ Rr. In fact, we can take P̃ (x) := P (xr). In particular, ΨP is regular or diagonal monomial if and only if ΨP̃ is 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 K be a field. With the same notations as above, if k is the radiality index of the system A, assume that k < q and that Ak is conjugated to A11 k ⊕A22 k , where the characteristic polynomials χA11 k (λ) and χA22 k (λ) are coprime, both of positive degrees, say n1 and 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 ni for i = 1, 2. Moreover, q(B) = q, k(B) = k and, writing B(x) = x−(q+1)( ∑ j≥0 x jBj), we have Aj = Bj for j = 0, 1, . . . , k and for any m > k, the truncation JmB only depends on Jm−kT and JmA. In other words, if à is another system with the same Poincaré rank q(Ã) = q and satisfying Jmà = JmA then JmΨT (m−k) [Ã] = 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 q̃. 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)q0-form. We proceed by induction on the dimension n of 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 Ak has 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 Ak has 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 B is 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 JNA of the system, where N = N(n, q, k), after a finer preparation of A by a regular polynomial gauge transformation. If g is 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 g and r > 1, we perform first the ramification Rr and then the shearing associated with the numerator h does 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 k where A11 k and A22 k are 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 ΨP such 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, JNB has the same Poincaré rank, the same radiality index and the same k-truncation than A. In particular, if qi = q(Bii) and ki = k(Bii) then qi ≤ q and qi − ki ≤ q − k. Taking into account that n1 and n2 are 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 com- position ψii of transformations in dimension ni (as in statement (i)) such that B̃ii = ψii(Bii) is in (TRS)qi0 -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 Rri with ri ∈ N≥1 (including the case R1 = id). Now, for {i, j} = {1, 2} write, using Lemma 2.1, Rrj ◦ ψii = ϕ̃ii ◦Rr1r2 , EJDE-2023/79 TURRITTIN’S THEOREM 9 where ϕ̃ii is a composition of transformations in dimension ni, either regular poly- nomial 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 (B ii), for which the Poincaré rank and radiality index are equal to rjqi and rjki, respectively. Writting ϕ̃ii = ΨQi , where Qi ∈Mni(K[x]), we put r = r1r2 and 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 ψ satisfies 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 B is any system then we have q(Rr(B)) = rq(B), k(Rr(B)) = rk(B) and for any M ≥ 0, the truncation JrMRr(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 JNA for N = N(n, q, k) and the inequality N(ni, qi, ki) ≤ N for i = 1, 2 proved above. 2.1.2. Case with a single eigenvalue. Suppose now that Ak has a single eigenvalue λk ∈ K. Recall that we are assuming that K = K. So, up to a constant gauge transformation, we may suppose that Ak is in Jordan normal form. Explicitly, there exists a (unique) sequence 1 ≤ n1 ≤ n2 ≤ · · · ≤ n` ≤ n with 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 Ak is 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≥0 the degree, as a polynomial in λ, of the g.c.d. of the family of all i× i minors of the characteristic matrix Ak−λIn. I n particular γn(Ak) = n. For such a system A (when Ak has 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 γi for a perturbation of the matrix Ak in the case where Ak has at least two blocks. Lemma 2.3. Consider Ak as 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 10 M. BARKATOU, F. A. CARNICERO, F. SANZS EJDE-2023/79 with the same diagonal blocks as Ak (that is, G(ii) = A (ii) k and G(ij) = 0 if i < j). Then we have γt(G) ≤ γt(Ak) for each t ∈ {1, 2, . . . , n} and the inequality is strict for at least one t if G 6= Ak (that is, G(ij) 6= 0 for at least a pair (i, j) with i > j). Step 2. Special matrices