# Turrittin’s Theorem revisited. The real case

M. Barkatou  XLIM, Université de Limoges ; CNRS,
123, Av. A. Thomas, 87060 Limoges cedex, France<EMAIL_ADDRESS>,
F. A. Carnicero  Departamento de Álgebra, Análisis Matemático, Geometría y
Topología. Universidad de Valladolid. Facultad de Ciencias. Paseo de Belén, 7,
E-47011. Valladolid, Spain<EMAIL_ADDRESS>and  F. Sanz Sánchez
Departamento de Álgebra, Análisis Matemático, Geometría y Topología.
Universidad de Valladolid. Facultad de Ciencias. Paseo de Belén, 7, E-47011.
Valladolid, Spain<EMAIL_ADDRESS>
###### Abstract.

We establish a real version of Turrittin’s result on polynomial and formal
normal forms of linear systems of ODEs with meromorphic coefficients. Both the
normal forms or the transformations used have only real coefficients. In order
to adapt the proofs to the real case, we make a review of the result in the
complex case.

###### Key words and phrases:

Linear systems of meromorphic ODEs, Formal Normal Forms, Turrittin’s Theorem

This work was supported by Ministerio de Ciencia e Innovación
(MTM2016-77642-C2-1-P and PID2019-105621GB-I00) and by Junta de Castilla y
León (VA083G19). The first author thanks UVa for the support during several
research stays at the Departamento de Álgebra, Análisis Matemático, Geometría
y Topología.

##  1\. Preliminaries and statements

Let $K$ be a field of characteristic zero and let $L_{K}=K[[x]][x^{-1}]$ 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 $\nu:L_{K}\to\mathbb{Z}\cup\\{\infty\\}$ 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 $n\in\mathbb{N}_{\geq 1}$,
$\mathcal{M}_{n}(R)$ denotes the ring of square matrices of size $n$ with
entries in $R$.

A matrix $A\in\mathcal{M}_{n}(L_{K})$ is identified with the formal
meromorphic linear system of ODEs

$[A]\;\;\;\;\;Y^{\prime}=AY,$

where $Y=(Y_{1},...,Y_{n})^{t}$ is a column vector of $n$ variables. Define
the order of $A$ to be $\nu(A):=\min\\{\nu(a_{ij})\,:\,1\leq i,j\leq n\\}$,
where $A=(a_{ij})$. 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

(1) $A=x^{\nu(A)}(A_{0}+xA_{1}+\cdots),\mbox{ where
}A_{i}\in\mathcal{M}_{n}(K)\mbox{ for any }i\mbox{ and }A_{0}\neq 0.$

Also, if $N$ is a non-negative integer, the truncated system up to degree $N$
is defined by

$J_{N}A:=x^{\nu(A)}(A_{0}+xA_{1}+\cdots+x^{N}A_{N}).$

The system $A$ is called singular (at $x=0$) if $\nu(A)<0$. The Poincaré rank
of the system is defined as the non-negative integer
$q=q(A):=\max\\{-\nu(A)-1,0\\}$. We usually rewrite $A$ as the system of
formal linear ODEs

$x^{q+1}Y^{\prime}=\widetilde{A}Y,\;\;\mbox{ where
}\;\;\widetilde{A}=A_{0}+xA_{1}+\cdots.$

A singular system with Poincaré rank $q=0$ (resp. $q>0$) is usually referred
to be of first kind (resp. of second kind).

Denote by $I_{n}$ the identity matrix of size $n$. Define the radiality index
of $A$ as the non-negative integer

$k=k(A):=\min(\\{j\,:A_{j}\not\in KI_{n}\\}\cup\\{q\\}).$

The truncation $J_{k-1}A$ 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:

  1. (i)

Given $P\in GL_{n}(L_{K})$, the linear change of variables $Y=PZ$ transforms
the system $[A]:Y^{\prime}=AY$ into the system $[B]:Z^{\prime}=BZ$ where

$B=P^{-1}AP-P^{-1}P^{\prime}.$

The map $\Psi_{P}:\mathcal{M}_{n}(L_{K})\to\mathcal{M}_{n}(L_{K})$ sending $A$
to $\Psi_{P}[A]:=P^{-1}AP-P^{-1}P^{\prime}$ is bijective. It is called the
gauge transformation associated to $P$. A gauge transformation $\Psi_{P}$ will
be called: regular, if $\nu(P)=0$ and $\det P(0)\neq 0$; polynomial, if each
entry of $P$ belongs to $K[x]$; diagonal monomial, if
$P=\mbox{diag}(x^{k_{1}},\cdots,x^{k_{n}})$ with $k_{j}\in\mathbb{N}_{\geq 0}$
for each $j$.

  2. (ii)

Given $r\in\mathbb{N}_{\geq 1}$, the change of the independent variable
$x=z^{r},$ transforms the system $\frac{dY}{dx}=A(x)Y$ into a system
$\frac{dY}{dz}=B(z)Y$, where

$B(z)=rz^{r-1}A(z^{r}).$

Re-written with the same letter $x$, we define the map
$R_{r}:\mathcal{M}_{n}(L_{K})\to\mathcal{M}_{n}(L_{K})$ given by
$R_{r}[A]:=rx^{r-1}A(x^{r})$, called the ramification of order $r$. It is an
injective map but not bijective for $r>1$.

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 $K=\mathbb{C}$ (or more generally, $K$ algebraically closed) is
classical and treated with different approaches in the literature (Birkhoff
[7], Hukuhara [10, 11], Turrittin [17], Wasow [19], Moser [15], Balser-
Jurkart-Lutz [3], Babbitt-Varadarajan [1], Barkatou [4, 5], Barkatou-Pflügel
[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.

Our objective in this paper is to extend Turrittin’s Theorem to the real case
$K=\mathbb{R}$, or more generally to the case where $K$ is a real closed
field. As far as we know, this case has not been treated yet (except, of
course, in the situation of a constant system $A\in\mathcal{M}_{n}(K)$ for
which the usual well known real Jordan canonical form of $A$ was proposed by
Turrittin himself in [18]). We present versions of real (formal and
polynomial) normal forms for any system, in such a way that they can be
obtained by transformations written in the base field $K$, without passing
through the algebraic closure $\overline{K}=K(\sqrt{-1})$.

###  1.1. The complex case

In order 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 [19]. 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
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 5, (b) below).

We start by defining the normal forms that we expect to get.

######  Definition 1 (Turrittin-Ramis-Sibuya form).

Let $A\in\mathcal{M}_{n}(L_{K})$ be a system with Poincaré rank $q=q(A)$ and
let $\mu\in\mathbb{N}_{\geq 0}$. We say that $A$ is in Turrittin-Ramis-Sibuya
form of degree $\mu$ (and of rank $q$), or in $(TRS)_{\mu}^{q}$-form for
short, if it is written as

$A(x)=x^{-(q+1)}\left(D(x)+x^{q}C+O(x^{q+\mu+1})\right),$

where $D(x)={\rm diag}(d_{1}(x),...,d_{n}(x))$ is a diagonal matrix with
polynomial entries $d_{j}(x)\in K[x]$ of degree at most $q-1$ (equal zero iff
$q=0$) and $C\in\mathcal{M}_{n}(K)$ is a constant matrix commuting with
$D(x)$. In this case, the truncated system $J_{q}(A)=x^{-(q+1)}(D(x)+x^{q}C)$
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 $C=A_{0}$. On the other hand, if $A$ is in
$(TRS)_{\mu}^{q}$-form for some $\mu$ and $q>0$ then its exponential part
$D(x)$ is not zero; in fact, it satisfies $D(0)\neq 0$.

The names Ramis and Sibuya in the definition above come from those authors
paper [16], 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 [12, 13] (where the name of “Ramis-Sibuya form” is
used).

######  Definition 2.

Let $C\in\mathcal{M}_{n}(K)$ be a constant square matrix with entries in $K$.
We say that $C$ is non-resonant111Other authors, for instance Balser in his
book [2], use instead the terminology “$C$ has good spectrum”. if for any pair
of distinct eigenvalues $\lambda,\lambda^{\prime}\in\overline{K}$ of $C$ we
have $\lambda-\lambda^{\prime}\not\in\mathbb{Z}$.

Now, Turrittin’s results for the case where $K=\overline{K}$ can be stated in
the two following theorems.

######  Theorem 3 (Complex Polynomial Normal Form).

Suppose that $K$ is an algebraically closed field and let
$A\in\mathcal{M}_{n}(L_{K})$ be a singular system with Poincaré rank equal to
$q=q(A)$.

  1. (i)

There exist some $r\in\mathbb{N}_{\geq 1}$ and finitely many polynomial gauge
transformations $\varphi_{1},...,\varphi_{m}$, either regular or diagonal
monomial, such that, denoting

$\psi=\varphi_{m}\circ\cdots\circ\varphi_{1}\circ R_{r},$

the transformed system $\widetilde{A}=\psi[A]$ is in
$(TRS)_{0}^{\tilde{q}}$-form, where $\tilde{q}=q(\widetilde{A})$. Moreover, if
$B\in\mathcal{M}_{n}(L_{K})$ is another singular system with $q(B)=q$ and
$J_{nq}A=J_{nq}B$ then $\widetilde{B}=\psi[B]$ is also in
$(TRS)_{0}^{\tilde{q}}$-form with the same principal part as $\widetilde{A}$;
i.e, $q(\widetilde{B})=\tilde{q}$ and
$J_{\tilde{q}}\psi[A]=J_{\tilde{q}}\psi[B].$

  2. (ii)

Assume that $A$ is in $(TRS)^{q}_{0}$-form and that its residual matrix is
non-resonant. Then, for any given $\mu\geq 0$, there exists a regular
polynomial gauge transformation $\phi_{\mu}=\Psi_{P^{\mu}}$, where
$P^{\mu}(0)=I_{n}$, such that $\phi_{\mu}[\widetilde{A}]$ is in
$(TRS)_{\mu}^{\tilde{q}}$-form with the same principal part as the original
system $A$. Moreover, the family $\\{P^{\mu}\\}_{\mu}$ can be chosen such that
$P^{\mu}$ is of degree at most $q+\mu$ and satisfying that
$J_{q+\mu}P^{\mu^{\prime}}=J_{q+\mu}P^{\mu}$ for any $\mu^{\prime}>\mu$.

  3. (iii)

Assume that $A$ is in $(TRS)^{q}_{0}$-form. Then there exists a gauge
transformation $\phi$, given by a finite composition of regular polynomial or
diagonal monomial transformations, such that $\phi[A]$ is in
$(TRS)^{q}_{0}$-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 4 (Complex Formal Normal Form).

Suppose that $K$ is an algebraically closed field and let
$A\in\mathcal{M}_{n}(L_{K})$ be a singular system with Poincaré rank equal to
$q=q(A)$. There exists a formal gauge transformation $\Psi_{P}$ and a
ramification $R_{r}$ such that the transformed system $F=(\Psi_{P}\circ
R_{r})[A]$ has Poinaré rank equal to $\tilde{q}=q(F)$ and can be written in a
Formal Normal Form

$[F]\hskip 28.45274ptY^{\prime}=x^{-(\tilde{q}+1)}(D(x)+x^{\tilde{q}}C)Y,$

where $D(x)$ and $C$ satisfy the requirements in Definition 1; i.e., $F$ is in
$(TRS)^{\tilde{q}}_{0}$-form and $J_{\tilde{q}}F=F$. Moreover, the
transformation $\Psi_{P}$ can be chosen to be equal to
$\Psi_{P}=\Psi_{Q}\circ\psi$, where $Q\in\mathcal{M}_{n}(K[[x]])$ with
$Q(0)=I_{n}$ and $\psi$ is a finite composition of regular polynomial or
diagonal monomial gauge transformations.

