# Comment on “Tests of general relativity with GW170817”

Anatoly A. Svidzinsky1 and Robert C. Hilborn2 1Department of Physics &
Astronomy, Texas A&M University, College Station, TX 77843
2American Association of Physics Teachers, One Physics Ellipse, College Park,
MD 20740

###### Abstract

In a recent paper “Tests of general relativity with GW170817”
(arXiv:1811.00364 [gr-qc]) the authors claimed overwhelming evidence in favor
of tensor gravitational wave (GW) polarization over vector by analyzing GW
signals measured by the LIGO-Virgo network. Here we show that the measured
LIGO-Livingston signal is substantially reduced at certain frequency intervals
which can be attributed to noise filtering. We found that if these regions are
excluded from the analysis then data are consistent with vector GW
polarization and not with tensor. We show that if the signal accumulation
method is applied over the entire detector bandwidth, including the regions in
which the signal is depleted by noise subtraction, the result underestimates
the LIGO-Livingston signal amplitude. That smaller amplitude then leads to an
erroneous conclusion that favors tensor polarization over vector polarization
for the GW.

In a recent preprint Max18  the authors reported results of the gravitational
wave (GW) polarization test with GW170817 performed using a Bayesian analysis
of the signal properties with the three LIGO-Virgo interferometer outputs. The
authors found overwhelming evidence in favor of pure tensor polarization over
pure vector with an exponentially large Bayes factor. This result is opposite
to the conclusion of our analysis based on a direct comparison of the GW
signals measured by the three detectors Svid18a . Namely, we found that the
measured signal ratios are inconsistent with the predictions of general
relativity, but consistent with the recently proposed vector theory of gravity
Svid17 ; Svid18 . Here we explain why we came to the opposite conclusion and
argue that the results reported in Max18  must be reconsidered by removing
“corrupted” frequency intervals from the LIGO-Livingston strain time series.

For the GW170817 event involving low mass neutron stars the gravitational
waveform is known analytically for almost the entire time interval when the
signal passes through the detector’s sensitivity band (apart from the last
second before merger). This is the case because the energy loss by the binary
system is accurately described by the quadrupole formula upto the last second.
Both in general relativity and vector gravity Svid17  the waveform is given by

$s(t)\propto\frac{1}{\left(t_{c}-t\right)^{1/4}}\cos\left[2\phi(t)\right],$
(1)

where

$\phi(t)=-\left(\frac{c^{3}}{5GM_{c}}\right)^{5/8}\left(t_{c}-t\right)^{5/8}+\phi_{0},$
(2)

$t_{c}$ is the coalescence time and $M_{c}$ is the chirp mass of the binary
system.

We use published strain time series for the three detectors LV17  which are
not normalized to the detector’s noise. The best fit to the LIGO-Livingston
and LIGO-Hanford data yields $M_{c}=1.188$ $M_{\odot}$ and $t_{c}=0.296$ s
(here we made an adjustment for different arrival times of the signal at the
detector locations).

Next we introduce an integrated complex interferometer response

$I(t_{0},\Delta t)=\int_{t_{0}}^{t_{0}+\Delta
t}\left(t_{c}-t\right)^{1/4}e^{-2i\phi(t)}h(t)dt,$ (3)

where $\Delta t$ is the signal collection time and $h(t)$ is the strain
measured by the interferometer that contains both signal and noise. According
to Eqs. (1), (2) and (3), the signal contribution to $I$ is proportional to
$\Delta t$ provided we disregard a small correction produced by the fast
oscillating term. Thus, the signal accumulates with an increase of the
collection time $\Delta t$. In contrast, noise does not accumulate with
$\Delta t$ and for large enough $\Delta t$ the noise contribution to $I$ can
be disregarded.

Thus, the theory predicts that the ratio

$u=\frac{I(t_{0},\Delta t)}{\Delta t}$ (4)

should be independent of the signal collection time interval
$[t_{0},t_{0}+\Delta t]$. This ratio can be interpreted as a signal per unit
time. It depends, in particular, on GW polarization which allows us to
distinguish between pure tensor and pure vector polarizations using the three
interferometer network Svid18a . In addition, $u$ should be used to determine
how much signal is present in the interferometer data stream at different
times. If noise filtering has not altered the signal at certain frequencies
then $u$ must have the same (complex) value for any collection time interval.

Figure 1: Absolute value of the signal per unit time $|u|$ measured by LIGO-
Hanford detector for the GW170817 event for different collection time
intervals. As in Ref. Abbo17b , times are shown relative to August 17, 2017
12:41:04 UTC. Figure 2:  The same as in Fig. 1, but for the LIGO-Livingston
detector. The scale of the vertical axis is arbitrary but the same in both
figures. Frequency regions in which signal is reduced by noise filtering are
indicated as dashed bars.

In Fig. 1 we plot $|u|$ for LIGO-Hanford detector for different collection
time intervals. The result is shown as a set of rectangular bars. The length
of a bar is equal to the collection time $\Delta t$, while the height
corresponds to the uncertainty produced by the detector noise. Namely, the
half-height is equal to one standard deviation. We calculated the error by
injecting a test signal into the measured strain time series. The average
value of $|u|$ for each interval $\Delta t$ is indicated by a dashed line. The
figure shows that $|u|$ for the LIGO-Hanford detector is consistent with a
constant for the entire time when the signal is present in the data stream.
Therefore, the signal in the LIGO-Hanford strain time series appears not to be
affected by the noise subtraction.

In Fig. 2 we plot $|u|$ for LIGO-Livingston detector. The error bars for this
detector are smaller due to lower noise. Solid bars indicate regions for which
signal is clearly visible in the LIGO-Livingston spectrogram Svid18a . The
vertical position of the solid bars is consistent with $u$ being a constant,
as predicted by the theory. Dashed bars correspond to gap regions in the
spectrogram Svid18a . Fig. 2 shows that signal in these regions is
substantially smaller than the signal content in other intervals.

The signal reduction can be attributed to noise filtering. For example, a
glitch which occurred in LIGO-Livingston detector about $1.1$ s before
coalescence can explain why there is less signal in the data stream during the
glitch duration. Namely, the glitch removal from the data stream led to some
of the GW signal being subtracted off unintentionally, as one can see from
Fig. 2. Signal reduction in other frequency regions of the LIGO-Livingston
detector can be attributed to filtering fan noise (see Fig. 3 in Cost18 ).

The GW170817 event allows us to determine the amount by which GW signal is
suppressed by the noise filtering. Perhaps noise filtering yields substantial
signal reduction at certain frequencies for GW events for which signal per
unit time is very weak, which is the case for GW170817. Thus, for such events
one should perform a consistency check of Figs. 1 and 2, and remove
“corrupted” frequency intervals from the data analysis. This must be taken
into account in the analysis of future GW events produced by inspiral of low-
mass objects.

Contrary to the amplitude situation, we found that signal phase was not
altered in both detectors, namely the phase of $u$ is consistent with a
constant. This explains why sky location of the source is predicted correctly
from the measured strain time series. The sky localization is constrained
primarily by the differences in arrival times of the signal in various
detectors Fair18 . For the GW170817 event the time delays are obtained with a
high accuracy by fitting the signal phase $\phi(t)$ for hundreds GW cycles.
More importantly, the GW source location is known to high accuracy thanks to
the observation of concomitant electromagnetic emission.

As we showed in Svid18a , the ratio of LIGO-Hanford and LIGO-Livingston signal
amplitudes $|H/L|$ is crucial for distinguishing between tensor and vector GW
polarizations. In Ref. Svid18a  we obtained this ratio by accumulating the
LIGO-Livingston signal only from the regions shown by the solid bars in Fig.
2. In these regions the signal is not corrupted. We found
$0.6<\left|H/L\right|<0.8$. This estimate, combined with other constraints, is
compatible with vector theory of gravity Svid17 ; Svid18  but rules out
general relativity Svid18a .

However, signal accumulation from the entire frequency band erroneously
underestimates the LIGO-Livingston signal amplitude yielding
$1<\left|H/L\right|<1.2$ Remark . This incorrect estimate results in the
opposite conclusion about GW polarization. Namely, it rules out vector
polarization, but is consistent with tensor polarization (see Fig. 17 in
Svid18a ). This is what the authors of Ref. Max18  have found. We believe that
if the LIGO-Virgo analysis were done taking into account the Livingston signal
amplitude reduction in certain frequency regions, they would obtain the
opposite result for the GW polarization, which would agree with our findings
Svid18a .

## References

  * (1) B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Tests of General Relativity with GW170817, arXiv:1811.00364 [gr-qc] (2018).
  * (2) A. A. Svidzinsky and R. C. Hilborn, GW170817 event rules out general relativity in favor of vector gravity, arXiv:1804.03520v2 [physics.gen-ph] (2018).
  * (3) A. A. Svidzinsky, Vector theory of gravity: Universe without black holes and solution of dark energy problem, Physica Scripta 92, 125001 (2017).
  * (4) A. A. Svidzinsky, Simplified equations for gravitational field in the vector theory of gravity and new insights into dark energy, arXiv:1810.07541 [physics.gen-ph] (2018).
  * (5) LIGO-Virgo Collaboration, Data release for event GW170817, Gravitational-wave strain data 16384 Hz sampling rate, https://losc.ligo.org/events/GW170817
  * (6) B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett. 119, 161101 (2017).
  * (7) C. F. Da Silva Costa, C. Billman, A. Effler, S. Klimenko and H. P. Cheng, Regression of non-linear coupling of noise in LIGO detectors, Class. Quantum Grav. 35, 055008 (2018).
  * (8) S. Fairhurst, Localization of transient gravitational wave sources: beyond triangulation, Class. Quantum Grav. 35, 105002 (2018).
  * (9) To obtain this estimate we applied data whitening procedure which assigns larger weight to frequency regions with better detector strain sensitivity.