and choice of g ∈ Q. Given an n-tuple σ = (n1, n2, . . . , n`) of positive integers as above, denote mj := ∑j u=1 nu for j = 1, . . . , `. A matrix T = (tuv) ∈ Mn(K) will be called a special matrix of type σ if tuv = 0 for every (u, v) such that u 6∈ {m1,m2, . . . ,m`}. A result in Wasow ([23, Lemma 19.2]) assures that for any N ≥ k + 1 there ex- ists a regular polynomial gauge transformation ΨPN , where PN is of degree N and PN (0) = In, such that the transformed system B = ΨPN (A) satisfies that all coef- ficients of the truncation JN (B) are special matrices of type σ = (n1, n2, . . . , n`), where the nj are the sizes of Jordan blocks of the matrix Ak. We will put N = N(n, q, k) and, renaming B again as the system A, we may assume the following assumption For each k + 1 ≤ j ≤ N , the coefficient Aj is a special matrix of type σ. (2.2) Now we write xq+1A(x) = A0 + xA1 + · · · = k∑ i=0 λix iIn + xkA, with A = Ak − λkIn + O(x) = (auv(x)). We put αuv := ν(auv(x)). Notice that αuv ∈ N≥0 for any u, v, that αuu > 0 for any u and that αu,u+1 = 0 for at least one index u ∈ {1, . . . , n}. Definition 2.4. With the conditions above, we define the shearing order (of A) as the rational number g = g(A) := min ( { αuv 1 + u− v : u > v} ∪ {αuu : 1 ≤ u ≤ n} ∪ {q − k} ) . Geometrically, as discussed in Wasow [23, pp.104-105], the shearing order is the smallest abscissa in which a line of the family {y = (v − u)x+ αuv}v 0 but Bk′ has at least two different eigenvalues. Thus, assume that q′ − k′ > 0 and that Bk′ has a single eigenvalue, say equal to λ′k′ . It is enough, by recurrence, to prove in this case that the tuple I(B) satisfies I(B) < I(A), for the lexicographical order. Notice that Bk′ = B̃k+h, after the discussion above concerning the value of q′ and k′. Consider the matrix Bk′ written in blocks according to the Jordan structure of Ak: Bk′ = ( B (ij) k′ ) 1≤i,j≤` , B (ij) k′ ∈Mnj×nj (K). Using the property (b), we have that Bk′ = H(n1) ⊕ · · · ⊕H(n1) + T where T is a lower triangular matrix. In particular, for i ∈ {1, . . . , `}, since Spec(B (ii) k′ ) = {λ′k′}, the matrix B (ii) k′ − λ′k′Ini is nilpotent and of rank ni − 1. This implies that the Jordan decomposition of B (ii) k′ has a single block and hence B (ii) k′ is conjugated to Hni + λ′k′Ini . Consequently, Bk′ is conjugated to a block-lower-triangular matrix with diagonal blocks equal to Hni + λ′k′Ini , those of the matrix Ak. Thus, either Bk′ is conjugated to Ak (when ` = 1) or we are in the situation where we can apply Lemma 2.3. From this lemma, and taking into account the equation (2.5), we deduce that I(B) < I(A), as wanted. Step 4. The case where g is not an integer. Assume now that the shearing order g is not an integer and put g = h r where h, r are positive integers without common factor and such that r ≥ 2. We remark that the condition that g is not an integer implies that g is given by one of the quotients αuv/(1 − u − v) in Definition 2.4, while αuu > g for any u, as well as q − k > g. Consider the ramification Rr and put à = Rr(A). This system has Poincaré rank and radiality index equal to q(Ã) = rq and k(Ã) = rk, respectively. Moreover, if 12 M. BARKATOU, F. A. CARNICERO, F. SANZS EJDE-2023/79 we denote Ñ = N(n, rq, rk) and N = N(n, q, r), then Ñ = rN but the truncation JÑ (Ã) only depends on JN (A), showing that if item (i)’ of Theorem 1.3 holds for à then it also holds for A. We notice moreover that the first non-radial coefficient of à is equal to Ak and that à also satisfies assumption (2.2) with respect to the same type σ = (n1, . . . , n`) given by the Jordan structure of Ak. On the other hand, the shearing order of à is given by g(Ã) = rg = h, a natural number (the new valuations αuv for à are all multiplied by r). Hence we are, for Ã, in the situation of step 3. However, the last component q − k of the tuple I(Ã) has increased and we can not conclude automatically. To finish, we put B = ΨSh(Ã), we denote by q′, r′ the Poincaré rank and the radiality index of B, respectively, and, writing B = x−(q′+1) (B0 + xB1 + . . . ), we show again that assuming that q′ − k′ > 0 and that Bk′ has