######  Remarks 5.

Concerning the statements in Theorem 3 and Theorem 4, we have the following.

  1. (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 [14].
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 $A$.

  2. (b)

To obtain the formal normal form $[F]$ for the system $A$ in Theorem 4 is
equivalent to say that there exists a matrix $P(t)\in\mathcal{M}_{n}(K[[t]])$
and some $r\in\mathbb{N}_{\geq 1}$ such that

(2)
$Z(x)=P(x^{1/r})\exp\left(\int\frac{D(x^{1/r})}{x^{(\tilde{q}+1))/r}}\right)x^{\frac{C}{r}}$

is a fundamental matrix of formal solutions of the system $A$.

  3. (c)

Another consequence of the expression (2) is that the ratio $\tilde{q}/r$ 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 $A$ and $B$ such that $B=\Psi_{T}[A]$ with $T\in
GL_{n}(K[[x]][x^{-1}])$ and we get (TRS)-normal forms of $A$ and $B$ as in
item (i) of Theorem 3 with resulting Poincaré ranks $\tilde{q}_{A}$ and
$\tilde{q}_{B}$ and exponential parts $D_{A}(x)$ and $D_{B}(x)$, respectively,
then there are integers $r_{1},r_{2}$ such that
$\tilde{q}_{A}/r_{1}=\tilde{q}_{B}/r_{2}$ and
$D_{A}(x^{1/r_{1}})=D_{B}(x^{1/r_{2}})$. In particular, $\tilde{q}=0$ iff the
system $A$ is equivalent to a system of first kind under a formal gauge
transformation.

  4. (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.

  5. (e)

In the proof below, we propose a bound, in terms of the eigenvalues of the
residual matrix $C$, for the degree of the transformation $\phi$ in item
(iii).

###  1.2. The real case

Suppose that $K$ is a real closed field, i.e., $K\varsubsetneq
K(i)=\overline{K}$, where $i=\sqrt{-1}$. Given $\lambda=a+bi\in\overline{K}$,
denote by

$\Lambda_{\lambda}=\left(\begin{array}[]{cc}a&-b\\\ b&a\end{array}\right).$

Recall that the characteristic polynomial of $\Lambda_{\lambda}$ has roots
$a\pm bi$ and is irreducible if and only if $\lambda\not\in K$, i.e., $b\neq
0$. For any $m\in\mathbb{N}_{\geq 1}$, define the monomorphism of $K$-algebras

$\Theta_{m}:\mathcal{M}_{m}(\overline{K})\to\mathcal{M}_{2m}(K),$

sending a matrix $C=(c_{uv})\in\mathcal{M}_{m}(\overline{K})$ to the $(2\times
2)$-block matrix $(\Lambda_{c_{uv}})\in\mathcal{M}_{2m}(K)$. A square matrix
in the image of $\Theta_{m}$ will be called a complex matrix over $K$, or a
$\mathbb{C}$-matrix, for short.

We extend $\Theta_{m}$ to a monomorphism of $K$-algebras, denoted with the
same letter, from $\mathcal{M}_{m}(L_{\overline{K}})$ into
$\mathcal{M}_{2m}(L_{K})$; that is, from formal meromorphic linear systems
over $\overline{K}$ 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 $\mathbb{C}$-system.

In what follows, if $U,V$ are two square matrices of sizes $k,l$,
respectively, we denote by $U\oplus V$ the square matrix of size $k+l$ given
in blocks

$U\oplus V:=\left(\begin{array}[]{cc}U&0\\\ 0&V\end{array}\right).$

######  Definition 6 (Real Turrittin-Ramis-Sibuya form).

Suppose thar $K$ is a real closed field. Let $A\in\mathcal{M}_{n}(L_{K})$ be a
system with Poincaré rank $q=q(A)$ and let $\mu\in\mathbb{N}_{\geq 0}$. We say
that $A$ is in Real Turrittin-Ramis-Sibuya form of degree $\mu$ (and of rank
$q$), or in $(\mathbb{R}TRS)_{\mu}^{q}$-form for short, if it can be written
in the form

$A(x)=x^{-(q+1)}\left(D_{1}(x)\oplus D_{2}(x)+x^{q}(C_{1}\oplus
C_{2})+O(x^{q+\mu+1})\right),$

where

  * •

$D_{1}(x)={\rm diag}(e_{1}(x),...,e_{n_{1}}(x))$ is diagonal polynomial with
entries $e_{j}(x)\in K[x]$ of degree at most $q-1$ (equal to zero if $q=0$).

  * •

$D_{2}(x)=\Theta_{n_{2}}\left({\rm diag}(d_{1}(x),...,d_{n_{2}}(x))\right)$ is
a diagonal $2\times 2$-block complex matrix such that the entries $d_{j}(x)$
belong to $\overline{K}[x]\setminus K[x]$ and are of degree at most $q-1$
(equal to zero if $q=0$).

  * •

$C_{1}$ and $C_{2}$ are constant matrices with entries in $K$ of sizes $n_{1}$
and $2n_{2}$, respectively, and $C_{2}$ is a $\mathbb{C}$-matrix.

  * •

$[D_{1}(x),C_{1}]=0$ and $[D_{2}(x),C_{2}]=0$.

In this case, the truncated system $J_{q}(A)=x^{-(q+1)}(D_{1}(x)\oplus
D_{2}(x)+x^{q}(C_{1}\oplus C_{2}))$ is called the principal part of $A$, while
$D_{1}(x)\oplus D_{2}(x)$, resp. $C_{1}\oplus C_{2}$, is called the
exponential part, resp. the residual matrix.

Our main result in the paper is the following real versions of Theorems 3 and
4.

######  Theorem 7 (Real Polynomial Normal Forms).

Suppose that $K$ is a real closed field and let $A\in\mathcal{M}_{n}(L_{K})$
be a singular system with Poincaré rank equal to $q=q(A)$.

  1. (i)

There exists $r\in\mathbb{N}_{\geq 1}$ and there exists finitely many
polynomial gauge transformations $\varphi_{1},...,\varphi_{m}$ (with
coefficients in $K$), either regular or diagonal monomial, such that, denoting

$\psi=\varphi_{m}\circ\cdots\circ\varphi_{1}\circ R_{r},$

the transformed system $\widetilde{A}=\psi[A]$ is in
$(\mathbb{R}TRS)_{0}^{\tilde{q}}$-form. Moreover, if
$B\in\mathcal{M}_{n}(L_{K})$ is another singular system with $q(B)=q$ and
$J_{nq}A=J_{nq}B$ then, with the same transformation $\psi$, the transformed
system $\widetilde{B}=\psi[B]$ is also in $(\mathbb{R}TRS)_{0}^{\tilde{q}}$
with the same principal part as $\widetilde{A}$, i.e.,
$J_{\tilde{q}}\psi[A]=J_{\tilde{q}}\psi[B].$

  2. (ii)

Assume that $A$ is in $(\mathbb{R}TRS)^{q}_{0}$-form and that its residual
matrix is non-resonant. Then, for any $\mu\geq 0$ there exists a regular
polynomial gauge transformation $\phi_{\mu}=\Psi_{P^{\mu}}$ where
$P^{\mu}(0)=I_{n}$ such that $\phi_{\mu}[A]$ is in
$(\mathbb{R}TRS)_{\mu}^{q}$-form and with the same principal part as the
system $A$. Moreover, the family $\\{P^{\mu}\\}_{\mu}$ can be chosen such that
$P^{\mu}$ is of degree at most $q+\mu$ and satisfying that
$J_{\mu}P^{\mu^{\prime}}=J_{\mu}P^{\mu}$ for any $\mu^{\prime}>\mu$.

  3. (iii)

Assume that $A$ is in $(\mathbb{R}TRS)^{q}_{0}$-form. Then there exists a
polynomial gauge transformation $\phi$, given by a finite composition of
regular or diagonal monomial transformations, such that $\phi[A]$ is in
$(\mathbb{R}TRS)^{q}_{0}$-form with non-resonant residual matrix (and the same
exponential part as $A$).

######  Theorem 8.

Let $K$ be a real closed field and let $A\in\mathcal{M}_{n}(L_{K})$ be a
singular system. Then there exists a formal gauge transformation $\psi_{P}$,
with $P\in L_{K}$, and a ramification $R_{r}$ such that the transformed system
$F^{\mathbb{R}}=(\psi_{P}\circ R_{r})[A]$ has Poincaré rank equal to
$\tilde{q}$ and is written as:

$[F^{\mathbb{R}}]\hskip 28.45274ptY^{\prime}=x^{-(\tilde{q}+1)}(D_{1}(x)\oplus
D_{2}(x)+x^{\tilde{q}}(C_{1}\oplus C_{2}))Y,$

where $D_{1},D_{2},C_{1},C_{2}$ satisfy the conditions in Definition 6; that
is $F^{\mathbb{R}}$ is in $(\mathbb{R}TRS)$-form and
$F^{\mathbb{R}}=J_{\tilde{q}}F^{\mathbb{R}}$. Moreover, the transformation
$\psi_{P}$ can be chosen to be equal to $\Psi_{P}=\Psi_{Q}\circ\psi$, where
$Q\in\mathcal{M}_{n}(K[[x]])$ satisfies $Q(0)=I_{n}$ and $\psi$ is a finite
composition of regular polynomial or diagonal monomial gauge transformations
(with coefficients in $K$).

##  2\. Proof of the Complex Turrittin’s Theorem

Let $A(x)\in\mathcal{M}_{n}(L_{K})$ by a system with Poincaré rank $q=q(A)$,
written as in (1). Let $k=k(A)$ be the radiality index. Since the radial part
$A_{0}+xA_{1}+\cdots+x^{k-1}A_{k-1}$ is preserved by any gauge transformation,
the coefficient $A_{k}$ 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 $A_{k}$
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.

Denote $N(n,q,k)=n(q-k)+k$ as in Remark 5, (a) and consider the statement (i)’
to be the same as item (i) in Theorem 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 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 9.

Fix any field $\kappa$. Let $r\in\mathbb{N}_{\geq 1}$. Let
$P(x)\in\mathcal{M}_{n}(\kappa[x])$ and let $\Psi_{P}$ be the associated
polynomial gauge transformation. Then there exists another polynomial gauge
transformation $\Psi_{\tilde{P}}$ satisfying
$R_{r}\circ\Psi_{P}=\Psi_{\tilde{P}}\circ R_{r}$. In fact, we can take
$\tilde{P}(x):=P(x^{r})$. In particular, $\Psi_{P}$ is regular or diagonal
monomial iff $\Psi_{\tilde{P}}$ 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 [19, 2, 5]), but it has a
finitely determined nature in terms of truncations of the system.