Reduced Forms of Linear Differential Systems and the Intrinsic Galois–Lie
Algebra of Katz

Reduced Forms of Linear Differential Systems
and the Intrinsic Galois–Lie Algebra of Katz

Moulay BARKATOU †, Thomas CLUZEAU †, Lucia DI VIZIO ‡ and Jacques-Arthur WEIL
†

M. Barkatou, T. Cluzeau, L. Di Vizio and J.-A. Weil

† XLIM, UMR7252, Université de Limoges et CNRS,
† 123 avenue Albert Thomas, 87060 Limoges Cedex, France
<EMAIL_ADDRESS><EMAIL_ADDRESS>jacques-
<EMAIL_ADDRESS>http://www.unilim.fr/pages_perso/moulay.barkatou/,
http://www.unilim.fr/pages_perso/thomas.cluzeau/,
http://www.unilim.fr/pages_perso/jacques-arthur.weil/

‡ Université Paris-Saclay, UVSQ, CNRS, Laboratoire de mathématiques de
Versailles,
‡ 78000, Versailles, France<EMAIL_ADDRESS>http://divizio.perso.math.cnrs.fr/

Received January 20, 2020, in final form June 04, 2020; Published online June
17, 2020

Generalizing the main result of [Aparicio-Monforte A., Compoint E., Weil
J.-A., J. Pure Appl. Algebra 217 (2013), 1504–1516], we prove that a linear
differential system is in reduced form in the sense of Kolchin and Kovacic if
and only if any differential module in an algebraic construction admits a
constant basis. Then we derive an explicit version of this statement. We
finally deduce some properties of the Lie algebra of Katz’s intrinsic Galois
group.

linear differential systems; differential Galois theory; Lie algebras; reduced
forms

34M03; 34M15; 34C20

##  1 Introduction

Let us consider the field of rational functions ${\mathbb{C}}(x)$, with the
derivation $\partial=\frac{{\rm d}}{{\rm d}x}$, and a linear differential
system $\partial\vec{y}=A\vec{y}$, where $A$ is a square matrix of order $n$
with coefficients in ${\mathbb{C}}(x)$. One can attach to such an object an
algebraic group, called the differential Galois group, whose geometric
properties encode the algebraic properties of the solutions of the linear
differential system. The problem of calculating explicitly the differential
Galois group of $\partial\vec{y}=A\vec{y}$ is old and still difficult. Among
the several references, we cite [1, 8, 10, 12, 20] that do not make any
assumption on the order $n$ of the system. Implemented (or implementable)
algorithms exist only for small dimensions $n$. See for example [7, 11, 14,
15, 16, 19, 23, 22].

Instead of calculating directly the differential Galois group of
$\partial\vec{y}=A\vec{y}$, one can try to study, or calculate, the Lie
algebra of the differential Galois group, called Galois–Lie algebra in what
follows. The Galois–Lie algebra already contains a significant part of the
information. Kolchin and Kovacic have proved that one can transform
$\partial\vec{y}=A\vec{y}$ into an equivalent system
$\partial\vec{y}=B\vec{y}$ defined over a finite extension $k$ of
${\mathbb{C}}(x)$, such that $B$ belongs to the set of $k$-rational points of
the Galois–Lie algebra (see [21, Proposition 1.31]). One can even prove that
the Galois–Lie algebra is then “generated” by the entries of the matrix $B$.
These ideas are formalized in [3, Section 2.3]. The linear differential system
$\partial\vec{y}=B\vec{y}$ is called a reduced form of
$\partial\vec{y}=A\vec{y}$. A linear differential system
$\partial\vec{y}=A\vec{y}$ is said to be in reduced form if $A$ is a
$k$-rational point of the Galois–Lie algebra.

Main results. In the present work, we prove three results on reduced forms and
their relation to the Galois–Lie algebra. First of all, we prove that a system
is in reduced form if and only if any differential module in a construction
admits a constant basis (see Theorem 3.2, in particular the equivalence
$1\Leftrightarrow 3$). This extends the criterion for reduced form from [3]
which concerned only invariant lines (see the equivalence $1\Leftrightarrow 2$
in Theorem 3.2).

Our second contribution is Theorem 3.9 that gives an effective
characterization of a gauge transformation that transforms a linear
differential system into a reduced one. It may be considered as an effective
counterpart of Theorem 3.2, based on the “local” data of the semi-invariants.
Compared to the original result by Kolchin and Kovacic on the existence of
reduced forms, we have to perform an algebraic extension of the base field
$k$, which may not be optimal, in order to gain the effectivity. Theorem 3.9
extends a result which only appeared in the course of the proof of [3,
Proposition 27], under the assumption that the differential system is
completely reducible.

Finally, we prove Theorem 4.5 on the Lie algebra of the intrinsic Galois
group, introduced in [13], where the idea of focusing on the Galois–Lie
algebra rather than on the differential Galois group itself is pursued.
Indeed, Katz introduces another Galois group for the linear differential
system $\partial\vec{y}=A\vec{y}$, called the generic or the intrinsic Galois
group. Then he considers the Lie algebra of such a group, for which he gives a
conjectural description equivalent to a well-known conjecture of Grothendieck
on the algebraicity of the solutions of a linear differential system. We will
call it the Katz algebra. In the last section, we gather material from [2, 5,
13, 21] and show how our criteria for reduced forms, combined with standard
Tannakian tools, clarify the structure of the Katz algebra. Namely, for a
reductive group, Theorem 4.5 shows that the Katz algebra is a $k$-form of the
Galois–Lie algebra.

The algorithm in [4]. In the latter reference, we showed how one can compute
the Galois–Lie algebra of an (absolutely) irreducible linear differential
system and hence (a good part of) its differential Galois group. Notice that
in [9], it is shown how to derive the (connected) Galois group of the reduced
form from the Galois–Lie algebra, so that one can actually recover the
connected component of the Galois group of the original system.

The algorithm selects a Lie algebra that is potentially the Katz algebra and
checks that there exists a gauge transformation that transforms its generators
into a set of constant generators of a Lie algebra, that is a candidate for
being the Lie algebra of the Galois group. To do so, it uses Theorem 3.9 in
the particular case of a completely reducible differential system, hence in
the case considered in [3]. See [4, Lemma 5.1]. Theorem 4.5 completes the
mathematical background of the algorithm, although it is technically not
needed in it.

Organization of the paper. In Section 2, we recall some notions on
differential modules, tensor constructions and differential Galois theory. In
Section 3, we prove our two theorems on the criteria for a linear differential
system to be in reduced form. In Section 4, we apply the previous results to
the study of the Katz algebra.

##  2 Notation and definitions

We consider a characteristic zero differential field $(k,\partial)$, that is a
characteristic zero field $k$ with a derivation $\partial\colon k\to k$, such
that $\partial(a+b)=\partial(a)+\partial(b)$ and
$\partial(ab)=\partial(a)b+a\partial(b)$, for any $a,b\in k$. We suppose that
the subfield of constants $C:=k^{\partial}=\\{f\in k\colon\partial f=0\\}$ is
algebraically closed.

###  2.1 Differential modules

A differential module ${\mathcal{M}}=(M,\nabla)$ over $k$ (of rank $n$) is a
$k$-vector space $M$ of dimension $n$, with a $C$-linear map $\nabla\colon
M\to M$ such that $\nabla(fm)=\partial(f)m+f\nabla(m)$ for any $f\in k$ and
any $m\in M$. For a detailed exposition on differential modules, see [21,
Section 2.2]. We denote by $M^{\nabla}$ or $\ker\nabla$ the set of horizontal
elements of ${\mathcal{M}}$, that is elements $m\in M$ satisfying
$\nabla(m)=0$. This is a $C$-vector space of dimension at most $n$.

Main properties of differential modules. Given a basis (denoted as a row)
$\underline{e}:=(e_{1},\dots,e_{n})$ of $M$ over $k$, the action of $\nabla$
with respect to the basis $\underline{e}$ is described by a square matrix
$A\in M_{n}(k)$ as follows

$\displaystyle\nabla\underline{e}=-\underline{e}A.$

For any $\vec{y}\in k^{n}$ such that $\underline{e}\vec{y}$ represents an
element of $M$, we have
$\nabla(\underline{e}\vec{y})=\underline{e}(\partial\vec{y}-A\vec{y})$. Thus
horizontal elements of ${\mathcal{M}}$ correspond to solutions over $k$ of the
differential system

$\displaystyle[A]\colon\ \partial\vec{y}=A\vec{y}.$

We say that $[A]\colon\partial\vec{y}=A\vec{y}$ is the linear differential
system associated to ${\mathcal{M}}$ with respect to the basis
$\underline{e}$.