a single eigenvalue, we have I(B) < I(A). Write Bk′ = ( B (ij) k′ ) as a block matrix in the same block structure as Ak. We have (cf. property (b) in step 3 above) that Bk′ is block-lower triangular with diagonal blocks given by B (ii) k′ = H(ni) + Ti, where Ti is a lower triangular matrix of size ni. Moreover, we must have that all elements in the diagonal of Ti are zero (since all values αuu for à are greater than h = rg, as mentioned above). On the other hand, by assumption (2.2), the entries of Ti on any row except possibly the last one are also zero. On the other hand, since we have assumed that each block B (ii) k′ , as the entire matrix Bk′ , has a single eigenvalue, being the trace of B (ii) k′ equal to zero, such eigenvalue is equal to zero. That is, each B (ii) k′ is nilpotent. But this implies that Ti = 0 and we conclude that B (ii) k′ = H(ni), for each i ∈ {1, 2, . . . , `}. (2.7) Furthermore, by the definition of the shearing order of Ã, we must have at least one non-zero entry below the principal diagonal of the matrix Bk′ . Together with the equation (2.7), this implies that ` > 1 and that Bk′ is a matrix G in the situation of Lemma 2.3 with a non-diagonal block B (ij) k′ 6= 0 for at least one pair (i, j) with i > j. We conclude that γt(Bk′) < γt(Ak) for at least one index t and thus I(B) < I(A), as wanted. Let us show now the statement in Remark 1.5, (c) concerning this item (i). In other words, we have to justify that all along the above process for obtaining a TRS-form of degree 0, the Poincaré rank can only increase after a ramification and never after a shearing transformation ΨSg with g ∈ N. First, notice that this property goes through the induction arguments discussed in paragraph 2.1.1. Thus, we may assume that we are in the case where Ak has a unique eigenvalue. If the shearing order g = g(A) is an integer, the required property is already established by equation (2.4). On the contrary, if the shearing order is g = h/r, with r > 1 and h not divisible by r, the procedure consists in the shearing ΨSh after the ramification Rr. We conclude using the same equation (2.4) once we observe that the system à := Rr[A] has as first non-radial term the same matrix Ak and satisfies g(Ã) = h ∈ N (just check that à satisfies already the property (2.2) and that the new values αuv in Definition 2.4 are the old ones multiplied by r). EJDE-2023/79 TURRITTIN’S THEOREM 13 2.2. Proof of Theorem 1.3, (ii). Getting (TRS)-form of higher degree µ when C is non-resonant. A proof of item (ii) when q = 0 can be found for instance in Wasow [23, Thm. 5.1], Coddington-Levinson [11, Thm. 4.1, Ch IV] or Balser [2, Thm. 5]. The general case q > 0 is not really different from those references because the exponential part D(x), being diagonal and commuting with C will play no essential role to achieve (ii). For the sake of completeness, let us indicate here the steps of the general proof. Assume that the system A is already in (TRS)q0-form with principal part D(x)+ xqC and write it in the form xq+1Y ′ = (A0 + xA1 + . . . )Y , so that xq+1JqA = A0 + · · ·+ xqAq = D(x) + xqC. We will need the following result from Linear Algebra (which is actually the core of the proof of the Splitting Lemma 2.2). See [23, Thm. 4.1] for a proof. Lemma 2.5. Fix a field K and let R,S be two square matrices with coefficients in K and of sizes n× n and m×m, respectively. Assume that R,S have no common eigenvalue in the algebraic closure K of K. Then the linear map X 7→ RX −XS from Mn×m(K) to itself is an isomorphism. Up to reorder the variables of Y , we write D(x) = D11(x)⊕D22(x)⊕ · · · ⊕D``(x), (2.8) where Djj(x) is a radial matrix of size nj (that is Djj(x) = Qj(x)Inj , where Qj(x) ∈ K[x]q−1) for each j = 1, . . . , `, and such that Dii(x) 6= Djj(x) if i 6= j. Write also the residual matrix C, as well as any coefficient matrix Aj with j ≥ q + 1, as block matrices C = (Cuv)1≤u,v≤` and Aj = (Auvj )1≤u,v≤`, where Cuv, Auvj ∈ Mnu×nv (K). Using the commutativity property [D(x), C] = 0, the assumption Dii(x) 6= Djj(x) for i 6= j and Lemma 2.5, we obtain that Cuv = 0 if u 6= v. We eliminate