######  Lemma 10 (Splitting Lemma).

Let $K$ be any field. With the same notations as above, if $k$ is the
radiality index of the system $A$, assume that $k<q$ and that $A_{k}$ is
conjugated to $A^{11}_{k}\oplus A^{22}_{k}$, where the characteristic
polynomials $\chi_{A^{11}_{k}}(\lambda)$ and $\chi_{A^{22}_{k}}(\lambda)$ are
coprime, both of positive degrees, say $n_{1}$ and $n_{2}$, respectively. Then
there exists a formal regular gauge transformation $\Psi_{T}$, where
$T\in\mathcal{M}_{n}(K[[x]])$ satisfies $T(0)=I_{n}$, such that the
transformed system $B=\Psi_{T}[A]$ writes as $B=B^{11}\oplus B^{22}$, where
$B^{ii}\in\mathcal{M}_{n_{i}}(L_{K})$ is a system of dimension $n_{i}$ for
$i=1,2$. Moreover, $q(B)=q$, $k(B)=k$ and, writing
$B(x)=x^{-(q+1)}(\sum_{j\geq 0}x^{j}B_{j})$, we have $A_{j}=B_{j}$ for
$j=0,1,...,k$ and for any $m>k$, the truncation $J_{m}B$ only depends on
$J_{m-k}T$ and $J_{m}A$. In other words, if $\widetilde{A}$ is another system
with the same Poincaré rank $q(\widetilde{A})=q$ and satisfying
$J_{m}\widetilde{A}=J_{m}A$ then
$J_{m}\Psi_{T^{(m-k)}}[\widetilde{A}]=J_{m}B$, where $T^{(m-k)}=J_{m-k}T$.

###  2.1. Proof of Theorem 3, (i)’. Getting a $(TRS)$-form of degree $0$ and
some rank $\tilde{q}$

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 $(TRS)_{0}^{q}$-form.

We proceed by induction on the dimension $n$ of the system. The starting case
$n=1$ is also trivial. Assume then that $n>1$.

####  2.1.1. Case with different eigenvalues

Suppose that we are in the case where $A_{k}$ 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
$A_{k}=A^{11}_{k}\oplus A^{22}_{k}$ where $A^{11}_{k}$ and $A^{22}_{k}$ are
matrices of respective sizes $n_{1},n_{2}$, both smaller than $n$, and having
no common eigenvalue. Using Lemma 10 for $N=N(n,q,k)$, there exists a regular
polynomial gauge transformation $\Psi_{P}$ such that the $N$-truncation of
$B=\Psi_{P}(A)$ is written as $J_{N}B=B^{11}\oplus B^{22}$, where, for
$i=1,2$, $B^{ii}$ is a system of dimension $n_{i}$. Moreover, $J_{N}B$ has the
same Poincaré rank, the same radiality index and the same $k$-truncation than
$A$. In particular, if $q_{i}=q(B^{ii})$ and $k_{i}=k(B^{ii})$ then $q_{i}\leq
q$ and $q_{i}-k_{i}\leq q-k$. Taking into account that $n_{1}$ and $n_{2}$ are
both positive and hence strictly smaller than $n$, we obtain for $i=1,2$ that

$N(n,q,k)\geq n_{i}(q-k)+q\geq n_{i}(q_{i}-k_{i})+q_{i}\geq
n_{i}(q_{i}-k_{i})+k_{i}=N(n_{i},q_{i},k_{i}).$

Using the induction hypothesis to system $B^{ii}$ for $i=1,2$, there is a
finite composition $\psi^{ii}$ of transformations in dimension $n_{i}$ (as in
statement (i)) such that $\widetilde{B}^{ii}=\psi^{ii}(B^{ii})$ is in
$(TRS)_{0}^{q_{i}}$-form and such that the second part of statement (i)’ holds
for the truncation order $N(n_{i},q_{i},k_{i})$ in the place of $n_{i}q_{i}$.
Moreover, in the composition $\psi^{ii}$ there is but a single ramification
$R_{r_{i}}$ (with $r_{i}\in\mathbb{N}_{\geq 1}$, including the case
$R_{r_{i}}=id$ ($r_{i}=1$). Now, for $\\{i,j\\}=\\{1,2\\}$ write, using Lemma
9,

$R_{r_{j}}\circ\psi^{ii}=\widetilde{\varphi}^{ii}\circ R_{r_{1}r_{2}},$

where $\widetilde{\varphi}^{ii}$ is a composition of transformations in
dimension $n_{i}$, either regular polynomial or diagonal monomial (that is, no
ramification). Notice that the composition $R_{r_{j}}\circ\psi^{ii}$ satisfies
the requirements of Theorem 3, (i)’ for the system $R_{r_{j}}(B^{ii})$, for
which the Poincaré rank and radiality index are equal to $r_{j}q_{i}$ and
$r_{j}k_{i}$, respectively. Writting $\widetilde{\varphi}^{ii}=\Psi_{Q_{i}}$,
where $Q_{i}\in\mathcal{M}_{n_{i}}(K[x])$, we put $r=r_{1}r_{2}$ and define

$\psi=\Psi_{Q_{1}\oplus Q_{2}}\circ R_{r}\circ\Psi_{P}=\varphi\circ R_{r},$

where $\varphi=\Psi_{Q_{1}\oplus Q_{2}}\circ\Psi_{P(x^{r})}$ is a composition
of gauge transformations, either regular polynomial or diagonal monomial. We
check that $\psi$ satisfies the requirements of statement (i)’ for the initial
system $A$. To be convinced, we need to observe two facts. In one hand, for
$\\{i,j\\}=\\{1,2\\}$, the composition $R_{r_{j}}\circ\psi^{ii}$ satisfies all
the requirements of (i)’ for the system $B^{ii}$, since so does $\psi^{ii}$ by
construction (notice that if $B$ is any system then we have
$q(R_{r}(B))=rq(B)$, $k(R_{r}(B))=rk(B)$ and for any $M\geq 0$, the truncation
$J_{rM}R_{r}(B)$ is univocally determined by $J_{M}B$). On the other hand, use
the second part of Lemma 10 to conclude that $J_{N}\Psi_{P}(A)$ only depends
on $J_{N}A$ for $N=N(n,q,k)$ and the inequality $N(n_{i},q_{i},k_{i})\leq N$
for $i=1,2$ proved above.

####  2.1.2. Case with a single eigenvalue

Suppose now that $A_{k}$ has a single eigenvalue $\lambda_{k}\in\overline{K}$.
Recall that we are assuming that $K=\overline{K}$. So, up to a constant gauge
transformation, we may suppose that $A_{k}$ is in Jordan normal form.
Explicitly, there exists a (unique) sequence $1\leq n_{1}\leq
n_{2}\leq\cdots\leq n_{\ell}\leq n$ with $n=n_{1}+\cdots+n_{\ell}$ such that

(3)
$A_{k}=(\lambda_{k}I_{n_{1}}+H^{(n_{1})})\oplus\cdots\oplus(\lambda_{k}I_{n_{\ell}}+H^{(n_{\ell})}),$

where

$H^{(n_{j})}=\left(\begin{array}[]{ccccc}0&1&&\cdots&0\\\ 0&0&1&\cdots&0\\\
&&\vdots&&\\\ 0&0&\cdots&0&1\\\ 0&0&\cdots&0&0\end{array}\right)_{n_{j}\times
n_{j}}$

(each such matrix will be called in the sequel a shifting matrix). Notice also
that, since $A_{k}$ is not a radial matrix, we have $n_{j}>1$ 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
$\gamma_{i}(A_{k})\in\mathbb{N}_{\geq 0}$ the degree, as a polynomial in
$\lambda$, of the g.c.d. of the family of all $i\times i$ minors of the
characteristic matrix $A_{k}-\lambda I_{n}$. In particular
$\gamma_{n}(A_{k})=n$. For such a system $A$ (when $A_{k}$ has a single
eigenvalue), we define the following tuple of non-negative integer numbers

$I(A):=(\gamma_{1}(A_{k}),\cdots,\gamma_{n}(A_{k}),q-k).$

Remark that each $\gamma(A_{k})$ depends only on the conjugation class of
$A_{k}$. We need to recall also the following result on Linear Algebra (see in
Wasow [19, Lemma 19.4] for a proof) concerning the behaviour of the values
$\gamma_{i}$ for a perturbation of the matrix $A_{k}$ in the case where
$A_{k}$ has at least two blocks.

######  Lemma 11.

Consider $A_{k}$ as a block-diagonal Jordan matrix
$A_{k}=\left(A_{k}^{(ij)}\right)$, where the diagonal blocks are given by
$A_{k}^{(ii)}=\lambda_{k}I_{n_{i}}+H^{(n_{i})}$. Assume that $\ell\geq 2$.
With the same block structure, let
$G=\left(G^{(ij)}\right)\in\mathcal{M}_{n}(K)$ be a block-lower-triangular
matrix with the same diagonal blocks as $A_{k}$ (that is,
$G^{(ii)}=A_{k}^{(ii)}$ and $G^{(ij)}=0$ if $i<j$). Then we have

$\gamma_{t}(G)\leq\gamma_{t}(A_{k})\;\;\;\mbox{ for any }t\in\\{1,2,...,n\\}$

and the inequality is strict for at least one $t$ if $G\neq A_{k}$ (that is,
$G^{(ij)}\neq 0$ for at least a pair $(i,j)$ with $i>j$).

Step 2. Special matrices and choice of $g\in\mathbb{Q}$.- Given a tuple
$\sigma=(n_{1},n_{2},\cdots,n_{\ell})$ of positive integers as above, denote
$m_{j}:=\sum_{u=1}^{j}n_{u}$ for $j=1,...,\ell$. A matrix
$T=(t_{uv})\in\mathcal{M}_{n}(K)$ will be called a special matrix of type
$\sigma$ if $t_{uv}=0$ for every $(u,v)$ such that
$u\not\in\\{m_{1},m_{2},...,m_{\ell}\\}$.

A result in Wasow ([19, Lemme 19.2]) assures that for any $N\geq k+1$ there
exists a regular polynomial gauge transformation $\Psi_{P_{N}}$, where $P_{N}$
is of degree $N$ and $P_{N}(0)=I_{n}$, such that the transformed system
$B=\Psi_{P_{N}}(A)$ satisfies that all coefficients of the truncation
$J_{N}(B)$ are special matrices of type
$\sigma=(n_{1},n_{2},\cdots,n_{\ell})$, where the $n_{j}$ are the sizes of
Jordan blocks of the matrix $A_{k}$. We will put $N=N(n,q,k)$ and, renaming
$B$ again as the system $A$, we may assume the following assumption

> (*) For any $k+1\leq j\leq N$, the coefficient $A_{j}$ is a special matrix
> of type $\sigma$.

Write now

$x^{q+1}A(x)=A_{0}+xA_{1}+\cdots=\sum_{i=0}^{k-1}\lambda_{i}x^{i}I_{n}+x^{k}\overline{A},\;\mbox{
with }\overline{A}=A_{k}+O(x)=(\overline{a}_{uv}(x)).$

Put $\alpha_{uv}:=\nu(\overline{a}_{uv}(x))$. Notice that
$\alpha_{uv}\in\mathbb{N}_{\geq 0}$ for any $u,v$, that $\alpha_{uu}>0$ for
any $u$ and that $\alpha_{u,u+1}=0$ for at least one index
$u\in\\{1,...,n\\}$.