If $\underline{f}=\underline{e}P$, with $P\in{\rm GL}_{n}(k)$, is another
basis of $M$, then the horizontal elements of ${\mathcal{M}}$ are of the form
$\underline{f}\vec{z}$, with $\vec{z}\in k^{n}$, where $\vec{z}$ verifies the
linear differential system

$\displaystyle\partial(\vec{z})=P[A]\vec{z},\qquad
P[A]:=P^{-1}AP-P^{-1}\partial(P).$

We say that two matrices $A,B\in M_{n}(k)$ are _equivalent over $k$_ if there
exists a _gauge transformation_ $P\in{\rm GL}_{n}(k)$ such that $B=P[A]$.

Notice that one can extend the scalars of $M$ to a field extension
$k^{\prime}$ of $k$, equipped with an extension of $\partial$. The Leibnitz
rule allows to extend $\nabla$ to $M\otimes_{k}k^{\prime}$, so that it makes
sense to consider gauge transformations in ${\rm GL}_{n}(k^{\prime})$, as
counterpart of basis changes of
${\mathcal{M}}\otimes_{k}k^{\prime}=(M\otimes_{k}k^{\prime},\nabla)$.

Algebraic constructions. Let us start by formalizing what we mean by
construction of linear algebra.

######  Definition 2.1.

A _construction of linear algebra_ is a finite iteration of the basic
constructors $\oplus$ (direct sum), $\otimes$ (tensor product), $\ast$ (dual),
${\rm Sym}^{r}$ ($r$-th symmetric power, for $r\in{\mathbb{N}}$) and
$\wedge^{r}$ ($r$-th exterior power). Given a construction of linear algebra
and a vector space $M$, we denote by ${\rm Constr}(M)$ the finite-dimensional
$k$-vector space obtained by applying the construction to $M$.

Given vector spaces $M_{1}$ and $M_{2}$ with respective bases $\underline{e}$
and $\underline{f}$, an application of each of the above basic constructors
produces _canonically_ a new basis: for instance,
$(e_{1},\ldots,e_{n},f_{1},\ldots,f_{m})$ is a basis for $M_{1}\oplus M_{2}$,
$(e_{i}\otimes f_{j}\,|\,i=1,\ldots,n;j=1,\ldots,m)$ is a basis for
$M_{1}\otimes M_{2}$, and so on (see [21, Section 2.2, p. 42], and [3, Section
3]). This way, given a vector space $M$ with basis $\underline{e}$ and a
construction ${\rm Constr}$, we iteratively construct a _canonical_ basis,
denoted ${\rm Constr}(\underline{e})$, of ${\rm Constr}(M)$. Two different
constructions may produce two isomorphic vector spaces. In this case, there is
a canonical isomorphism between the two vector spaces which identifies the
canonical bases.

######  Remark 2.2.

We need to make some comments on our lists of algebraic constructions.

  1. 1.

Notice that this list implicitly contains $\mathop{\rm Hom}$ (homomorphisms)
and $\mathop{\rm End}$ (endomorphisms) via the canonical identifications
$\mathop{\rm Hom}(M_{1},M_{2})\cong M_{1}\otimes M_{2}^{\ast}$ and
$\mathop{\rm End}(M)\cong M\otimes M^{\ast}$. See also Example 2.3 below.

  2. 2.

Some authors use additional basic constructors such as quotients and
subspaces. We choose not to do so to avoid the rising of apparent
singularities (see [3, Remark 18]) and because this list is sufficient for our
purposes. See Section 3.2, where we need an ordinary point for all
constructions.

Let $H$ denote a linear algebraic group, with Lie algebra $\mathfrak{h}$,
acting on $M$. Let the linear map $\sigma\in H$ have a matrix $U$ in the basis
$\underline{e}$. The morphism induced by $\sigma$ on ${\rm Constr}(M)$ is
denoted by ${\rm Constr}(\sigma)$. Its matrix in the basis ${\rm
Constr}(\underline{e})$ is denoted by ${\rm Constr}(U)$ and the map
$U\mapsto{\rm Constr}(U)$ is a group morphism. Similarly, for
$h\in\mathfrak{h}$, it acts on $M$ as a linear derivation $D_{h}$ with matrix
$N$. The action of this linear derivation $D_{h}$ on the basis ${\rm
Constr}(\underline{e})$ of ${\rm Constr}(M)$ induces a matrix
$\mathfrak{c}\mathfrak{o}\mathfrak{n}\mathfrak{s}\mathfrak{t}\mathfrak{r}(N)$;
the map
$N\mapsto\mathfrak{c}\mathfrak{o}\mathfrak{n}\mathfrak{s}\mathfrak{t}\mathfrak{r}(N)$
is a Lie algebra morphism. In what follows, we will use the gothical letters
$\mathfrak{c}\mathfrak{o}\mathfrak{n}\mathfrak{s}\mathfrak{t}\mathfrak{r}$ for
these constructions “in the sense of Lie algebras”.

Note that the entries of ${\rm Constr}(U)$ are _polynomials_ in the entries of
$U$ and in $1/\det(U)$; the $1/\det(U)$ is needed to have duals as, when ${\rm
Constr}(M)=M^{\ast}$, ${\rm Constr}(U)=\big{(}U^{-1}\big{)}^{\rm T}$. The
entries of
$\mathfrak{c}\mathfrak{o}\mathfrak{n}\mathfrak{s}\mathfrak{t}\mathfrak{r}(N)$
are _linear forms_ in the entries of $N$. See [21, Section 2.4, p. 53] and [3,
Sections 3.1 and 3.2].

Let $P\in{\rm GL}_{n}(k)$. If we consider a change of basis
$\underline{f}=\underline{e}P$ in the vector space $M$, then the corresponding
change-of-basis matrix in ${\rm Constr}(M)$ will be given by ${\rm Constr}(P)$
and we have ${\rm Constr}(\underline{f})={\rm Constr}(\underline{e}){\rm
Constr}(P)$. The constructions of linear algebra apply functorially to
differential modules. Let ${\mathcal{M}}=(M,\nabla)$ be a differential module
over $k$. The operator $\nabla$ induces a $C$-linear map from ${\rm
Constr}(M)$ to ${\rm Constr}(M)$, that we will also denote by $\nabla$,
defining a differential module structure over ${\rm Constr}(M)$. We will call
the latter a _construction of ${\mathcal{M}}$_ (or _tensor construction_) and
denote it ${\rm Constr}({\mathcal{M}})=({\rm Constr}(M),\nabla)$.

Let $-A$ be the matrix of $\nabla$ with respect to a basis $\underline{e}$ as
defined above. For any construction ${\rm Constr}({\mathcal{M}})$, the matrix
of $\nabla$ with respect to ${\rm Constr}(\underline{e})$ will be
$-\mathfrak{c}\mathfrak{o}\mathfrak{n}\mathfrak{s}\mathfrak{t}\mathfrak{r}(A)$.
For example, in the case ${\rm Constr}({\mathcal{M}})={\mathcal{M}}^{*}$,
$\mathfrak{c}\mathfrak{o}\mathfrak{n}\mathfrak{s}\mathfrak{t}\mathfrak{r}(A)=-A^{\rm
T}$, while if ${\rm
Constr}({\mathcal{M}})={\mathcal{M}}\otimes_{k}{\mathcal{M}}^{\ast}$, then we
have
$\mathfrak{c}\mathfrak{o}\mathfrak{n}\mathfrak{s}\mathfrak{t}\mathfrak{r}(A)=A\otimes
I_{n}-I_{n}\otimes A^{\rm T}$. See [3, Sections 3.1 and 3.2].

######  Example 2.3.

Let ${\mathcal{M}}=(M,\nabla)$ be a differential module over $k$ and let $-A$
be the matrix of $\nabla$ with respect to a basis $\underline{e}$. We consider
the differential module $\mathop{\rm End}({\mathcal{M}})=(\mathop{\rm
End}_{k}(M),\nabla)$. If $\varphi\in\mathop{\rm End}_{k}(M)$ then
$\nabla(\varphi)$ is the endomorphism of $M$ defined by
$\nabla(\varphi)(m)=\nabla(\varphi(m))-\varphi(\nabla(m))$, for all $m\in M$.
If $F$ is the matrix of $\varphi$ with respect to the basis $\underline{e}$,
one can check that we have
$\nabla(\varphi)(\underline{e})=\underline{e}(\partial F-AF+FA)$. If we denote
by square matrices $F$ the elements of $\mathop{\rm End}(M)$ with respect to
the basis induced by $\underline{e}$, then the linear differential system
associated to $\mathop{\rm End}({\mathcal{M}})$ with respect to the basis
induced by $\underline{e}$ is $\partial F=AF-FA$.

The horizontal elements of $\mathop{\rm End}({\mathcal{M}})$ are the elements
$\varphi\in\mathop{\rm End}_{k}(M)$ such that we have
$\nabla(\varphi(m))=\varphi(\nabla(m))$, $\forall\,m\in M.$ Thus the
horizontal elements of the differential module $\mathop{\rm
End}({\mathcal{M}})$ are exactly the $k$-endomorphisms of $M$ which commute
with $\nabla$. They are called _differential module endomorphisms_ of
${\mathcal{M}}$. They form a $C$-algebra denoted by
${\mathcal{E}}{({\mathcal{M}})}$ which is called the _eigenring_ of
${\mathcal{M}}$.