all coefficients Aj for j ≥ q + 1 by means of a regular formal trans- formation in two steps. First step. We eliminate all non-diagonal blocks {Auvj }u 6=v. We proceed by induc- tion with respect to q = q(A). If q = 0 (that is, D(x) = 0), the block structure is the trivial one and there are no non-diagonal blocks, so that there is nothing to prove. Suppose that q > 0. We consider a coarser block structure D(x) = D 11 (x)⊕ · · · ⊕D`1`1 (x) (2.9) in such a way that D jj (0) is a radial constant matrix for every j = 1, . . . , `1 and D jj (0) 6= D ii (0) if i 6= j. Notice that, up to reordering, each block D jj (x) is formed by several of the diagonal blocks of the decomposition (2.8). We consider a matrix T (x) ∈ Mn(K[[x]]) of the form T (x) = In + xq+1T q+1 + xq+2T q+2 + . . . such that, writing each coefficient T j = (T uv j ) in the same block structure as the one in (2.9), we have T uu j = 0 for any u ∈ {1, . . . , `1}. The regular gauge transformation ψT transforms the system A into a system B := ψT [A] which is also in (TRS)q0- form with the same principal part D(x) + xqC. Write xq+1B = D(x) + xqC +∑ j≥q+1 x jBj , and, for any j, consider each coefficient of A or B written in the 14 M. BARKATOU, F. A. CARNICERO, F. SANZS EJDE-2023/79 block structure (2.9) as Aj = (A uv j ) and Bj = (B uv j ), respectively. We obtain recursively, for any couple of indices u, v ∈ {1, . . . , `1} with u 6= v: B uv q+1 = A uv q+1 +D uu 0 T uv q+1 − T uv q+1D vv 0 B uv q+2 = A uv q+2 +D uu 0 T uv q+2 − T uv q+2D vv 0 + θuv2 (A0, A1, T q+1) · · · B uv q+j = A uv q+j +D uu 0 T uv q+j − T uv q+jD vv 0 + θuvj ({As, T q+s}s Re(λj2) > · · · > Re(λjsj ) in case sj > 1. Let m(C) := ∑ {j:sj>1} λj1 − λjsj ∈ N. (2.13) Notice that m(C) > 0 if and only if C is resonant. In order to prove item (iii) in this case, we show that, if m(C) > 0, there is a constant regular transformation ΨP0 and a monomial diagonal transformation ΨS such that the transformed system ΨS ◦ ΨP0 [A] has Poincaré rank equal to q, the same exponential part D(x) and a residual matrix C ′ satisfying m(C ′) < m(C). The matrix P0 is chosen so that C := P−1 0 CP0 is block diagonal of the form C = (C11 ⊕ · · · ⊕ C1s1)⊕ · · · ⊕ (Cr1 ⊕ · · · ⊕ Crsr ), (2.14) where Spec(Cji) = {λji}. The transformed system A = ΨP0 [A] = P−1 0 AP0 has the same exponential part D(x) (since this is a radial matrix) and residual matrix equal to C. Put xq+1A = ∑ l≥0Alx l and use the block structure given by equation (2.14) for each coefficient Al = (A uv l ), where (uv) run in the set {(ji) : j = 1, . . . , r, i = 1, . . . , sj}. Assume for instance that Ω1 has at least two elements. Then we consider the diagonal monomial matrix S := xIt ⊕ In−t, (2.15) where t is equal to the size of C11. Consider B := ΨS [A] and write xq+1B =∑ l≥0Blx l and Bl = (Buvl ) with the same block structure as in (2.14). A calculation 16 M. BARKATOU, F. A. CARNICERO, F. SANZS EJDE-2023/79 shows that Buul = A uu l , if l 6= q, Buvl = A uv l , if u 6= (11) and v 6= (11), B (11)v l = A (11)v l+1 , if v 6= (11), B u(11) l = A u(11) l−1 , if u 6= (11) B(11)(11) q = A (11)(11) q − It. (2.16) In particular, we obtain q(B) = q, B is in (TRS)q0-form with the same exponential part D(x) and the residual matrix C ′ = Bq is upper triangular with respect to the block structure (Buvq ) and with diagonal equal to diag(C ′) = ((C11 − It)⊕ · · · ⊕ C1s1)⊕ · · · ⊕ (Cr1 ⊕ · · · ⊕ Crsr ). We deduce that m(C ′) = 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 trans- formation ψ such 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 = ΨP and we put P := Jm+qP , ψ := ΨP , we have that ψ is a polynomial regular transformation of degree at most m+ q that satisfies Jm+q(ψ[A]) = Jm+q(A 11(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 ϕj such that ϕj [A jj ] has the same exponential part as Ajj(x) and a non-resonant residual matrix. Moreover, as already noticed, the degree of each ϕj is bounded by m(Cjj) and the principal part of ϕj [A jj ] only depends on Jm(Cjj)+q(A jj). We conclude that the composition φ = (ϕ1 ⊕ · · · ⊕ ϕ`) ◦ ψ is a polynomial gauge transformation so that φ[A] is