######  Definition 12.

With the conditions above, we define the shearing order (of $A$) as the
rational number

$g=g(A):=\min\left(\\{\frac{\alpha_{uv}}{1+u-v}\,:\,u>v\\}\cup\\{\alpha_{uu}\,:\,1\leq
u\leq n\\}\cup\\{q\\}\right).$

Geometrically, as discussed in Wasow [19, pp.104-105], the shearing order is
the smallest abscissa in which a line of the following family

$\\{y=(v-u)x+\alpha_{uv}\\}_{v<u}\cup\\{y=\alpha_{uu}\\}_{u}\cup\\{y=q-k\\}$

cuts the diagonal $y=x$.

Notice also that $g\in\mathbb{Q}$ and that $0<g\leq q-k$. Write
$g=\frac{h}{r}$ where $h,r$ are positive integers with no common factor.

The name “shearing” comes from Wasow’s name for the gauge “ramified”
transformation with matrix
$S_{g}=\text{diag}\,(1,x^{g},x^{2g},...,x^{(n-1)g})$. In our definitions, such
a transformation has only sense if $g=h$ is an integer. Otherwise, we perform
instead the composition $\Psi_{S_{h}}\circ R_{r}(=:``\Psi_{S_{g}}")$.

Step 3. The case where $g=h$ is integer. We consider the monomial diagonal
gauge transformation $\Psi_{S_{h}}$ where
$S_{h}=\text{diag}\,(1,x^{h},x^{2h},...,x^{(n-1)h})$. If $A=(a_{uv}(x))$ then
the transformed system
$B:=\Psi_{S_{h}}[A]=S_{h}^{-1}AS_{h}-S_{h}^{-1}S^{\prime}_{h}$ writes as

(4)
$B=\left(\begin{array}[]{cccc}a_{11}(x)&x^{h}a_{12}(x)&\cdots&x^{(n-1)h}a_{1n}(x)\\\
x^{-h}a_{21}(x)&a_{22}(x)&\cdots&x^{(n-2)h}a_{2n}(x)\\\ &\vdots&&\\\
x^{-(n-1)h}a_{n1}(x)&x^{-(n-2)h}a_{12}(x)&\cdots&a_{nn}(x)\end{array}\right)+x^{-1}K,$

where $K$ is a constant diagonal matrix (in fact
$K=\text{diag}(0,h,2h,\dots,(n-1)h)$).

By the choice of the shearing order $g=h$, one can see that we can write
$B(x)=x^{-(q+1)}(\widetilde{B}_{0}+x\widetilde{B}_{1}+\cdots)$, where the
$\widetilde{B}_{j}$ are constant matrices satisfying

  1. (a)

$\widetilde{B}_{j}$ is diagonal for $j\in\\{0,1,...,k+h-1\\}$, with
$\widetilde{B}_{j}=A_{j}$ if $j<k$, $\widetilde{B}_{k}=\lambda_{k}I_{n}$ and
$\widetilde{B}_{j}=0$ if $k<j<k+h$.

  2. (b)

The entries of $\widetilde{B}_{k+h}$ above the principal diagonal coincide
with those of the matrix $H^{(n_{1})}\oplus\cdots\oplus H^{(n_{\ell})}$.
Moreover, $\widetilde{B}_{k+h}$ has a non-zero entry on or above the principal
diagonal, except, possibly, if $h=q-k$.

  3. (c)

For any $M\geq k$, the matrix $\widetilde{B}_{M}$ depends only on the
truncation $J_{M+(n-1)h}(A)$.

As a consequence, the Poincaré rank after the shearing transformation does not
increase. That is,

(5) $q(B)\leq q.$

Moreover, using (a) and (b), and denoting $q^{\prime}=q(B)$,
$k^{\prime}=k(B)$, we have that if $q^{\prime}=q$ then $k^{\prime}=k+h$,
while, if $q^{\prime}<q$ then $k=0$, $q^{\prime}=q-h$ and $k^{\prime}=0$. In
any case, we deduce

(6) $q^{\prime}-k^{\prime}<q-k\;\;\mbox{ and
}\;\;N(n,q^{\prime},k^{\prime})+(n-1)h\leq N(n,q,k).$

Thus, using the property (c) and the second part of equation (6), it will
suffice to prove item (i)’ of Theorem 3 for the system $B$.

Write, accordingly to our main notations,
$B(x)=x^{-(q^{\prime}+1)}\left(B_{0}+xB_{1}+\cdots\right)$. We will be done if
$B$ is in the trivial case $q^{\prime}-k^{\prime}=0$ or if
$q^{\prime}-k^{\prime}>0$ but $B_{k^{\prime}}$ has at least two different
eigenvalues. Thus, assume that $q^{\prime}-k^{\prime}>0$ and that
$B_{k^{\prime}}$ has a single eigenvalue, say equal to
$\lambda_{k^{\prime}}^{\prime}$. It is enough, by recurrence, to prove in this
case that the tuple $I(B)$ satisfies $I(B)<I(A)$.

Notice that $B_{k^{\prime}}=\widetilde{B}_{k+h}$, after the discussion above
concerning the value of $q^{\prime}$ and $k^{\prime}$. Consider the matrix
$B_{k^{\prime}}$ written in blocks according to the Jordan structure of
$A_{k}$:

$B_{k^{\prime}}=\left(B_{k^{\prime}}^{(ij)}\right)_{1\leq
i,j\leq\ell},\;\;B_{k^{\prime}}^{(ij)}\in\mathcal{M}_{n_{j}\times n_{j}}(K).$

Using the property (b), we have that
$B_{k^{\prime}}=H^{(n_{1})}\oplus\cdots\oplus H^{(n_{1})}+T$ where $T$ is a
lower triangular matrix. In particular, for $i\in\\{1,...,\ell\\}$, since
$\mbox{Spec}(B_{k^{\prime}}^{(ii)})=\\{\lambda_{k^{\prime}}^{\prime}\\}$, the
matrix $B_{k^{\prime}}^{(ii)}-\lambda_{k^{\prime}}^{\prime}I_{n_{i}}$ is
nilpotent and of rank $n_{i}-1$. This implies that the Jordan decomposition of
$B_{k^{\prime}}^{(ii)}$ has a single block and hence $B_{k^{\prime}}^{(ii)}$
is conjugated to $H^{n_{i}}+\lambda_{k^{\prime}}^{\prime}I_{n_{i}}$.
Consequently, $B_{k^{\prime}}$ is conjugated to a block-lower-triangular
matrix with diagonal blocks equal to
$H^{n_{i}}+\lambda_{k^{\prime}}^{\prime}I_{n_{i}}$, those of the matrix
$A_{k}$. Thus, either $B_{k^{\prime}}$ is conjugated to $A_{k}$ (when
$\ell=1$) or we are in the situation where we can apply Lemma 11. From this
lemma, and taking into account the first part of equation (6), 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 $g=\frac{h}{r}$ where $h,r$ are positive
integers without common factor and such that $r\geq 2$. We remark that the
condition that $g$ is not an integer implies that $g$ is given by one of the
quotients $\alpha_{uv}/(1-u-v)$ in Definition 12, while $\alpha_{uu}>g$ for
any $u$, as well as $q-k>g$.

Consider the ramification $R_{r}$ and put $\widetilde{A}=R_{r}(A)$. This
system has Poincaré rank and radiality index equal to $q(\widetilde{A})=rq$
and $k(\widetilde{A})=rk$, respectively. Moreover, if we denote
$\widetilde{N}=N(n,rq,rk)$ and $N=N(n,q,r)$, then $\widetilde{N}=rN$ but the
truncation $J_{\widetilde{N}}(\widetilde{A})$ only depends on $J_{N}(A)$,
showing that if item (i)’ of Theorem 3 holds for $\widetilde{A}$ then it also
holds for $A$.

We notice moreover that the first non-radial coefficient of $\widetilde{A}$ is
equal to $A_{k}$ and that $\widetilde{A}$ also satisfies assumption (*) with
respect to the same type $\sigma=(n_{1},...,n_{\ell})$ given by the Jordan
structure of $A_{k}$. On the other hand, the shearing order of $\widetilde{A}$
is given by $g(\widetilde{A})=rg=h$, a natural number (the new valuations
$\alpha_{uv}$ for $\widetilde{A}$ are all multiplied by $r$). Hence we are,
for $\widetilde{A}$, in the situation of step 3. However, the last component
$q-k$ of the tuple $I(\widetilde{A})$ has increased and we can not conclude
automatically. To finish, we put $B=\Psi_{S_{h}}(\widetilde{A})$, we denote by
$q^{\prime},r^{\prime}$ the Poincaré rank and the radiality index of $B$,
respectively, and, writing
$B=x^{-(q^{\prime}+1)}\left(B_{0}+xB_{1}+\cdots\right)$, we show again that
assuming that $q^{\prime}-k^{\prime}>0$ and that $B_{k^{\prime}}$ has a single
eigenvalue, we have $I(B)<I(A)$.

Write $B_{k^{\prime}}=\left(B^{(ij)}_{k^{\prime}}\right)$ as a block matrix in
the same block structure as $A_{k}$. We have (cf. property (b) in step 3
above) that $B_{k^{\prime}}$ is block-lower triangular with diagonal blocks
given by $B^{(ii)}_{k^{\prime}}=H^{(n_{i})}+T_{i}$, where $T_{i}$ is a lower
triangular matrix of size $n_{i}$. Moreover, we must have that all elements in
the diagonal of $T_{i}$ are zero (since all values $\alpha_{uu}$ for
$\widetilde{A}$ are greater than $h=rg$, as mentioned above). On the other
hand, by assumption (*), the entries of $T_{i}$ on any row except possibly the
last one are also zero. On the other hand, since we have assumed that each
block $B^{(ii)}_{k^{\prime}}$, as the entire matrix $B_{k^{\prime}}$, has a
single eigenvalue, being the trace of $B^{(ii)}_{k^{\prime}}$ equal to zero,
such eigenvalue is equal to zero. That is, each $B^{(ii)}_{k^{\prime}}$ is
nilpotent. But this implies that $T_{i}=0$ and we conclude that

(7) $B^{(ii)}_{k^{\prime}}=H^{(n_{i})},\;\;\mbox{ for any
}i\in\\{1,2,...,\ell\\}.$

Furthermore, by the definition of the shearing order of $\widetilde{A}$, we
must have at least one non-zero entry below the principal diagonal of the
matrix $B_{k^{\prime}}$. Together with the equation (7), this implies that
$\ell>1$ and that $B_{k^{\prime}}$ is a matrix $G$ in the situation of Lemma
11 with a non-diagonal block $B^{(ij)}_{k^{\prime}}\neq 0$ for at least one
pair $(i,j)$ with $i>j$. We conclude that
$\gamma_{t}(B_{k^{\prime}})<\gamma_{t}(A_{k})$ for at least one index $t$ and
thus $I(B)<I(A)$, as wanted.

Let us show now the statement in Remark 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 $\Psi_{S_{g}}$ with
$g\in\mathbb{N}$.

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 $A_{k}$ has a unique eigenvalue. If the shearing order $g=g(A)$ is an
integer, the required property is already established by equation (5). 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 $\Psi_{S_{h}}$ after the
ramification $R_{r}$. We conclude using the same equation (5) once we observe
that the system $\widetilde{A}:=R_{r}[A]$ has as first non-radial term the
same matrix $A_{k}$ and satisfies $g(\widetilde{A})=h\in\mathbb{N}$ (just
check that $\widetilde{A}$ satisfies already the property (*) and that the new
values $\alpha_{uv}$ in Definition 12 are the old ones multiplied by $r$).