Among all the possible constructions, the differential module
${\mathcal{M}}\otimes_{k}{\mathcal{M}}^{\ast}$ will play a special role in the
exposition below. If we identify it canonically to $\mathop{\rm
End}({\mathcal{M}})$, then the linear differential system associated to
${\mathcal{M}}\otimes_{k}{\mathcal{M}}^{\ast}$ in the basis induced by
$\underline{e}$ is exactly $\partial F=AF-FA$. The set ${\mathcal{E}}([A])$ of
matrices $F\in M_{n}(k)$ satisfying the above matrix differential equation is
called the _eigenring_ of the system $[A]\colon\partial\vec{y}=A\vec{y}$. It
is isomorphic (as a $C$-algebra) to ${\mathcal{E}}({\mathcal{M}})$.

###  2.2 Picard–Vessiot extensions

We introduce very briefly some notions of differential Galois theory, with the
main purpose of fixing the notation. There exists several detailed
introduction to the topic. We refer to [21] for a general introduction and to
[3] for more specific notions which are needed in this paper.

Let us consider the linear differential system
$[A]\colon\partial\vec{y}=A\vec{y}$. To any such system we can attach a
$k$-algebra $R$, with an extension of $\partial$, having the following
properties:

  1. 1)

there exists $U\in{\rm GL}_{n}(R)$ such that $\partial U=AU$;

  2. 2)

the entries of $U$ plus $\det U^{-1}$ generate $R$ over $k$, namely
$R=k\big{[}U,\det U^{-1}\big{]}$;

  3. 3)

$R$ has no proper non-trivial ideals stable by $\partial$, i.e., it is a
simple differential ring.

We say that $R$ is a Picard–Vessiot ring of $k$ for $[A]$. It is an integral
domain and its ring of constants $R^{\partial}$ coincides with $C$. Its
quotient field $K={\rm Frac}(R)$ is generated (as a field) by the entries of
$U$ and its subfield of constant is again $C$. We call $K$ a Picard–Vessiot
extension of $k$ for $[A]$.

######  Remark 2.4.

We are going to use several properties of Picard–Vessiot rings and extensions,
namely:

  1. 1.

If $[A]$ and $[B]$ are two equivalent systems over $k$ then any Picard–Vessiot
extension of $k$ for $[A]$ is a Picard–Vessiot extension of $k$ for $[B]$.
Hence one can define a Picard–Vessiot extension of $k$ for a given
differential module ${\mathcal{M}}=(M,\nabla)$ as a Picard–Vessiot extension
of $k$ for the differential system associated to ${\mathcal{M}}$ with respect
to a basis of $M$.

  2. 2.

Let $K$ be a Picard–Vessiot extension of $k$ for differential module
${\mathcal{M}}=(M,\nabla)$. The Leibnitz rule allows to endow $M\otimes_{k}K$
with a natural structure of differential module over $K$, which will be
denoted by ${\mathcal{M}}\otimes_{k}K$. The definition of $K$ implies that
${\mathcal{M}}\otimes_{k}K$ is trivial, i.e., ${\mathcal{M}}\otimes_{k}K$
admits a basis over $K$ of horizontal elements. One can show that if a module
is trivial, then all its algebraic constructions and their subquotients are
trivial. See [21, Exercice 2.12,5].

  3. 3.

Let $V:=({\mathcal{M}}\otimes_{k}K)^{\nabla}$ be the $C$-vector space of the
horizontal elements of ${\mathcal{M}}\otimes_{k}K$. As already pointed out, it
has dimension $n$.

We give now a definition that we will use in the main theorem.

######  Definition 2.5.

Let ${\mathcal{M}}=(M,\nabla)$ be a differential module and $K$ a
Picard–Vessiot extension. A semi-invariant of ${\mathcal{M}}$ is a horizontal
element $m\otimes g$ contained in some construction of the form ${\rm
Constr}({\mathcal{M}}\otimes_{k}K)\cong({\rm Constr}(M)\otimes_{k}K,\nabla)$,
i.e., an element $m\otimes g$ with $m\in{\rm Constr}(M)$ and $g\in K$ such
that $\nabla(m\otimes g)=0$. If $m$ is a horizontal element in some
construction ${\rm Constr}(M)$, then it is called an invariant of
${\mathcal{M}}$.

For the convenience of the reader, we reprove the following classical lemma
that we will use in this work.

######  Lemma 2.6.

Let ${\mathcal{M}}=(M,\nabla)$ be a differential module and $K$ a
Picard–Vessiot extension of $k$ for ${\mathcal{M}}$. Fix a basis
$\underline{e}$ of ${\mathcal{M}}$ and let $[A]\colon\partial\vec{y}=A\vec{y}$
be the associated linear differential system. The following statements are
equivalent:

  1. $1)$

there exists $m\in M$ such that $\nabla(m)=fm$, for some $f\in k$ $($i.e., $m$
generates a $\nabla$-stable line$)$;

  2. $2)$

there exists a solution $g\vec{v}$ over $K$ of $\partial\vec{y}=A\vec{y}$,
with $g\in K$, such that $\partial(g)/g=-f\in k$, and $\vec{v}\in k^{n}$.

###### Proof.

Let us assume that there exists $m\in M$ such that $\nabla(m)=fm$, for some
$f\in k$. Then the line $L$ generated by $m$ over $k$ is a differential module
${\mathcal{L}}$, and
${\mathcal{L}}\otimes_{k}K\subset{\mathcal{M}}\otimes_{k}K$ is a trivial
differential module (see Remark 2.4). Thus there exists $g\in K$, $g\neq 0$,
such that $m\otimes g$ is a horizontal element of ${\mathcal{L}}\otimes_{k}K$.
It follows that we have

$\displaystyle 0=\nabla(m\otimes g)=fm\otimes
g+m\otimes\partial(g)=m\otimes(fg+\partial(g)),$

which implies $\partial(g)/g=-f\in k$. If we define $\vec{v}\in k^{n}$ by the
relation $m={\underline{e}}\vec{v}$, then we can check that $g\vec{v}$ is a
solution vector of $\partial\vec{y}=A\vec{y}$.

Let us now suppose that we are in the situation described in the second
assertion. If we define $m:={\underline{e}}\vec{v}$, then we have
$\nabla(m)={\underline{e}}(\partial\vec{v}-A\vec{v})$. Moreover, we have
$\partial(g\vec{v})=\partial(g)\vec{v}+g\partial\vec{v}=Ag\vec{v}$ so that
$\partial\vec{v}-A\vec{v}=-(\partial(g)/g)\vec{v}=f\vec{v}$. Finally
$\nabla(m)=fm$, with $f\in k$ which ends the proof. ∎

######  Remark 2.7.

In particular, Lemma 2.6 above implies that, if $m\otimes g$ is a semi-
invariant, then necessarily $\nabla(m)=(-\partial(g)/g)m$, or equivalently $m$
generates a $\nabla$-stable line $L$ contained in a construction ${\rm
Constr}(M)$ of $M$. If $m^{\prime}=hm$, for some $h\in k$, then
$\nabla(m^{\prime})=(-\partial(g)/g+\partial(h)/h)m^{\prime}$ and
$\nabla(m^{\prime}\otimes(g/h))=0$. Moreover, if $m\otimes g$ is a semi-
invariant of ${\mathcal{M}}$, such that $m$ generates a line $L$ in a
construction of $M$, and $c\in C$ is a nonzero constant, then $c(m\otimes g)$
is another semi-invariant, corresponding to the same line $L$. One can prove
that all the semi-invariants can be obtained in this way. Roughly speaking,
semi-invariants of ${\mathcal{M}}$ correspond to exponential solutions of
$\partial\vec{y}=\mathfrak{c}\mathfrak{o}\mathfrak{n}\mathfrak{s}\mathfrak{t}\mathfrak{r}(A)\vec{y}$
and invariants correspond to rational solutions. For more details on these
definitions, see [3, Section 3.4].

For further reference we recall the following lemma:

######  Lemma 2.8 ([3, Lemma 29]).

Let $[A]\colon\partial\vec{y}=A\vec{y}$ be the linear differential system
associated to ${\mathcal{M}}$ with respect to a fixed basis $\underline{e}$.
We suppose that there exist an algebraic extension $k^{\prime}/k$ and a matrix
$P\in{\rm GL}_{n}(k^{\prime})$ such that, for any invariant of ${\mathcal{M}}$
given by a horizontal element in some construction ${\rm
Constr}({\mathcal{M}})$ of coordinates $\vec{v}$ with respect to the basis
${\rm Constr}(\underline{e})$, the vector ${\rm Constr}(P)^{-1}\vec{v}$ has
constant coordinates. Then the same property holds for any invariant of
${\mathcal{M}}\otimes_{k}\overline{k}$, where $\overline{k}$ is the algebraic
closure of $k$.