a system in (TRS)q0-form with non-resonant residual matrix. Moreover, the degree of φ can 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 transfor- mations used in the process are only those of the form ΨS , where S is as in equation (2.15). As we have already observed from equations (2.16), such transformations preserve the Poincaré 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 K and 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 ofK-algebras fromMm(LK) to M2m(LK) sending an m-dimensional system B = x−(q+1) ∑ j≥0 x jBj with co- efficients in K to the system Θm(B) = x−(q+1) ∑ j≥0 x jΘm(Bj). Notice that Θm preserves the Poincaré rank but not necessarily the radiality index of the system. On the other hand, one can check easily that Θm commutes with the gauge transfor- mations 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 r is 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 Poincaré rank equal to q and written as A = x−(q+1) (A0 + xA1 + . . . ). Let k be the radiality index of A and assume that k < q and that the spectrum of Ak in K consists in a pair of conjugated values a ± ib with a, b ∈ K and b 6= 0 (thus in particular n = 2m is 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)( ∑ j≥0 x jBj), we have Aj = Bj ∈ KIn for j = 0, 1, . . . , k − 1 and for any µ ≥ k, the truncation JµB only depends on Jµ−kT and JµA. In other words, if à is another system with q(Ã) = q and satisfying JµÃ = JµA, then Jµ(ΨJµ−kT [Ã]) = 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 \K and 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 + S is a C-matrix. Proof. Put λ = a+ ib with a, b ∈ K and b 6= 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 S and, taking into account that b 6= 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 non- singular matrix T0 with entries in K such that T−1 0 AkT0 = Λ +H where Λ = Λλ ⊕ Λλ ⊕ · · · ⊕ Λλ, H =  0 ε1I2 0 ε2I2 . . . εm−1I2 0  , with εj ∈ {0, 1}. Up to replacing A by ΨT0 [A], we may assume that the original coefficient Ak has already the form above Ak = Λ +H, a C-matrix. We look for a regular formal gauge transformation ΨT with 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 A and 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 Qj is a matrix which depends polynomially only on the matrices of the family {Ak+s, Ts}s k we have already constructed T1, . . . , Tj−k−1 such 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` = (Y uv` )1≤u,v≤m in a block structure of 2 × 2 matrices. We construct the different blocks Tuvj−k in the following order. We start by the bottom of the first column: the block Tm1 j−k satisfies, after equation (3.4), [Λλ, T m1 j−k] +Am1 j +Qm1 j = Bm1 j . Using Lemma 3.2, we choose Tm1 j−k in such a way that Bm1 j is a C-matrix. Then we continue with the block Tm−1,1 j−k which satisfies [Λλ, T m−1,1 j−k ] + εm−1T m1 j−k +Am−1,1 j +Qm−1,1 j = Bm−1,1 j . Taking into account that Tm1 j−k has already been chosen and using Lemma 3.2, we choose Tm−1,1 j−k such that Bm−1,1 j is 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−k in inverse order for u from u = m to u = 1. After that, we construct the blocks in the second column Tu2 j−k, again from u = m to u = 1: by (3.4) we obtain [Λλ, T u,2 j−k] + εuT u+1,2 j−k − ε1Tu,1j−k +Au2 j +Qu2 j = Bu2 j (with εm = 0), and we choose Tu2 j−k such that Bu2 j is a C-matrix, once the blocks Tu+1,2 j−k , Tu1 j−k in 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 Tuvj−k so that any block Buvj (and hence the whole matrix Bj) is a C-matrix. � 3.2. Proof of Theorem 1.7, (i). Getting a (RTRS)-form of degree 0. Fix a singular system A ∈ Mn(LK) with Poincaré rank q = q(A) and write A = x−(q+1) (A0 + xA1 + . . . ) as in (1.1) with A0 6= 0. Denote by k = k(A) the ra- diality 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 0 A0T0 = C1⊕C2, where C1 is a matrix with eigenvalues in K and C2 is a C-matrix. The radial part A0+xA1+· · ·+xq−1Aq−1 is 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)q0- form and (i’) follows (notice that N = q in 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: A is already in (RTRS)q0-form with n1 = n = 1 and n2 = 0 and N = q in this case. Suppose then that n > 1. Suppose first that the first non-radial term Ak has 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 Aiik is a square matrix with positive size ni with entries in K and Spec(A11 k )∩Spec(A22 k ) = ∅. Apply the Splitting Lemma to A up to order N = N(n, q, k). That is, there exists a regular polynomial gauge transformation ΨT where T = I + xT1 + · · · + xN−kTN−k such 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 Biik . 