###  2.2. Proof of Theorem 3, (ii). Getting $(TRS)$-form of higher degree
$\mu$ when $C$ is non-resonant

A proof of this item (ii) when $q=0$ can be found for instance in Wasow [19,
Thm. 5.1], Coddington-Levinson [9, 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 $(TRS)^{q}_{0}$-form with principal
part $D(x)+x^{q}C$ and write it in the form
$x^{q+1}Y^{\prime}=\left(A_{0}+xA_{1}+\cdots\right)Y,$ so that

$x^{q+1}J_{q}A=A_{0}+\cdots+x^{q}A_{q}=D(x)+x^{q}C.$

We will need the following result from Linear Algebra (which is actually the
core of the proof of the Splitting Lemma 10). See [19, Thm. 4.1] for a proof.

######  Lemma 13.

Fix a field $K$ and let $R,S$ be two square matrices with coefficients in $K$
and of sizes $n\times n$ and $m\times m$, respectively. Assume that $R,S$ have
no common eigenvalue in the algebraic closure $\overline{K}$ of $K$. Then the
linear map $X\mapsto RX-XS$ from $\mathcal{M}_{n\times m}(K)$ to itself is an
isomorphism.

Up to reorder the variables of $Y$, we write

(8) $D(x)=D^{11}(x)\oplus D^{22}(x)\oplus\cdots\oplus D^{\ell\ell}(x),$

where $D^{jj}(x)$ is a radial matrix of size $n_{j}$ (that is
$D^{jj}(x)=Q_{j}(x)I_{n_{j}}$, where $Q_{j}(x)\in K[x]_{q-1}$) for each
$j=1,..,\ell$, and such that $D^{ii}(x)\neq D^{jj}(x)$ if $i\neq j$. Write
also the residual matrix $C$, as well as any coefficient matrix $A_{j}$ with
$j\geq q+1$, as block matrices $C=(C^{uv})_{1\leq u,v\leq\ell}$ and
$A_{j}=(A_{j}^{uv})_{1\leq u,v\leq\ell}$, where
$C^{uv},A_{j}^{uv}\in\mathcal{M}_{n_{u}\times n_{v}}(K)$. Using the
commutativity property $[D(x),C]=0$, the assumption $D^{ii}(x)\neq D^{jj}(x)$
for $i\neq j$ and Lemma 13, we obtain that

$C^{uv}=0\;\;\mbox{ if }u\neq v.$

We eliminate all coefficients $A_{j}$ for $j\geq q+1$ by means of a regular
formal transformation in two steps.

First step. We eliminate all non-diagonal blocks $\\{A_{j}^{uv}\\}_{u\neq v}$.
We proceed by induction 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

(9)
$D(x)=\overline{D}^{11}(x)\oplus\cdots\oplus\overline{D}^{\ell_{1}\ell_{1}}(x)$

in such a way that $\overline{D}^{jj}(0)$ is a radial constant matrix for
every $j=1,..,\ell_{1}$ and $\overline{D}^{jj}(0)\neq\overline{D}^{ii}(0)$ if
$i\neq j$. Notice that, up to reordering, each block $\overline{D}^{jj}(x)$ is
formed by several of the diagonal blocks of the decomposition (8). We consider
a matrix $\overline{T}(x)\in\mathcal{M}_{n}(K[[x]])$ of the form
$\overline{T}(x)=I_{n}+x^{q+1}\overline{T}_{q+1}+x^{q+2}\overline{T}_{q+2}+\cdots$
such that, writing each coefficient $\overline{T}_{j}=(\overline{T}_{j}^{uv})$
in the same block structure as the one in (9), we have
$\overline{T}_{j}^{uu}=0$ for any $u\in\\{1,...,\ell_{1}\\}$. The regular
gauge transformation $\psi_{\overline{T}}$ transforms the system $A$ into a
system $\overline{B}:=\psi_{\overline{T}}[A]$ which is also in
$(TRS)^{q}_{0}$-form with the same principal part $D(x)+x^{q}C$. Write
$x^{q+1}\overline{B}=D(x)+x^{q}C+\sum_{j\geq q+1}x^{j}\overline{B}_{j}$, and,
for any $j$, consider each coefficient of $A$ or $\overline{B}$ written in the
block structure (9) as $A_{j}=(\overline{A}_{j}^{uv})$ and
$\overline{B}_{j}=(\overline{B}_{j}^{uv})$, respectively. We obtain
recursively, for any couple of indices $u,v\in\\{1,...,\ell_{1}\\}$ with
$u\neq v$:

(10)
$\begin{array}[]{c}\overline{B}_{q+1}^{uv}=\overline{A}_{q+1}^{uv}+\overline{D}_{0}^{uu}\overline{T}_{q+1}^{uv}-\overline{T}_{q+1}^{uv}\overline{D}_{0}^{vv}\\\
\overline{B}_{q+2}^{uv}=\overline{A}_{q+2}^{uv}+\overline{D}_{0}^{uu}\overline{T}_{q+2}^{uv}-\overline{T}_{q+2}^{uv}\overline{D}_{0}^{vv}+\theta^{uv}_{2}(A_{0},A_{1},\overline{T}_{q+1})\\\
\vdots\\\
\overline{B}_{q+j}^{uv}=\overline{A}_{q+j}^{uv}+\overline{D}_{0}^{uu}\overline{T}_{q+j}^{uv}-\overline{T}_{q+j}^{uv}\overline{D}_{0}^{vv}+\theta^{uv}_{j}(\\{A_{s},\overline{T}_{q+s}\\}_{s<j})\\\
\vdots\end{array}$

where each $\theta^{uv}_{j}$ is a polynomial matrix acting on the block
entries of the explicitly indicated matrices. Using Lemma 13 with
$R=\overline{D}^{uu}_{0}$ and $S=\overline{D}^{vv}_{0}$, we obtain recursively
blocks $\overline{T}^{uv}_{q+1},\overline{T}^{uv}_{q+2},\ldots$ (i.e. the
entire matrix $\overline{T}(x)$) so that $\overline{B}^{uv}_{j}=0$ for every
$j\geq q+1$ and for every $u,v\in\\{1,...,\ell_{1}\\}$ with $u\neq v$. In
other words, $\overline{B}=\psi_{\overline{T}}[A]$ is a block-diagonal formal
system
$\overline{B}=\overline{B}^{11}\oplus\cdots\oplus\overline{B}^{\ell_{1}\ell_{1}}$.
Now, we consider separately each one of the subsystems $\overline{B}^{jj}$,
for $j=1,...,\ell_{1}$. By construction, this system is in (TRS)-form of
degree $0$ and with exponential part equal to
$\overline{D}^{jj}(x)=\overline{D}^{jj}_{0}+\widetilde{D}^{jj}(x)$. Put
$\widetilde{B}^{jj}:=\overline{B}^{jj}-x^{-(q+1)}\overline{D}^{jj}_{0}$, a
system with Poincaré rank equal to
$q(\widetilde{B}^{jj})=q-\nu(\widetilde{D}^{jj}(x))<q$, and also in (TRS)-form
of degree $0$ and exponential part equal to
$x^{-\nu_{j}}\widetilde{D}^{jj}(x)$, where
$\nu_{j}:=\nu(\widetilde{D}^{jj}(x))$. Consider a block decomposition

(11)
$\widetilde{D}^{jj}(x)=\widetilde{D}^{jj,1}(x)\oplus\cdots\oplus\widetilde{D}^{jj,\ell^{j}}(x)$

analogous to that of $D(x)$ in equation (8); that is, each block in (11) is a
radial matrix and two such blocks are different. By recurrence on $q$, there
is a formal regular gauge transformation with matrix $T^{j}(x)=I+\cdots$ (with
size equal to the size of the system $\overline{B}^{jj}$) such that
$\Psi_{T^{j}}[\widetilde{B}^{jj}]$ is block-diagonal in the same block
structure as (11). Taking into account that $\overline{D}^{jj}_{0}$ is a
radial matrix, the same is true for $\Psi_{T^{j}}[\overline{B}^{jj}]$. Put

$T(x)=(T^{1}(x)\oplus\cdots\oplus
T^{\ell_{1}})\cdot\overline{T}(x)=I+\cdots\in\mathcal{M}_{n\times n}(K[[x]]).$

Then, $B:=\Psi_{T}[A]$ is a system in $(TRS)^{q}_{0}$-form with the same
principal part as $A$ and block-diagonal with respect to the concatenation of
the different block-structures given by (11); i.e.,
$D(x)=\bigoplus_{j=1}^{\ell_{1}}\widetilde{D}^{jj}(x)=\bigoplus_{j=1}^{\ell_{1}}\bigoplus_{i=1}^{\ell^{j}}\widetilde{D}^{jj,i}.$
We are done, since this last decomposition is, up to reordering, the same as
the initial one (8).

Step 2. Eliminating the diagonal terms. Consider the block-diagonal system
$B=\Psi_{T}[A]=B^{11}\oplus\cdots\oplus B^{\ell\ell}$ as in the step 1, in the
block structure given by (8). We propose a second formal regular gauge
transformation associated to $U\in\mathcal{M}_{n}(K[[x]])$ of the form

$U=(I_{n_{1}}+U^{11})\oplus\cdots\oplus(I_{n_{\ell}}+U^{\ell\ell}),$

where $U^{uu}$ has size $n_{u}$ and $U^{uu}(0)=0$ for any $u$. Write
$U^{uu}=xU^{uu}_{1}+x^{2}U^{uu}_{2}+\cdots$. The transformed system
$E:=\psi_{U}[B]$ is again block-diagonal and $x^{q+1}J_{q}E=D(x)+x^{q}C$. We
write $E=E^{11}\oplus\cdots\oplus E^{\ell\ell}$ and
$x^{q+1}E^{uu}=D^{uu}+x^{q}C^{uu}+x^{q+1}E^{uu}_{q+1}+\cdots$ for each block
$E^{uu}$. Taking into account that each block $D^{uu}$ is radial and hence
commutes with any matrix, we obtain recursively for any $u\in\\{1,...,\ell\\}$
and for any $j\geq 1$

(12)
$E^{uu}_{q+j}=B^{uu}_{q+j}+C^{uu}U^{uu}_{j}-U^{uu}_{j}(C^{uu}+jI_{n_{u}})+\eta^{uu}_{j}(\\{U^{uu}_{s},B^{uu}_{q+s}\\}_{s<j}),$

where, similarly as in equation (10), $\eta^{uu}_{j}$ is a polynomial acting
over the expressed matrices, with $\eta^{uu}_{1}=0$. Using again Lemma 13 with
matrices $R=C^{uu}$ and $S=C^{uu}+jI_{n_{u}}$ (notice that these two matrices
have no common eigenvalue by the non resonance condition for $C$), we can
construct recursively from (12) the coefficients
$U^{uu}_{q+1},U^{uu}_{q+2},\ldots$ such that $E^{uu}_{q+j}=0$ for any
$u\in\\{1,...,\ell\\}$ and for any $j\geq 1$.

The conclusion from the two steps above is that the composition
$\Psi_{UT}=\Psi_{U}\circ\Psi_{T}$ satisfies

$\Psi_{UT}[A]=x^{-(q+1)}(D(x)+x^{q}C).$

To finish, notice from the expression of the transform of a system by a
regular gauge transformation that, for every $\mu\geq 0$, the $(q+\mu)$-jet of
$\Psi_{UT}[A]$ only depends on the $(q+\mu)$-jet of the product $UT$ (and on
$J_{q+\mu}A$). Thus, we put $P^{\mu}:=J_{q+\mu}(UT)$ and Theorem 3, (ii)
follows.