###  2.3 The differential Galois group

The differential Galois group of $[A]$ (or, equivalently, of ${\mathcal{M}}$)
is defined as

${\rm Gal}_{\partial}([A]):={\rm Aut}^{\partial}(K/k),$

where $K$ is a Picard–Vessiot extension of $k$ for $[A]$ and

$\displaystyle{\rm Aut}^{\partial}(K/k):=\big{\\{}\varphi\colon K\to
K,\text{~{}field automorphism s.t.~{}}\forall\,f\in k,$
$\displaystyle\hphantom{{\rm Aut}^{\partial}(K/k):=\\{}{}\text{we
have~{}}\varphi(f)=f\text{~{}and~{}}[\varphi,\partial]=0\big{\\}}.$

The first result of the differential Galois theory is that any fundamental
matrix of solutions $U\in{\rm GL}_{n}(R)$ of $[A]$ determines a faithful
representation of ${\rm Gal}_{\partial}([A])$ as a _linear algebraic group_
defined over $C$:

$\displaystyle{\rm Gal}_{\partial}([A])$ $\displaystyle\to{\rm GL}_{n}(C),$
$\displaystyle\varphi$ $\displaystyle\mapsto U^{-1}\varphi(U).$

In fact, $\varphi(U)$ is a fundamental matrix of solutions of $[A]$, therefore
$U^{-1}\varphi(U)$ must be an invertible matrix with constant coefficients.
The choice of another fundamental matrix of solutions leads to a conjugated
representation. We will sometimes simply call $G$ the differential Galois
group ${\rm Gal}_{\partial}([A])$, identifying it with its image via the
morphism above and without mentioning the matrix $U$, unless the context makes
it necessary.

We are not explaining here any result on the Galois correspondence and we
refer the interested reader to the literature. For the purpose of this paper,
we mostly need to know that, if $k^{\circ}$ is the relative algebraic closure
of $k$ in $K$, then $K/k^{\circ}$ is a Picard–Vessiot extension for $[A]$ over
$k^{\circ}$. Moreover the field of constants is still $C$ and we have ${\rm
Aut}^{\partial}(K/k^{\circ})=G^{\circ}$, where $G^{\circ}$ is the connected
component of $G$ containing $1$. This means that the differential Galois group
of $[A]$ over $k^{\circ}$ coincides with the automorphisms of $G$ that fix
$k^{\circ}$ and can be identified with $G^{\circ}$.

The Galois–Lie algebra
$\boldsymbol{\mathfrak{g}\mathfrak{a}\mathfrak{l}([A])}$. Since the
differential Galois group $G={\rm Gal}_{\partial}([A])$ is an algebraic group
over $C$, one can naturally consider its Lie algebra
${\mathfrak{g}}:=\mathfrak{g}\mathfrak{a}\mathfrak{l}([A])$ called Galois–Lie
algebra, that is the tangent space to $G^{\circ}$ at $1$. If we look at its
$C$-rational points, we have

$\displaystyle\mathfrak{g}\mathfrak{a}\mathfrak{l}([A])(C)=\\{N\in
M_{n}(C)\colon 1+\varepsilon N\in{\rm
Gal}_{\partial}([A])(C[\varepsilon])\\},$

where ${\rm Gal}_{\partial}([A])(C[\varepsilon])$ are the rational points of
${\rm Gal}_{\partial}([A])$ over the $C$-algebra $C[\varepsilon]$, with
$\varepsilon^{2}=0$.

As it is an algebraic Lie algebra over $C$,
$\mathfrak{g}\mathfrak{a}\mathfrak{l}([A])$ is generated as a $C$-vector space
by a finite subset of $M_{n}(C)$. It will be useful to notice that the same
subset of matrices of $M_{n}(C)$ generates, as a $k$-vector space, the algebra
$\mathfrak{g}\mathfrak{a}\mathfrak{l}([A])(k)$ of $k$-rational points of
$\mathfrak{g}\mathfrak{a}\mathfrak{l}([A])$. For further reference we recall
the following result:

######  Proposition 2.9 ([21, Proposition 1.31]).

Let $\mathfrak{h}$ be an algebraic Lie algebra defined over $C$ and such that
$A\in\mathfrak{h}(k)$. Then
$\mathfrak{g}\mathfrak{a}\mathfrak{l}([A])\subset\mathfrak{h}$.

##  3 Reduced forms

###  3.1 Characterization of reduced forms

We keep the notation of the previous section.

######  Definition 3.1.

We say that a linear differential system $[A]\colon\partial\vec{y}=A\vec{y}$
is in _reduced form_ when $A$ belongs to
$\mathfrak{g}\mathfrak{a}\mathfrak{l}([A])(k)$. Let ${\mathcal{M}}=(M,\nabla)$
be a differential module and $\underline{e}$ be a $k$-basis of $M$. We say
that $\underline{e}$ is a  _reduced basis_ if the system associated to
${\mathcal{M}}$ with respect to the basis $\underline{e}$ is in reduced form.

As pointed out in the introduction, Kolchin and Kovacic has proved that a
reduced form always exists on a finite extension of $k$. See [21, Proposition
1.31]. Criteria for reduced forms have been studied by Aparicio, Compoint and
Weil in [3]. Their main result is generalized below. Our contribution in this
theorem is the new characterization of reduced form stated in item (3).

######  Theorem 3.2.

Let $[A]\colon\partial\vec{y}=A\vec{y}$ be the linear differential system
associated to a differential module ${\mathcal{M}}=(M,\nabla)$ over $k$, with
respect to a fixed basis $\underline{e}$. The following assertions are
equivalent:

  1. $1)$

$[A]$ is in reduced form;

  2. $2)$

for any construction ${\rm Constr}({\mathcal{M}})$ of ${\mathcal{M}}$, every
$\nabla$-stable line of ${\rm Constr}(M)$ admits a constant basis $($i.e., a
basis whose elements have constant coordinates with respect to the basis
induced by $\underline{e})$;

  3. $3)$

for any construction ${\rm Constr}({\mathcal{M}})$ of ${\mathcal{M}}$, every
$\nabla$-stable sub-$k$-vector space of ${\rm Constr}(M)$ admits a constant
basis.

If moreover ${\mathcal{M}}$ is completely reducible $($i.e., it is direct sum
of irreducibles$)$, then the assertions above are equivalent to

  1. $4)$

any invariant of ${\mathcal{M}}$ has constant coordinates.

###### Proof.

Lemma 2.6 and Remark 2.7 show that the constant bases of $\nabla$-stable lines
correspond to the semi-invariants considered in [3]. Therefore
“$(1)\Leftrightarrow(2)$” is a reformulation of [3, Theorem 1]. Since
“$(3)\Rightarrow(2)$” is tautological, it is enough to prove that
“$(2)\Rightarrow(3)$”. Let ${\mathcal{N}}=(N,\nabla)$ be a construction of
${\mathcal{M}}$ and $W$ be a $\nabla$-stable sub-$k$-vector space of $N$ of
dimension $d$. It follows that $\wedge^{d}W$ is $\nabla$-stable sub-$k$-vector
space of dimension $1$ of $\wedge^{d}N$. By assumption, there exists a non-
zero element of $\wedge^{d}W$, whose coordinates $\vec{w}$ with respect to the
basis induced by $\underline{e}$ on $\wedge^{d}N$ are in $C$. Hence, in the
basis induced by $\underline{e}$, the map

$\displaystyle\Psi\colon\ N$ $\displaystyle\to\wedge^{d+1}N,$
$\displaystyle\vec{v}$ $\displaystyle\mapsto\vec{w}\wedge\vec{v},$

is represented by a matrix with coefficients in $C$. It follows that
$\ker\Psi$ has a basis of vectors with coordinates in $C$, with respect to the
basis induced by $\underline{e}$. Since $\ker\Psi=W$, we have proved
“$(2)\Rightarrow(3)$”.

By definition, a horizontal element of a construction ${\rm
Constr}({\mathcal{M}})$ corresponds to a solution of the associated
differential system
$\partial\vec{y}=\mathfrak{c}\mathfrak{o}\mathfrak{n}\mathfrak{s}\mathfrak{t}\mathfrak{r}(A)\vec{y}$
with respect to the basis ${\rm Constr}(\underline{e})$. Definition 2.5 and
[21, Exercice 2.38] imply that the equivalence “$(1)\Leftrightarrow(4)$”
coincides with [3, Proposition 27] which ends the proof of Theorem 3.2. ∎

######  Remark 3.3.

Let $m\in{\rm Constr}(M)$ generate a line which is invariant by $\nabla$ and
let $\underline{e}$ be a reduced basis. By Theorem 3.2 above, we can choose
$m$ such that $m={\rm Constr}(\underline{e})\vec{v}$, with $\vec{v}\in C^{n}$.
Moreover since $\nabla(m)=fm$ for some $f\in k$, there exists $g\in K$
verifying $\partial(g)=fg$. This means that $g\vec{v}$ is a solution vector of
the system associated to ${\rm Constr}({\mathcal{M}})$ with respect to ${\rm
Constr}(\underline{e})$. This is what we mean when we say that invariants and
semi-invariants have constant coordinates with respect to this reduced basis.

######  Example 3.4.

Theorem 3.2 has as corollary the following known fact: in a reduced basis, the
eigenring of ${\mathcal{M}}$ (see Example 2.3 for the definition) can be
identified to the ring of matrices with coefficients in $C$ which commute with
$A$, hence with all the elements of a Wei–Norman decomposition of $A$ (see [3,
Section 2.2] and [24]).

The Kolchin–Kovacic theorem implies in particular that there exists a finite
extension $k^{\prime}/k$ such that ${\mathcal{M}}\otimes_{k}k^{\prime}$ admits
a reduced basis. See [21, Proposition 1.31 and Corollary 1.32] and [3, Remark
31].

######  Definition 3.5.

We say that $k^{\prime}$ is a _reduction field_ when
${\mathcal{M}}\otimes_{k}k^{\prime}$ admits a reduced basis.