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: λk 6= λk. Notice that n = 2m is even in this case. We apply Proposi- tion 3.1 to the system A. Notably, let ΨT be a formal regular gauge transformation with T ∈ Mn(K[[x]]) such that B = ΨT [A] is a C-system. Let B be the system with coefficients in K satisfying B = Θm(B). Apply Theorem 1.3, (i) to B: we obtain a natural number r ≥ 1 and gauge transformations ϕ1, . . . , ϕs, either reg- ular polynomial or monomial diagonal (with coefficients in K) such that, putting ψ = ϕs ◦ . . . ◦ ϕ1 ◦Rr, we have (a) The system ψ[B] is in (TRS)q̃0-form for some q̃ ≥ 0. (b) Being N = N(n, q, k), if E is another system with q(E) = q and JNE = JNB, the system ψ[E] is also in (TRS)q̃0-form with the same principal part as ψ[B]. Now, for any i = 1, . . . , s, if ϕi = ΨPi with Pi ∈ Mn(K[[x]]), we put ϕ̃i := ΨΘm(Pi). Notice that ϕ̃i is a gauge transformation with coefficients in K, either regular polynomial or diagonal monomial. Denote by ψ̃ = ϕ̃s ◦ · · · ◦ ϕ̃1 ◦Rr and let 20 M. BARKATOU, F. A. CARNICERO, F. SANZS EJDE-2023/79 us see that the composition ψ̃ ◦ΨJNT = ϕ̃s ◦ · · · ◦ ϕ̃1 ◦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)]) = ψ̃[JN (Θm(B))] = ψ̃[JN (B)] is in (RTRS)q̃0-form. On the other hand, we have the following Claim. If E ∈ Mn(LK) is any system with Poincaré rank equal to q and JNE = JNB then Jq̃(ψ̃[E]) = Jq̃(ψ̃[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 E is a C-system). To be convinced that it is true for any system E in 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 Jq̃(ψ̃[E]) = Jq̃(ψ̃[JME]) for such systems and that the map HM : JME 7→ Jq̃(ψ̃[JME]) is a polynomial map in the entries of the coefficient matrices E0, . . . , EM of JME. Necessarily the minimum M with this property must be greater or equal than N . But, as we have just said, property (b) implies that the value of HM does not depend on the entries of EN+1, . . . , EM if E is a C-system. Since the set of JM -jets of C-systems with a fixed Poincaré rank q has non-empty interior in the space of JM -jets of all systems (with that fixed Poincaré rank), we conclude that M = N satisfies the property above. The Claim follows. Case 2: λk = λk. In this case, Ak has a single eigenvalue λk ∈ K. After a constant gauge transformation with entries in K, we can write Ak in its Jor- dan 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 com- plex Turrittin theorem in paragraph 2.1 from the step in which Ak has the Jordan form (2.1). Notably, the terms in the truncation JNA, with N = N(n, q, k), de- termine a shearing order g = h/r ∈ Q (cf. Definition 2.4). Then, we consider the transformed system B = ΨSh ◦Rr[A], where Rr is the ramification of index r and Sh = diag(1, xh, . . . , x(n−1)h). Denoting by q′ and k′ the Poincaré rank and the radiality index of B respectively, one of the following situations occurs: • k′ = q (including the case q′ = 0): we finish since this the trivial case. • 0 ≤ k′ < q′ and Bk′ has at least two non-conjugated eigenvalues in K: we finish using splitting lemma and induction on n as above. • 0 ≤ k′ < q′ and Bk′ has a unique pair of conjugated eigenvalues that do not belong to K: we finish since we are in Case 1 above. • 0 ≤ k′ < q′ and Bk′ has 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 K and 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 K is algebraically closed in para- graph 2.2, with only some minor changes. Let us indicated them. Suppose that the system A is in (RTRS)q0-form with exponential part equal to D(x) = D1(x) ⊕ D2(x) and residual matrix C = C1 ⊕ C2 in 