###  2.3. Proof of Theorem 3, (iii). Getting a non-resonant residual matrix

A proof of this item, for $q=0$, can be found in the same references cited in
paragraph 2.2 above.

We assume that $A$ is in $(TRS)^{q}_{0}$ form with exponential part equal to
$D(x)$ and residual part equal to $C$.

Radial case.- We consider first the case where $D(x)$ is a radial matrix;
i.e., $D(x)=Q(x)I_{n}$, where $Q(x)$ is a polynomial (including the case $q=0$
for which $D(x)=0$).

Take the partition ${\rm Spec}(C)=\Omega_{1}\cup\cdots\cup\Omega_{r}$ of the
spectrum of $C$ in such a way that two eigenvalues differ by a non-zero
integer number if and only if they belong both to some $\Omega_{j}$. Denote
$\Omega_{j}=\\{\lambda^{j}_{1},...,\lambda^{j}_{s_{j}}\\}$ and assume that the
indices are chosen so that
$Re(\lambda^{j}_{1})>Re(\lambda^{j}_{2})>\cdots>Re(\lambda^{j}_{s_{j}})$ in
case $s_{j}>1$. Let

(13)
$m(C):=\sum_{\\{j\,:\,s_{j}>1\\}}\lambda_{1}^{j}-\lambda^{j}_{s_{j}}\in\mathbb{N}.$

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 $\Psi_{P_{0}}$ and a monomial diagonal transformation
$\Psi_{S}$ such that the transformed system $\Psi_{S}\circ\Psi_{P_{0}}[A]$ has
Poincaré rank equal to $q$, the same exponential part $D(x)$ and a residual
matrix $C^{\prime}$ satisfying $m(C^{\prime})<m(C)$.

The matrix $P_{0}$ is chosen so that $\overline{C}:=P_{0}^{-1}CP_{0}$ is block
diagonal of the form

(14) $\overline{C}=(C^{11}\oplus\cdots\oplus
C^{1s_{1}})\oplus\cdots\oplus(C^{r1}\oplus\cdots\oplus C^{rs_{r}}),$

where ${\rm Spec}(C^{ji})=\\{\lambda^{j}_{i}\\}$. The transformed system
$\overline{A}=\Psi_{P_{0}}[A]=P_{0}^{-1}AP_{0}$ has the same exponential part
$D(x)$ (since this is a radial matrix) and residual matrix equal to
$\overline{C}$. Put $x^{q+1}\overline{A}=\sum_{l\geq 0}\overline{A}_{l}x^{l}$
and use the block structure given by equation (14) for each coefficient
$\overline{A}_{l}=(\overline{A}_{l}^{uv})$, where $u,v$ run in the set
$\\{(ji)\,:\,j=1,..,r,i=1,...,s_{j}\\}\cup\\{\Lambda\\}$. Assume for instance
that $\Omega_{1}$ has at least two elements. The we consider the diagonal
monomial matrix

(15) $S:=xI_{t}\oplus I_{n-t},$

where $t$ is equal to the size of $C^{11}$. Consider
$B:=\Psi_{S}[\overline{A}]$ and write $x^{q+1}B=\sum_{l\geq 0}B_{l}x^{l}$ and
$B_{l}=(B_{l}^{uv})$ with the same block structure as in (14). A calculation
shows that

(16) $\begin{array}[]{ll}B_{l}^{uu}=\overline{A}_{l}^{uu},\mbox{ if }l\neq
q,&B_{l}^{uv}=\overline{A}_{l}^{uv},\mbox{ if }u\neq(11)\mbox{ and
}v\neq(11),\\\ B_{l}^{(11)v}=x^{-1}\overline{A}_{l}^{(11)v},\mbox{ if
}v\neq(11),&B_{l}^{u(11)}=x\overline{A}_{l}^{u(11)},\mbox{ if }u\neq(1,1),\\\
B_{q}^{(11)(11)}=\overline{A}_{q}^{(11)(11)}-I_{t}.&\end{array}$

In particular, we obtain $q(B)=q$, $B$ is in $(TRS)^{q}_{0}$-form with the
same exponential part $D(x)$ and the residual matrix $C^{\prime}=B_{q}$ is
upper triangular with respect to the block structure $(B_{q}^{uv})$ and with
diagonal equal to

${\rm diag}(C^{\prime})=((C^{11}-I_{t})\oplus\cdots\oplus
C^{1s_{1}})\oplus\cdots\oplus(C^{r1}\oplus\cdots\oplus C^{rs_{r}})\oplus
C^{\Lambda}.$

We deduce that $m(C^{\prime})=m(C)-1$ 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 5, (e)). Moreover, such transformation depends
only on the truncation $J_{m(C)+q}(A)$. 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 $D(x)=D^{11}(x)\oplus\cdots\oplus D^{\ell\ell}(x)$, as in equation
(8). Consider also $C=C^{11}\oplus\cdots\oplus C^{\ell\ell}$, decomposed into
the same block structure. Put

$m:=\max\\{m(C^{ii}),\>\,i=1,...,\ell\\}.$

Taking into account Step 1 in the proof of (ii), there is regular polynomial
gauge transformation $\psi$ such that the truncation $J_{m+q}(\psi[A])$
decomposes into several systems of smaller size

$J_{m+q}(\psi[A])=A^{11}(x)\oplus\cdots\oplus A^{\ell\ell}(x),$

where the principal part of $A^{jj}(x)$ is
$x^{-\nu(D^{jj})}(D^{jj}(x)+x^{q}C^{jj})$. In particular, $A^{jj}(x)$ is in
the radial case treated above, so that there is a finite composition of
constant regular and monomial diagonal transformations $\varphi_{j}$ such that
$\varphi_{j}[A^{jj}]$ has the same exponential part as $A^{jj}(x)$ and a non-
resonant residual matrix. We conclude that the composition
$\phi=(\varphi_{1}\oplus\cdots\oplus\varphi_{\ell})\circ\psi$ transforms $A$
into a system in $(TRS)^{q}_{0}$-form with non-resonant residual matrix. On
the other hand, as we have noticed in the radial case, the degree of each
$\varphi_{j}$ is bounded by $m(C^{jj})$, and hence the degree of the
polynomial gauge transformation $\phi$ can be bounded by $2m$. This ends the
proof of item (iii) of Theorem 3 and completes the statement in Remark 5, (e).
$\square$

One final comment on how to conclude Remark 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 $\Psi_{S}$,
where $S$ is as in equation (15). As we have already observed from equations
(16), such transformations preserve the Poincaré rank.

##  3\. Proof of the Real Turrittin’s Theorem

In this section we fix a real closed field $K$ and we prove Theorem 7, the
real version of Turrittin’s Theorem on polynomial normal forms. As mentioned,
the formal statement Theorem 8 will be a consequence of it.

We make use of the monomorphism of $K$-algebras defined in paragraph 1.2. That
is, for any $m\in\mathbb{N}$, we consider

$\Theta_{m}:\mathcal{M}_{m}(\overline{K})\to\mathcal{M}_{2m}(K),\;\;(c_{uv})\mapsto(\Lambda_{c_{uv}}).$

and (with the same name), its extension to a morphism of $K$-algebras from
$\mathcal{M}_{m}(L_{\overline{K}})$ to $\mathcal{M}_{2m}(L_{K})$ sending an
$m$-dimensional system $B=x^{-(q+1)}\sum_{j\geq 0}x^{j}B_{j}$ with
coefficients in $\overline{K}$ to the system
$\Theta_{m}(B)=x^{-(q+1)}\sum_{j\geq 0}x^{j}\Theta_{m}(B_{j})$. Notice that
$\Theta_{m}$ preserves the Poincaré rank but not necessarily the radiality
index of the system. On the other hand, one can check easily that $\Theta_{m}$
commutes with the gauge transformations and with ramifications. To be precise,
if $B,P\in\mathcal{M}_{m}(L_{\overline{K}})$ with $\text{det}(P)\neq 0$, we
have

(17) $\Theta_{m}(\Psi_{P}[B])=\Psi_{\Theta_{m}(P)}[\Theta_{m}(B)],$

and, if $r$ is a natural non-zero number, then

(18) $\Theta_{m}\circ R_{r}=R_{r}\circ\Theta_{m}.$

###  3.1. Propagating a $\mathbb{C}$-matrix to higher order coefficients

The key result for the proof of Theorem 7 is the following one.

######  Proposition 14.

Consider a system $A\in\mathcal{M}_{n}(L_{K})$ with Poincaré rank equal to $q$
and written as $A=x^{-(q+1)}\left(A_{0}+xA_{1}+\cdots\right)$. Let $k$ be the
radiality index of $A$ and assume that $k<q$ and that the spectrum of $A_{k}$
in $\overline{K}$ consists in a pair of conjugated values $a\pm ib$ with
$a,b\in K$ and $b\neq 0$ (thus in particular $n=2m$ is even). Then there
exists a formal regular gauge transformation $\Psi_{T}$, where
$T\in\mathcal{M}_{n}(K[[x]])$, such that the transformed system
$B=\Psi_{T}[A]$ is a $\mathbb{C}$-system. Moreover, writing
$B=x^{-(q+1)}(\sum_{j\geq 0}x^{j}B_{j})$, we have $A_{j}=B_{j}\in KI_{n}$ for
$j=0,1,...,k-1$ and for any $\mu\geq k$, the truncation $J_{\mu}B$ only
depends on $J_{\mu-k}T$ and $J_{\mu}A$. In other words, if $\widetilde{A}$ is
another system with $q(\widetilde{A})=q$ and satisfying
$J_{\mu}\widetilde{A}=J_{\mu}A$, then
$J_{\mu}(\Psi_{J_{\mu-k}T}[\widetilde{A}])=J_{\mu}B$.

The proof of Proposition 14 has a big similarity with the one of the Splitting
Lemma (cf. Lemma 10) or of Theorem 3, (ii). This time, it is based on the
following result for $\mathbb{C}$-matrices of size two:

######  Lemma 15.

Let $\lambda\in\overline{K}\setminus K$ and let
$\Lambda_{\lambda}=\Theta_{1}(\lambda)$ be the corresponding
$\mathbb{C}$-matrix of size $2$. Given $S\in\mathcal{M}_{2}(K)$ an arbitrary
matrix with coefficients in $K$, there exists a matrix
$X\in\mathcal{M}_{2}(K)$ such that $\Lambda_{\lambda}X-X\Lambda_{\lambda}+S$
is a $\mathbb{C}$-matrix.