######  Remark 3.6.

When $k^{\prime}$ is a reduction field, then the differential Galois group of
${\mathcal{M}}\otimes_{k}k^{\prime}$ is connected, see [3, Lemma 32], and the
Galois correspondence implies that $k^{\prime}\cap K=k^{\circ}$, the fixed
field of $G^{\circ}$ in the Picard–Vessiot extension $K$.

###  3.2 Gauge transformation to a reduced form with local conditions

######  Assumption 3.7.

In this section, we suppose that the field $k$ is a subfield of the field of
meromorphic functions over a region $D$ of ${\mathbb{C}}$ in the variable $x$
such that $x\in k$ and $\partial=\frac{{\rm d}}{{\rm d}x}$.

######  Remark 3.8.

The field of rational functions $k={\mathbb{C}}(x)$ satisfies the assumption
above, as well as most differential fields occurring in the concrete examples.
Indeed let $k$ be any differential field. Then the entries of the matrix $A$
of the linear differential system $[A]\colon\partial\vec{y}=A\vec{y}$,
generate a differential field $\widetilde{k}$, which is a finitely generated
differential extension of ${\mathbb{Q}}$. Seidenberg’s embedding theorem (see
[17, 18]) ensures that $\widetilde{k}$ can be embedded isomorphically in a
differential field of meromorphic functions on an open region $D$ of
${\mathbb{C}}$.

We consider a linear differential system $[A]\colon\partial\vec{y}=A\vec{y}$,
with coefficients in $k$. For all points $x_{0}\in D$ such that $x_{0}$ is not
a pole of $A$, the system $[A]$ has a fundamental matrix of solutions
$\widehat{U}_{x_{0}}(x)$ with the following properties:

  1. 1)

$\widehat{U}_{x_{0}}(x)\in{\rm GL}_{n}({\mathbb{C}}[[x-x_{0}]])$,

  2. 2)

$\widehat{U}_{x_{0}}(x_{0})={\rm Id}$, the identity matrix.

The columns of $\widehat{U}_{x_{0}}(x)$ generate over ${\mathbb{C}}$ the
vector space $V$ of solutions of $[A]$ (contained in
${\mathbb{C}}[[x-x_{0}]]^{n}$), and $K:=k(\widehat{U}_{x_{0}}(x))$ is a
Picard–Vessiot extension of $[A]$. Let ${\rm Constr}$ be a construction. Then
${\rm Constr}\big{(}\widehat{U}_{x_{0}}\big{)}(x_{0})$ is the identity matrix
(because ${\rm Constr}$ acts on matrices as a group morphism). Furthermore,
${\rm Constr}\big{(}\widehat{U}_{x_{0}}\big{)}$ is a fundamental matrix of
solutions for
$[\mathfrak{c}\mathfrak{o}\mathfrak{n}\mathfrak{s}\mathfrak{t}\mathfrak{r}(A)]$.
Consider an invariant of ${\mathcal{M}}$ given by a horizontal element $m$ in
${\rm Constr}({\mathcal{M}})$ such that $m$ has coordinates $\vec{v}(x)\in
k^{N}$ with respect to the basis ${\rm Constr}(\underline{e})$. The vector
$\vec{v}(x)$ is a solution of
$[\mathfrak{c}\mathfrak{o}\mathfrak{n}\mathfrak{s}\mathfrak{t}\mathfrak{r}(A)]$
and hence we have $\vec{v}(x)={\rm
Constr}\big{(}\widehat{U}_{x_{0}}\big{)}(x)\vec{w}$ for some $\vec{w}$ with
constant coefficients in ${\mathbb{C}}$. The fact that ${\rm
Constr}\big{(}\widehat{U}_{x_{0}}\big{)}(x_{0})$ is the identity matrix allows
to conclude that $\vec{w}=\vec{v}(x_{0})$.

In the theorem below, we give an algebraic characterization for a reduction
matrix. The original Kolchin–Kovacic theorem ensures that, when the base field
$k$ is a $C^{1}$-field, there exists a reduced basis defined on the relative
algebraic closure $k^{\circ}$ of $k$ in the Picard–Vessiot extension $K$.
However, its proof is not effective and it relies on finding rational points
on varieties and this part is not either, to our knowledge, algorithmic yet.
In order to gain effectiveness, we enlarge $k$ to an algebraic extension
$k^{\prime}$, in which we can compute (without $C^{1}$-assumptions on $k$) a
reduction matrix that is characterized by the property of transforming any
invariant in its “value at $x_{0}$”. The field $k^{\prime}$ may depend on the
choice of $x_{0}$. However, as seen in Remark 3.6, we have $k^{\prime}\cap
K=k^{\circ}$ so $k^{\prime}\cap K$ does not depend on the choice of $x_{0}$.

The spirit of this result and of our proof appears in the proof of [3, Theorem
3] in a particular case. It is extended here to all constructions and this is
useful for reduced form algorithms; see [4, Lemma 5.1], where this result was
alluded to, without a full proof for lack of space.

######  Theorem 3.9.

Let us consider a linear differential system $\partial\vec{y}=A\vec{y}$,
defined over a field $k$, associated to a completely reducible differential
module ${\mathcal{M}}$ with respect to a fixed basis $\underline{e}$. We
choose a point $x_{0}\in D$ such that $A$ does not have a pole at $x_{0}$.
Then, there exists a finite extension $k^{\prime}$ of $k$ and a matrix
$P_{x_{0}}\in{\rm GL}_{n}(k^{\prime})$ such that
$\underline{f}:=\underline{e}P_{x_{0}}$ is a reduced basis of
${\mathcal{M}}\otimes_{k}k^{\prime}$, having the following property:

> For any invariant of ${\mathcal{M}}$ given by a horizontal element $m$ in
> some construction ${\rm Constr}({\mathcal{M}})$ such that $m$ has
> coordinates $\vec{v}(x)\in k^{N}$ with respect to the basis ${\rm
> Constr}(\underline{e})$, we have $\vec{v}(x)={\rm
> Constr}(P_{x_{0}})\vec{v}(x_{0})$.

######  Remark 3.10.

In the last statement, we view ${\rm Constr}(\underline{e})$ as a basis of
${\rm Constr}(M\otimes_{k}k^{\prime})$ and identify an element $m\in{\rm
Constr}(M)$ with its image $m\otimes 1\in{\rm
Constr}(M)\otimes_{k}k^{\prime}\cong{\rm Constr}(M\otimes_{k}k^{\prime})$.

###### Proof.

In the new basis $\underline{f}$, if it exists, all invariants are constant
therefore it is a reduced basis, thanks to Lemma 2.8 and Theorem 3.2. We now
prove the existence. Let $X$ be a matrix with indeterminate entries. For any
invariant in some construction ${\rm Constr}({\mathcal{M}})$ having
coordinates $\vec{v}(x)\in k^{N}$ with respect to the basis ${\rm
Constr}(\underline{e})$, consider the equation $\vec{v}(x)={\rm
Constr}(X)\vec{v}(x_{0})$. It provides an infinite set of polynomial equations
over $k$ in the entries of $X$ which are all satisfied by the entries of a
fundamental matrix of solutions $\widehat{U}_{x_{0}}(x)$ of $[A]$ (see the
explanations above). As a consequence, the latter set of equations generates a
proper ideal in the ring of polynomials in $n^{2}$ variables, which is
finitely generated because of the noetherianity. The Nullstellensatz then
ensures that there exists a solution $P\in{\rm GL}_{n}(k^{\prime})$, where
$k^{\prime}/k$ is a finite extension of $k$. ∎

######  Remark 3.11.

Assume that ${\mathcal{M}}$ admits a reduced basis $\underline{e}$. Consider
an invariant $m$ in some construction ${\rm Constr}({\mathcal{M}})$ having
(constant) coordinates $\vec{v}\in C^{N}$ with respect to the basis ${\rm
Constr}(\underline{e})$. Then we have $\vec{v}={\rm
Constr}\big{(}{\widehat{U}_{x_{0}}}(x)\big{)}\vec{c}$ and the constant vector
$\vec{c}$ is an invariant (in the usual sense of representation theory) of the
Galois group ${\rm Gal}_{\partial}([A])$ in its representation induced by
$\widehat{U}_{x_{0}}(x)$. Now, by construction,
${\widehat{U}}_{x_{0}}(x_{0})={\rm Id}$ (the identity matrix); as ${\rm
Constr}$ acts as a group morphism, we see that ${\rm
Constr}\big{(}{\widehat{U}}_{x_{0}}\big{)}(x_{0})={\rm Id}$. Now, because
$\vec{v}$ is constant, we have $\vec{v}={\rm
Constr}\big{(}{\widehat{U}}_{x_{0}}\big{)}(x_{0})\vec{c}$ so
$\vec{v}=\vec{c}$. In a reduced basis, this observation allows to identify the
invariants of ${\mathcal{M}}$ and those of ${\rm Gal}_{\partial}([A])$.

##  4 The intrinsic Galois–Lie algebra of Katz

Let ${\mathcal{M}}=(M,\nabla)$ be a differential module over $k$.

######  Definition 4.1 ([13]).

The _intrinsic Galois group $G_{\rm Katz}$ of Katz_ is the set of
$\varphi\in{\rm GL}(M)$ such that, for any $\nabla$-stable sub-$k$-vector
space $N$ of a construction ${\rm Constr}(M)$, $N$ is set-wise stable under
${\rm Constr}(\varphi)$.