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 C is non-resonant, which is equivalent to say that both C1 and G2 are non-resonant matrices (of sizes n1 and 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(E tt(x)), (3.5) where • Each Djj(x) = fj(x)Imj is a radial matrix in Mmj (K[x]) (i.e., the coeffi- cients of fj(x) are in K). • Each Ejj(x) = gj(x)Ikj is a radial matrix in Mkj (K[x]) (i.e., the coeffi- cients 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+t andDjj(x) := Φkj−s(E j−s,j−s(x)) for j = s + 1, . . . , `. We write also A = x−(q+1) ∑ l x lAl as in (1.1) and each coefficient Al = (Auvl )1≤u,v≤` in the block structure of (3.5). Notice that the residual matrix C = C1 ⊕C2 is block-diagonal in this structure, since it commutes with D(x) = D1(x) ⊕ D2(x) and C2 is a C-matrix. Thus, Auvq+1 = Cuv = 0 if u 6= v. We want to eliminate all coefficients Aq+1, Aq+2, . . . by means of a formal regular gauge transformation ΨP with 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 Auvl , for u 6= v and l ≥ q+1, and then the diagonal blocks Auul , 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 D jj (x) with j ∈ {1, . . . , `1} is of maximal size such that its value D jj 0 := D jj (0) at zero is: (a) Either a radial matrix (i.e., D jj 0 = ajIhj with some aj ∈ K). (b) Or a radial C-matrix (i.e., D jj 0 = Φhj ((aj + ibj)Ihj ) for some aj + ibj ∈ K \K). Notice that a block D jj (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≤s and {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+1T q+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≤`1D jj (x). Hence, B = ⊕1≤j≤`1B jj , where B jj is of size hj when D jj 0 is in the case (a), or B jj is of size 2hj when D jj 0 is 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 B jj is a C-system (notice that bj 6= 0 in this case, so that k(B jj ) = 0 and q(B jj ) = q > 0). Put B̃jj := B jj − x−(q+1)D jj 0 for j = 1, . . . , `1, a system with Poincaré 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 transfor- mation ΨT j that applies and transform the subsystem B̃jj into a block-diagonal one with respect to the structure induced on the block B jj by (3.5). We only have to take care about the following: for any index j ∈ {1, . . . , `1} such that D jj 0 in the case (b) above, the matrix T j must be chosen to be a C-matrix, so that ΨT j pre- serves the system x−(q+1)D jj 0 and thus this transformation applied to B jj produces the same result. Finally, the second step (eliminating the diagonal blocks Auul for 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 U jj , 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 C is 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/2 of some system à ∈Mn/2(LK) with exponential part equal to D̃(x) = g(x)In/2. By Theorem 1.3, (iii), we transform à into 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 Investi- gación, Ministerio de Ciencia e Innovación, 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 Álgebra, Análisis Matemático, Ge- ometŕıa y Topoloǵıa. Authors thank the anonymous referees for their carefully reading of the article and for their comments. References [1] Babbitt, D. 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Thomas, 87060 Limoges cedex, France Email address: moulay.barkatou@unilim.fr Félix A. Carnicero Departamento de Álgebra, Análisis Matemático, Geometŕıa y Topoloǵıa. Universidad de Valladolid, Facultad de Ciencias, Paseo de Belén, 7, E-47011, Valladolid, Spain Email address: carnicero fac@hotmail.com Fernando Sanzs Departamento de Álgebra, Análisis Matemático, Geometŕıa y Topoloǵıa, Universidad de Valladolid, Facultad de Ciencias, Paseo de Belén, 7, E-47011. Valladolid, Spain Email address: fsanz@uva.es 1. Preliminaries 1.1. Complex case 1.2. Real case 2. Proof of the complex Turrittin's theorem 2.1. Proof of Theorem ??, (i)'. 2.2. Proof of Theorem ??, (ii). 2.3. Proof of Theorem ??, (iii). 3. Proof of the real Turrittin's theorem 3.1. Propagating a C-matrix to higher order coefficients 3.2. Proof of Theorem ??, (i) 3.3. Proof of Theorem ??, (ii): 3.4. Proof of Theorem ??, (iii) Acknowledgments References