Proof. Put $\lambda=a+ib$ with $a,b\in K$ and $b\neq 0$ and write $S=(s_{ij})$
and $X=(x_{ij})$ with $1\leq i,j\leq 2$. Computing we have

(19)
$\Lambda_{\lambda}X-X\Lambda_{\lambda}+S=\left(\begin{array}[]{cc}-u+s_{11}&v+s_{12}\\\
v+s_{21}&u+s_{22}\end{array}\right),$

where $u=b(x_{12}+x_{21})$ and $v=b(x_{11}+x_{22})$. The matrix in (19) is a
$\mathbb{C}$-matrix iff we have $-u+s_{11}=u+s_{22}$ and
$v+s_{12}=-(v+s_{21})$. These two last equations have solutions in $u,v$ once
we are given the entries $s_{ij}$ of $S$ and, taking into account that $b\neq
0$, we conclude the lemma. $\square$

Proof of Proposition 14.- First, using a real canonical form of $A_{k}$, there
is a non-singular matrix $T_{0}$ with entries in $K$ such that
$T_{0}^{-1}A_{k}T_{0}=\Lambda+H$ where

$\Lambda=\Lambda_{\lambda}\oplus\Lambda_{\lambda}\oplus\cdots\oplus\Lambda_{\lambda},\;H=\left(\begin{array}[]{ccccc}0&\epsilon_{1}I_{2}&&&\\\
&0&\epsilon_{2}I_{2}&&\\\ &&\ddots&&\\\ &&&&\epsilon_{m-1}I_{2}\\\
&&&&0\end{array}\right),\mbox{ with }\epsilon_{j}\in\\{0,1\\}.$

Up to replacing $A$ by $\Psi_{T_{0}}[A]$, we may assume that the original
coefficient $A_{k}$ has already the form above $A_{k}=\Lambda+H$, a
$\mathbb{C}$-matrix. We look for a regular formal gauge transformation
$\Psi_{T}$ with $T=I_{n}+xT_{1}+x^{2}T_{2}+\cdots$ satisfying the required
property. For that, we compute the coefficients of the transformed system
$B:=\Psi_{T}[A]$ in terms of the coefficients of $A$ and of $T$. With similar
computations as already done in the preceding section, if we write
$B=x^{-(q+1)}\left(B_{0}+xB_{1}+\cdots\right)$ then we get:

\- The radial part does not change; i.e., $B_{\ell}=A_{\ell}$ for
$\ell=0,1,...,k-1$.

\- $B_{k}=A_{k}=\Lambda+H$.

\- For $j\geq k+1$, we obtain

(20) $B_{j}=[A_{k},T_{j-k}]+A_{j}+Q_{j},$

where $Q_{j}$ is a matrix which depends polynomially only on the matrices of
the family $\\{A_{k+s},T_{s}\\}_{s<j-k}$.

Let us show that we can choose recursively $T_{1},T_{2},\ldots$ such that each
$B_{j}$ in equation (20) is a $\mathbb{C}$-matrix for any $j\geq k$. This will
finish the proof of Proposition 14.

The starting case $j=k$ is done since $B_{k}=A_{k}$ is already a
$\mathbb{C}$-matrix. Suppose that for $j>k$ we have already constructed
$T_{1},...,T_{j-1}$ such that $B_{\ell}$ is a $\mathbb{C}$-matrix for
$\ell<j$. For each value of the letter $Y\in\\{A,T,B,Q\\}$ and for each
$\ell\leq j$, we write $Y_{\ell}=(Y_{\ell}^{uv})_{1\leq u,v\leq m}$ in a block
structure of $2\times 2$ matrices. We construct the different blocks
$T_{j}^{uv}$ in the following order. We start by the bottom of the first
column: the block $T^{m1}_{j}$ satisfies, after equation (20),

$[\Lambda_{\lambda},T_{j}^{m1}]+A_{j}^{m1}+Q_{j}^{m1}=B_{j}^{m1}.$

Using Lemma 15, we choose $T_{j}^{m1}$ in such a way that $B_{j}^{m1}$ is a
$\mathbb{C}$-matrix. Then we continue with the block $T_{j}^{m-1,1}$ which
satisfies

$[\Lambda_{\lambda},T_{j}^{m-1,1}]+\epsilon_{m-1}T_{j}^{m1}+A_{j}^{m-1,1}+Q_{j}^{m-1,1}=B_{j}^{m-1,1}.$

Taking into account that $T_{j}^{m1}$ has already been chosen and using Lemma
15, we choose $T^{m-1,1}_{j}$ such that $B^{m-1,1}_{j}$ is a
$\mathbb{C}$-matrix. The process can be repeated in this way until we
construct all blocks in the first column, that is, those of the form
$T_{j}^{u1}$ in inverse order for $u$ from $u=m$ to $u=1$. After that, we
construct the blocks in the second column $T_{j}^{u2}$, again from $u=m$ to
$u=1$: by (20) we get

$[\Lambda_{\lambda},T_{j}^{u,2}]+\epsilon_{u}T_{j}^{u+1,2}-\epsilon_{1}T_{j}^{u,1}+A_{j}^{u2}+Q_{j}^{u2}=B_{j}^{u2}$

(with $\epsilon_{m}=0$), and we choose $T_{j}^{u2}$ such that $B^{u2}_{j}$ is
a $\mathbb{C}$-matrix, once the blocks $T^{u+1,2}_{j}$, $T^{u1}_{j}$ in the
above equation are already known. We continue in this way in order to complete
the construction of all blocks $T^{uv}_{j}$ so that any block $B^{uv}_{j}$
(and hence the whole matrix $B_{j}$) is a $\mathbb{C}$-matrix. $\square$

###  3.2. Proof of Theorem 7, (i). Getting a $(\mathbb{R}TRS)$-form of degree
$0$

Fix a singular system $A\in\mathcal{M}_{n}(L_{K})$ with Poincaré rank $q=q(A)$
and write $A=x^{-(q+1)}\left(A_{0}+xA_{1}+\cdots\right)$ as in (1) with
$A_{0}\neq 0$. 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 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 $A_{k}=A_{q}$, there is a non-singular
matrix $T_{0}\in\mathcal{M}_{n}(K)$ such that
$T_{0}^{-1}A_{0}T_{0}=C_{1}\oplus C_{2}$, where $C_{1}$ is a matrix with
eigenvalues in $K$ and $C_{2}$ is a $\mathbb{C}$-matrix. The radial part
$A_{0}+xA_{1}+\cdots+x^{q-1}A_{q-1}$ is preserved by $\Psi_{T_{0}}$ and can be
written in the form $D_{1}(x)\oplus\Theta_{n_{2}}(D_{2}(x))$ where
$D_{1}(x)\in\mathcal{M}_{n_{1}}(K[X])$ and
$D_{2}(x)\in\mathcal{M}_{n_{2}}(K[x])$ are both diagonal polynomial. Hence
$\Psi_{T_{0}}[A]$ is in $(\mathbb{R}TRS)^{q}_{0}$-form and (i’) follows
(notice that $N=q$ in this case).

Assume that $0\leq k<q$. We proceed by induction with respect to the size $n$
of the system. The case $n=1$ is also trivial: $A$ is already in
$(\mathbb{R}TRS)^{q}_{0}$-form with $n_{1}=n=1$ and $n_{2}=0$ and $N=q$ in
this case. Suppose then that $n>1$.

Suppose first that the first non-radial term $A_{k}$ has at least two non-
conjugated eigenvalues in $\overline{K}$. In this case, after a constant
regular gauge transformation $\Psi_{T_{0}}$, with $T_{0}\in GL_{n}(K)$, we can
assume that $A_{k}=A^{11}_{k}\oplus A^{22}_{k}$, where each $A^{ii}_{k}$ is a
square matrix with positive size $n_{i}$ with entries in $K$ and
$\text{Spec}(A^{11}_{k})\cap\text{Spec}(A^{22}_{k})=\emptyset$. Apply the
Splitting Lemma to $A$ up to order $N=N(n,q,k)$. That is, there exists a
regular polynomial gauge transformation $\Psi_{T}$ where
$T=I+xT_{1}+\cdots+x^{N-k}T_{N-k}$ such that $J_{N}(\Psi_{T}[A])=B^{11}\oplus
B^{22}$, where $B^{ii}$ is a (polynomial) system of size $n_{i}<n$ with
coefficients in $K$. By induction on the size, item (i’) holds for both
systems $B^{ii}_{k}$. 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 $\text{Spec}(A_{k})=\\{\lambda_{k},\overline{\lambda_{k}}\\}$
for some $\lambda_{k}\in\overline{K}$. We consider the two possible
situations:

Case 1: $\lambda_{k}\neq\overline{\lambda_{k}}$. Notice that $n=2m$ is even in
this case. We apply Proposition 14 to the system $A$. Notably, let $\Psi_{T}$
be a formal regular gauge transformation with $T\in\mathcal{M}_{n}(K[[x]])$
such that $B=\Psi_{T}[A]$ is a $\mathbb{C}$-system. Let $\overline{B}$ be the
system with coefficients in $\overline{K}$ satisfying
$B=\Theta_{m}(\overline{B})$. Apply Theorem 3, (i) to $\overline{B}$: we get a
natural number $r\geq 1$ and gauge transformations
$\varphi_{1},\ldots,\varphi_{s}$, either regular polynomial or monomial
diagonal (with coefficients in $\overline{K}$) such that, putting
$\psi=\varphi_{s}\circ\ldots\circ\varphi_{1}\circ R_{r}$, we have

  1. (a)

The system $\psi[\overline{B}]$ is in $(TRS)^{\tilde{q}}_{0}$-form for some
$\tilde{q}\geq 0$.

  2. (b)

Being $N=N(n,q,k)$, if $\overline{E}$ is another system with
$q(\overline{E})=q$ and $J_{N}\overline{E}=J_{N}\overline{B}$, the system
$\psi[\overline{E}]$ is also in $(TRS)^{\tilde{q}}_{0}$-form with the same
principal part as $\psi[\overline{B}]$.

Now, for any $i=1,...,s$, if $\varphi_{i}=\Psi_{P_{i}}$ with
$P_{i}\in\mathcal{M}_{n}(\overline{K}[[x]])$, we put
$\widetilde{\varphi}_{i}:=\Psi_{\Theta_{m}(P_{i})}$. Notice that
$\widetilde{\varphi}_{i}$ is a gauge transformation with coefficients in $K$,
either regular polynomial or diagonal monomial. Denote by
$\widetilde{\psi}=\widetilde{\varphi}_{s}\circ\cdots\circ\widetilde{\varphi}_{1}\circ
R_{r}$ and let us see that the composition

$\widetilde{\psi}\circ\Psi_{J_{N}T}=\widetilde{\varphi}_{s}\circ\cdots\circ\widetilde{\varphi}_{1}\circ\Psi_{J_{N}T(x^{r})}\circ
R_{r}$

satisfies the requirements of Theorem 7, (i’). Using the property (b) above
and equations (17) and (18) we have that

$\Theta_{m}(\psi[J_{N}(\overline{B})])=\widetilde{\psi}[J_{N}(\Theta_{m}(\overline{B}))]=\widetilde{\psi}[J_{N}(B)]$

is in $(\mathbb{R}TRS)^{\tilde{q}}_{0}$-form. On the other hand, we have the
following

Claim.- If $E\in\mathcal{M}_{n}(L_{K})$ is any system with Poincaré rank equal
to $q$ and $J_{N}E=J_{N}B$ then
$J_{\tilde{q}}(\widetilde{\psi}[E])=J_{\tilde{q}}(\widetilde{\psi}[J_{N}B])$.

Applying this claim to $E=\Psi_{J_{N}T}[A]$, taking into account that
$J_{N}(\Psi_{J_{N}T}[A])=J_{N}B$, we conclude Theorem 7, (i’) in this Case 1.