The _Katz algebra_ ${\mathfrak{g}}_{\rm Katz}$ is the Lie algebra of $G_{\rm
Katz}$.

######  Remark 4.2.

For $\varphi\in\mathop{\rm End}(M)$, the functor $\mathfrak{constr}$ acts on
$\varphi$ as a Lie algebra morphism, see the explanations before Example 2.3.
In particular, for any invariant $m\in{\rm Constr}(M)$ and
$\varphi\in{\mathfrak{g}}_{\rm Katz}$, we have
$\mathfrak{constr}(\varphi)(m)=0$.

By Noetherianity, ${\mathfrak{g}}_{\rm Katz}$ is a stabilizer of a finite
family of differential modules contained in some constructions of
${\mathcal{M}}$. This shows that ${\mathfrak{g}}_{\rm Katz}$ is an _algebraic_
Lie algebra. It is the central object of the famous Grothendieck–Katz
conjecture on $p$-curvatures. The following properties of ${\mathfrak{g}}_{\rm
Katz}$ are known, see [2, 13, 21], but are reproved for self-containedness.

######  Lemma 4.3.

Let ${\mathcal{M}}=(M,\nabla)$ be a differential module over $k$, $K$ a
Picard–Vessiot extension, $G$ the differential Galois group of ${\mathcal{M}}$
and ${\mathfrak{g}}$ the Galois–Lie algebra.

  1. $1.$

The Lie algebra ${\mathfrak{g}}_{\rm Katz}$ can be defined as the stabilizer
of a single line in an algebraic construction of $M$ $($which can be chosen to
be $\nabla$-invariant$)$.

  2. $2.$

The Lie algebra ${\mathfrak{g}}_{\rm Katz}$ is a differential module for the
structure induced by ${\mathcal{M}}\otimes_{k}{\mathcal{M}}^{*}$.

  3. $3.$

${\mathfrak{g}}_{\rm Katz}=({\mathfrak{g}}\otimes_{C}K)^{G}$.

###### Proof.

1\. Because of the Noetherianity, a theorem of Chevalley ensures that
${\mathfrak{g}}_{\rm Katz}$ is the stabilizer of a finite family $W_{i}$ of
$\nabla$-stable $k$-vector spaces contained in some constructions of $M$. A
classical argument then shows that ${\mathfrak{g}}_{\rm Katz}$ can be defined
as the stabilizer of $\wedge^{d}\oplus_{i}W_{i}$, where $d$ is the dimension
of $\oplus_{i}W_{i}$ over $k$. See [13, proof of Proposition 9.3]

2\. If $m$ generates a $\nabla$-stable line in some construction of $M$, which
defines ${\mathfrak{g}}_{\rm Katz}$ as a stabilizer, and
$\varphi\in{\mathfrak{g}}_{\rm Katz}$, then
$\mathfrak{constr}(\varphi)(m)=\alpha m$ and $\nabla(m)=\beta m$, for some
$\alpha,\beta\in k$. We conclude that

$\displaystyle\nabla(\mathfrak{constr}(\varphi))(m)$
$\displaystyle=\nabla(\mathfrak{constr}(\varphi)(m))-\mathfrak{constr}(\varphi)(\nabla(m))=\nabla(\alpha
m)-\mathfrak{constr}(\varphi)(\beta m)$
$\displaystyle=\partial(\alpha)m+\alpha\nabla(m)-\beta\mathfrak{constr}(\varphi)(m)=\partial(\alpha)m.$

Therefore $\nabla(\mathfrak{constr}(\varphi))$ belongs to ${\mathfrak{g}}_{\rm
Katz}$.

3\. The Tannakian correspondence will show that ${\mathfrak{g}}_{\rm
Katz}\cong\left({\mathfrak{g}}\otimes_{k}K\right)^{G}$. Indeed, we have seen
that ${\mathfrak{g}}_{\rm Katz}$ is the stabilizer of a line $L$. Then
$L_{C}:=(L\otimes_{C}K)^{\nabla}$ is a $C$-line defined in the corresponding
construction on the $C$-vector space of solutions
$V:=(M\otimes_{k}K)^{\nabla}$. It is stabilized by ${\mathfrak{g}}$ so
${\mathfrak{g}}\otimes_{C}K$ stabilizes $L_{C}\otimes_{C}K$. Hence
$({\mathfrak{g}}\otimes_{C}K)^{G}$ stabilises $(L_{C}\otimes_{C}K)^{G}$. Now
$(L_{C}\otimes_{C}K)^{G}=L$ so $({\mathfrak{g}}\otimes_{C}K)^{G}$ stabilises
$L$ and thus $({\mathfrak{g}}\otimes_{C}K)^{G}\subseteq{\mathfrak{g}}_{\rm
Katz}$. In fact, the same argument shows that
$({\mathfrak{g}}\otimes_{C}K)^{G}$ stabilizes any differential module in a
construction and thus $({\mathfrak{g}}\otimes_{C}K)^{G}={\mathfrak{g}}_{\rm
Katz}$.

This ends the proof. ∎

######  Definition 4.4.

An invariant $m$ of ${\mathcal{M}}$ in ${\rm Constr}({\mathcal{M}})$ is called
a _Chevalley invariant_ of ${\mathfrak{g}}_{\rm Katz}$ when
${\mathfrak{g}}_{\rm Katz}=\\{h\in\mathfrak{gl}(M)\,|\,h(m)=0\\}$.

######  Theorem 4.5.

Let ${\mathcal{M}}=(M,\nabla)$ be a differential module over $k$, $G$ the
differential Galois group of ${\mathcal{M}}$, ${\mathfrak{g}}$ the Galois–Lie
algebra and ${\mathfrak{g}}_{\rm Katz}$ the Katz algebra. Let $k^{\prime}$ be
a reduction field so that ${\mathcal{M}}\otimes_{k}k^{\prime}$ admits a
reduced basis $\underline{e}$. If ${\mathcal{M}}$ is completely reducible,
then ${\mathfrak{g}}\otimes_{C}k^{\prime}={\mathfrak{g}}_{\rm
Katz}\otimes_{k}k^{\prime}$.

###### Proof.

In a reduced basis of ${\mathcal{M}}\otimes_{k}k^{\prime}$, Remark 3.11 shows
that ${\mathfrak{g}}$ and ${\mathfrak{g}}_{\rm Katz}\otimes_{k}k^{\prime}$
have the same invariants. Now ${\mathcal{M}}$ is completely reducible
(equivalently, the Galois group $G$ is reductive) so both ${\mathfrak{g}}$ and
${\mathfrak{g}}_{\rm Katz}$ are completely reducible and this remains true
after extension of scalars. From [6, Chapter II, Section 5.5, p. 92] or [3,
Lemma 26], it follows that both ${\mathfrak{g}}$ and ${\mathfrak{g}}_{\rm
Katz}\otimes_{k}k^{\prime}$ are determined by their invariants. These two
observations show that a Chevalley invariant of ${\mathfrak{g}}$ is also a
Chevalley invariant of ${\mathfrak{g}}_{\rm Katz}\otimes_{k}k^{\prime}$. Let
$m$ be a Chevalley invariant of ${\mathfrak{g}}_{\rm
Katz}\otimes_{k}k^{\prime}$ in some construction ${\rm
Constr}(M)\otimes_{k}k^{\prime}$. In the reduced basis, $m$ has constant
coefficients: we have $m={\rm Constr}(\underline{e})\vec{v}$, with $\vec{v}\in
C^{n}$. Thus, a matrix $N=(n_{ij})_{i,j}$ is in ${\mathfrak{g}}_{\rm
Katz}\otimes_{k}k^{\prime}$ if and only if $\mathfrak{constr}(N)\vec{v}=0$.
This relation yields a system of linear equations
$\mathcal{L}((n_{i,j})_{i,j})=0$ for the entries $(n_{i,j})_{i,j}$ of $N$,
with coefficients in $C$. Now, as seen above, $\vec{v}$ is also a Chevalley
invariant of the Galois–Lie algebra ${\mathfrak{g}}$. So a matrix is in
${\mathfrak{g}}$ if and only if its entries satisfy the same linear equations
$\mathcal{L}((n_{i,j})_{i,j})=0$. Consequently,
${\mathfrak{g}}\otimes_{C}k^{\prime}$ and ${\mathfrak{g}}_{\rm
Katz}\otimes_{k}k^{\prime}$ share the same constant basis so that
${\mathfrak{g}}\otimes_{C}k^{\prime}={\mathfrak{g}}_{\rm
Katz}\otimes_{k}k^{\prime}$. ∎

######  Definition 4.6.

Let $k^{\prime}$ be an extension of $k$ and $W$ be a subspace of a
construction ${\rm Constr}(M)\allowbreak\otimes_{k}k^{\prime}$. We say that
$W$ is _defined over $C$_ when it is generated by constant matrices (i.e.,
when it is a $k^{\prime}$-form of a $C$-space).

######  Corollary 4.7.

Let $k^{\prime}$ be a reduction field so that
${\mathcal{M}}\otimes_{k}k^{\prime}$ admits a reduced basis $\underline{e}$.
Then ${\mathfrak{g}}_{\rm Katz}\otimes_{k}k^{\prime}$ is defined over $C$.
Moreover, if $A$ is the matrix of the associated linear differential system
with respect to $\underline{e}$, then $A\in{\mathfrak{g}}_{\rm
Katz}\otimes_{k}k^{\prime}$.