It remains to show the Claim. It is a consequence of the property (b) above
(using (17) and (18)) in the case that $E$ is a $\mathbb{C}$-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
$J_{\tilde{q}}(\widetilde{\psi}[E])=J_{\tilde{q}}(\widetilde{\psi}[J_{M}E])$
for such systems and that the map $H_{M}:J_{M}E\mapsto
J_{\tilde{q}}(\widetilde{\psi}[J_{M}E])$ is a polynomial map in the entries of
the coefficient matrices $E_{0},...,E_{M}$ of $J_{M}E$. 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 $H_{M}$ does not depend
on the entries of $E_{N+1},\ldots,E_{M}$ if $E$ is a $\mathbb{C}$-system.
Since the set of $J_{M}$-jets of $\mathbb{C}$-systems with a fixed Poincaré
rank $q$ has non-empty interior in the space of $J_{M}$-jets of all systems
(with that fixed Poincaré rank), we conclude that $M=N$ satisfies the property
above. The Claim follows.

Case 2: $\lambda_{k}=\overline{\lambda_{k}}$. In this case, $A_{k}$ has a
single eigenvalue $\lambda_{k}\in K$. After a constant gauge transformation
with entries in $K$, we can write $A_{k}$ in its Jordan canonical form as in
equation (3). We can define in this case the tuple
$I(A)=(\gamma_{1}(A_{k}),\ldots,\gamma_{n}(A_{k}),q-k)$ and proceed exactly as
in the proof of the complex Turrittin theorem in paragraph 2.1 from the step
in which $A_{k}$ has the Jordan form (3). Notably, the terms in the truncation
$J_{N}A$, with $N=N(n,q,k)$, determine a shearing order $g=h/r\in\mathbb{Q}$
(cf. Definition 12). Then, we consider the transformed system
$B=\Psi_{S_{h}}\circ R_{r}[A]$, where $R_{r}$ is the ramification of index $r$
and $S_{h}=\text{diag}(1,x^{h},...,x^{(n-1)h})$. Denoting by $q^{\prime}$ and
$k^{\prime}$ the Poincaré rank and the radiality index of $B$ respectively,
one of the following situations occurs:

  * •

$k^{\prime}=q$ (including the case $q^{\prime}=0$): we finish since this the
trivial case.

  * •

$0\leq k^{\prime}<q^{\prime}$ and $B_{k^{\prime}}$ has at least two non-
conjugated eigenvalues in $\overline{K}$: we finish using splitting lemma and
induction on $n$ as above.

  * •

$0\leq k^{\prime}<q^{\prime}$ and $B_{k^{\prime}}$ has a unique pair of
conjugated eigenvalues that do not belong to $K$: we finish since we are in
Case 1 above.

  * •

$0\leq k^{\prime}<q^{\prime}$ and $B_{k^{\prime}}$ 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 ends the proof of Theorem 7, (i’). $\square$

###  3.3. Proof of Theorem 7, (ii): getting $(\mathbb{R}TRS)$-form of higher
degree

The proof can be done similarly to the case where $K$ is algebraically closed
in paragraph 2.2, with only some minor changes. Let us indicated them.

Suppose that the system $A$ is in $(\mathbb{R}TRS)^{q}_{0}$-form with
exponential part equal to $D(x)=D_{1}(x)\oplus D_{2}(x)$ and residual matrix
$C=C_{1}\oplus C_{2}$ in the conditions of Definition 6. In particular,
$D_{2}(x)=\Theta_{n_{2}}(E_{2}(x))$ and $C_{2}=\Theta_{n_{2}}(G_{2})$, where
$G_{2}\in\mathcal{M}_{n_{2}}(\overline{K})$ and $E_{2}(x)={\rm
diag}(d_{1}(x),...,d_{n_{2}}(x))$ with
$d_{j}(x)\in\overline{K}[x]_{q-1}\setminus K[x]$. We also assume that $C$ is
non-resonant, which is equivalent to say that both $C_{1}$ and $G_{2}$ are
non-resonant matrices (of sizes $n_{1}$ and $n_{2}$, respectively).

We consider a block structure to write our system, similar to the one given in
(8), but compatible with the fact that $D_{2}(x)$ is a $\mathbb{C}$-matrix.
Notably, we write

(21) $D(x)=D^{11}(x)\oplus\cdots\oplus
D^{ss}(x)\oplus\Phi_{k_{1}}(E^{11}(x))\oplus\cdots\oplus\Phi_{k_{t}}(E^{tt}(x)),$

where

  * •

Each $D^{jj}(x)=f_{j}(x)I_{m_{j}}$ is a radial matrix in
$\mathcal{M}_{m_{j}}(K[x])$ (i.e., the coefficients of $f_{j}(x)$ are in $K$).

  * •

Each $E^{jj}(x)=g_{j}(x)I_{k_{j}}$ is a radial matrix in
$\mathcal{M}_{k_{j}}(\overline{K}[x])$ (i.e., the coefficients of $g_{j}(x)$
are in $\overline{K}$).

  * •

$D_{1}(x)=D^{11}(x)\oplus\cdots\oplus D^{ss}(x)$ and
$D_{2}(x)=\Phi_{n_{2}}(E^{11}(x)\oplus\cdots\oplus E^{tt}(x))$.

Put In accordance with the notation of equation (8), we denote $\ell:=s+t$ and
$D^{jj}(x):=\Phi_{k_{j-s}}(E^{j-s,j-s}(x))$ for $j=s+1,...,\ell$. We write
also $A=x^{-(q+1)}\sum_{l}x^{l}A_{l}$ as in (1) and each coefficient
$A_{l}=(A^{uv}_{l})_{1\leq u,v\leq\ell}$ in the block structure of (21).

Notice that the residual matrix $C=C_{1}\oplus C_{2}$ is block-diagonal in
this structure, since it commutes with $D(x)=D_{1}(x)\oplus D_{2}(x)$ and
$C_{2}$ is a $\mathbb{C}$-matrix. Thus, $A_{q+1}^{uv}=C^{uv}=0$ if $u\neq v$.

We want to eliminate all coefficients $A_{q+1},A_{q+2},...$ by means of a
formal regular gauge transformation $\Psi_{P}$ with
$P\in\mathcal{M}_{n}(K[[x]])$ and $P(0)=I_{n}$. We proceed as in paragraph 2.2
in two steps: first we eliminate the non-diagonal blocks $A_{l}^{uv}$, for
$u\neq v$ and $l\geq q+1$, and then the diagonal blocks $A_{l}^{uu}$, for
$u=1,...,\ell$ and $l\geq q+1$.

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 (8) More precisely, we consider a similar block
structure as the one in (9), where each block $\overline{D}^{jj}(x)$ with
$j\in\\{1,...,\ell_{1}\\}$ is of maximal size such that its value
$\overline{D}^{jj}_{0}:=\overline{D}^{jj}(0)$ at zero is:

  1. (a)

Either a radial matrix (i.e., $\overline{D}^{jj}_{0}=a_{j}I_{h_{j}}$ with some
$a_{j}\in K$).

  2. (b)

Or a radial $\mathbb{C}$-matrix (i.e.,
$\overline{D}^{jj}_{0}=\Phi_{h_{j}}((a_{j}+ib_{j})I_{h_{j}})$ for some
$a_{j}+ib_{j}\in\overline{K}\setminus K$).

Notice that a block $\overline{D}^{jj}(x)$ in the case (a) may contain several
of the blocks in the decomposition (21), even of the two different types
$\\{D^{ll}(x)\\}_{l\leq s}$ and $\\{D^{ll}(x)\\}_{l>s}$. In any case, using
the same equations (10) and Lemma 13, we can construct a formal matrix
$\overline{T}=I_{n}+x^{q+1}\overline{T}_{q+1}+\cdots\in K[[x]]$ such that the
system $\overline{B}=\Psi_{\overline{T}}(A)$ has zero non-diagonal blocks with
respect to this last structure $D(x)=\bigoplus_{1\leq
j\leq\ell_{1}}\overline{D}^{jj}(x)$. Hence, $\overline{B}=\bigoplus_{1\leq
j\leq\ell_{1}}\overline{B}^{jj}$, where $\overline{B}^{jj}$ is of size $h_{j}$
when $\overline{D}^{jj}_{0}$ is in the case (a), or $\overline{B}^{jj}$ is of
size $2h_{j}$ when $\overline{D}^{jj}_{0}$ is in the case (b). In this last
case, using Proposition 14, we can assume that $\overline{B}^{jj}$ is a
$\mathbb{C}$-system (notice that $b_{j}\neq 0$ in this case, so that
$k(\overline{B}^{jj})=0$ and $q(\overline{B}^{jj})=q>0$).

Put $\widetilde{B}^{jj}:=\overline{B}^{jj}-x^{-(q+1)}\overline{D}^{jj}_{0}$
for $j=1,...,\ell_{1}$, 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 transformation
$\Psi_{T^{j}}$ that applies and transform the subsystem $\widetilde{B}^{jj}$
into a block-diagonal one with respect to the structure induced on the block
$\overline{B}^{jj}$ by (21). We only have to take care about the following:
for any index $j\in\\{1,...,\ell_{1}\\}$ such that $\overline{D}^{jj}_{0}$ in
the case (b) above, the matrix $T^{j}$ must be chosen to be a
$\mathbb{C}$-matrix, so that $\Psi_{T^{j}}$ preserves the system
$x^{-(q+1)}\overline{D}^{jj}_{0}$ and thus this transformation applied to
$\overline{B}^{jj}$ produces the same result.

Finally, the second step (eliminating the diagonal blocks $A^{uu}_{l}$ for
$l\geq q+1$) is obtained exactly in the same way as in Step 2 of the proof of
Theorem 3, (ii) in paragraph 2.2: we have to solve recursively the same
equations (12) for the blocks $U^{jj}$, and this can be done independently of
the base field $K$, since we only need Lemma 13 (valid for any field) and the
hypothesis that $C$ is non-resonant.

###  3.4. Proof of Theorem 7, (iii). Getting a non-resonant matrix

The proof of this item is made entirely equal to the corresponding complex
case (cf. Theorem 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 $D(x)=f(x)I_{n}$ with some polynomial $f(x)\in
K[x]_{q-1}$), or we are in the $\mathbb{C}$-radial case (that is
$D(x)=\Theta_{n/2}(g(x)I_{n/2})$, where $g(x)\in\overline{K}[x]_{q-1}\setminus
K[x]$). In the second of these two cases, we have that $D(x)\neq 0$ and $0\leq
k(A)<q(A)$, so that we can apply Proposition 14 and assume, after a regular
polynomial gauge transformation of degree $m=m(C)$ (cf. equation (13)), that
the truncation $J_{q+m}(A)$ is a $\mathbb{C}$-system, image by $\Theta_{n/2}$
of some system $\widetilde{A}\in\mathcal{M}_{n/2}(L_{\overline{K}})$ with
exponential part equal to $\widetilde{D}(x)=g(x)I_{n/2}$. By Theorem 3, (iii),
we transform $\widetilde{A}$ into another one with the same exponential part
and non-resonant residual matrix by a regular gauge transformation $\Psi_{S}$,
where $S=I_{n/2}+xS_{1}+\cdots\in\mathcal{M}_{n/2}(\overline{K}[x]_{m})$. In
this case, the regular transformation $\Psi_{\Theta_{n/2}(S)}$ proves item
(iii) for the real system $A$.

The general case is done as in paragraph 2.3 by means of the decomposition
(21) of $D(x)$ and using step 1 of the proof of Theorem 7, (ii), already
discussed in the previous paragraph 3.3.

This ends the proof of Theorem 7. $\square$

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