###### Proof.

Let $L$ be a $\nabla$-invariant line in some ${\rm Constr}(M)$ such that
${\mathfrak{g}}_{\rm Katz}$ is the stabilizer of $L$. By Theorem 3.2, we can
choose a generator $m$ of $L\otimes_{C}k^{\prime}$ whose coordinates are
constant with respect to the reduced basis. This shows that
${\mathfrak{g}}_{\rm Katz}\otimes_{k}k^{\prime}$ (but not ${\mathfrak{g}}_{\rm
Katz}$ in general) is defined over $C$. The fact that $A\in{\mathfrak{g}}_{\rm
Katz}\otimes_{k}k^{\prime}$ follows from the definition of a reduced basis and
from Theorem 4.5. ∎

######  Example 4.8.

We illustrate the previous results on an example where everything can be
checked by hand calculations. Let

$A=\left(\begin{matrix}0&1\\\ x&\frac{1}{2x}\end{matrix}\right).$

The Galois group $G$ is a central extension of the infinite dihedral group
(see [3, Example 6.1]). The connected component $G^{\circ}$ of $G$ containing
$1$ is the multiplicative group $G_{m}$ with Lie algebra $\mathfrak{g}_{m}$
generated by

$\left(\begin{matrix}1&0\\\ 0&-1\end{matrix}\right).$

Furthermore, using [4], we can see that the Katz algebra is $1$-dimensional
and it is generated by

$N_{1}:=\left(\begin{matrix}0&\frac{1}{x}\\\ 1&0\end{matrix}\right).$

We have $\nabla(N_{1}):=N_{1}^{\prime}-[A,N_{1}]=-\frac{1}{2x}N_{1}$. Over
$k=C(x)$, no nonzero multiple of $N_{1}$ is conjugated to a constant matrix.
However, over the finite extension $k^{\prime}:=C(\sqrt{x})$, $\sqrt{x}N_{1}$
is conjugated to a constant matrix. Indeed, we have $N_{1}=P^{-1}DP$ with

$D=\left(\begin{matrix}{\frac{1}{\sqrt{x}}}&0\\\
0&-\frac{1}{\sqrt{x}}\end{matrix}\right)=\frac{1}{\sqrt{x}}\left(\begin{matrix}1&0\\\
0&-1\end{matrix}\right),\qquad P=\left(\begin{matrix}1&-1\\\
\sqrt{x}&\sqrt{x}\end{matrix}\right).$

We see that ${\mathfrak{g}}_{\rm Katz}\otimes_{k}k^{\prime}$ is now generated
by the (constant) generator of the Galois–Lie algebra $\mathfrak{g}_{m}$ and,
applying the gauge transformation $P$, we have the reduced form

$P[A]=\left(\begin{matrix}\sqrt{x}&0\\\
0&-\sqrt{x}\end{matrix}\right)=\sqrt{x}\left(\begin{matrix}1&0\\\
0&-1\end{matrix}\right).$

This shows that ${\mathfrak{g}}_{\rm
Katz}\otimes_{k}k^{\prime}=\mathfrak{g}_{m}\otimes_{C}k^{\prime}$ and
$P[A]\in{\mathfrak{g}}_{\rm Katz}\otimes_{k}k^{\prime}$. Note that
${\mathfrak{g}}_{\rm Katz}$ is _not_ defined over $C$ whereas
${\mathfrak{g}}_{\rm Katz}\otimes_{k}k^{\prime}$ is.

######  Remark 4.9.

Following the definition of the eigenring $\mathcal{E}([A])$ in Example 2.3,
any matrix in $\mathcal{E}([A])$ maps a solution of $[A]$ to a solution of
$[A]$. Example 3.4 showed that, in a reduced basis, the eigenring is generated
by constant matrices which commute with $A$. In reduced form, we have
$A=\sum\limits_{i=1}^{d}f_{i}N_{i}$ where $f_{i}\in k^{\prime}$, the $f_{i}$
are linearly independent over $C$ and the $N_{i}$ generate $\mathfrak{g}$ (see
[3, Definition 6 and Remark 7]). Theorem 4.5 shows that the $N_{i}$ also
generate ${\mathfrak{g}}_{\rm Katz}\otimes_{k}k^{\prime}$ as a Lie algebra.
Given a constant matrix $T\in\mathcal{E}([A])$, we have $[A,T]=0$, i.e.,
$\sum\limits_{i=1}^{d}f_{i}[N_{i},T]=0$. As the $f_{i}$ are linearly
independent over $C$, it follows that we have $[N_{i},T]=0$ for all $i$. So,
in a reduced basis, the eigenring becomes the set of constant matrices which
commute with ${\mathfrak{g}}_{\rm Katz}\otimes_{k}k^{\prime}$ (or,
equivalently, with $\mathfrak{g}$).

In [2], André warns that spaces which are stable under ${\mathfrak{g}}_{\rm
Katz}$ may not be stable under $\nabla$. We can see this easily using reduced
forms. Consider a linear differential system
$[A]\colon\partial\vec{y}=A\vec{y}$ having all its solutions algebraic over
$k$. Then ${\mathfrak{g}}_{\rm Katz}=0$ so that anything is stable under
${\mathfrak{g}}_{\rm Katz}$. In a reduced basis, the matrix of the linear
differential system is the zero matrix and $\nabla$ coincides with
$\partial=\frac{{\rm d}}{{\rm d}x}$. A random vector in $k^{n}$ is not stable
under $\nabla$ even though it is stable under ${\mathfrak{g}}_{\rm Katz}$.
However, any line defined over $C$ is clearly stable under $\partial$ and
hence under $\nabla$. The next result builds on this observation to
characterize which spaces, among those which are stable under
${\mathfrak{g}}_{\rm Katz}$, are stable under $\nabla$.

######  Proposition 4.10.

Let ${\mathcal{M}}=(M,\nabla)$ be a completely reducible differential module
over $k$ and $k^{\prime}$ denote a reduction field for ${\mathcal{M}}$. Let
$W$ be a subspace of a construction ${\rm Constr}(M)\otimes_{k}k^{\prime}$.
Then $W$ is stable under $\nabla$ if and only if both conditions below are
fulfilled:

  1. $1)$

$W$ is stable under ${\mathfrak{g}}_{\rm Katz}\otimes_{k}k^{\prime}$;

  2. $2)$

$W$ is defined over $C$.

###### Proof.

The “only if” part follows from the definition of the Katz algebra (1) and
from Theorem 3.2(2). Now, assume that $W$ is stable under ${\mathfrak{g}}_{\rm
Katz}\otimes_{k}k^{\prime}$ and defined over $C$. Let $C_{1},\ldots,C_{s}$
denote a constant basis of $W$. Let $A$ be the matrix of the linear
differential system associated with ${\mathcal{M}}\otimes_{k}k^{\prime}$ in a
reduced basis. We have $A=\sum_{j}f_{j}(x)N_{j}$, where $f_{j}(x)\in
k^{\prime}$ and the $N_{j}$ form a (constant) basis of ${\mathfrak{g}}_{\rm
Katz}$ (and of ${\mathfrak{g}}$). As the $C_{i}$ are constant, $\nabla$ acts
on them via $\nabla(C_{i})=-\mathfrak{constr}(A)C_{i}$ and thus
$\nabla(C_{i})=-\sum_{j}f_{j}(x)\mathfrak{constr}(N_{j})C_{i}$. Now, by
hypothesis $W$ is stable under ${\mathfrak{g}}_{\rm
Katz}\otimes_{k}k^{\prime}$ so that $\mathfrak{constr}(N_{j})C_{i}$ is a
linear combination (over $k^{\prime}$) of the $C_{l}$. It follows that
$\nabla(C_{i})$ is in $W$ as announced. ∎

Each space in ${\rm Constr}(V)$ which is invariant under $\mathfrak{g}$ is in
(Tannakian) correspondence with a submodule of ${\rm Constr}(M)$, invariant
under $\nabla$ and hence under ${\mathfrak{g}}_{\rm Katz}$. A reciprocal
property would be to characterize, among all subspaces in any Constr(M) that
are stable under ${\mathfrak{g}}_{\rm Katz}$, which ones are stable under
$\nabla$ (and hence are in Tannakian correspondence with a
$\mathfrak{g}$-module in ${\rm Constr}(V)$. Proposition 4.10 shows that this
it is possible to do that at the cost of extending scalars. When we study the
subspaces of ${\rm Constr}(M)\otimes_{k}k^{\prime}$ which are invariant under
${\mathfrak{g}}_{\rm Katz}\otimes_{k}k^{\prime}$, the ones which are invariant
under $\nabla$ (and hence in tannakian correspondence with a
$\mathfrak{g}$-module) are exactly those which admit a constant basis.

### Acknowledgements

We are grateful to the anonymous referees for their relevant suggestions which
helped us to improve the clarity and quality of this work.

## References

  * [1] Amzallag E., Minchenko A., Pogudin G., Degree bound for toric envelope of a linear algebraic group, arXiv:1809.06489.
  * [2] André Y., Sur la conjecture des $p$-courbures de Grothendieck–Katz et un problème de Dwork, in Geometric Aspects of Dwork Theory, Vols. I, II, Walter de Gruyter, Berlin, 2004, 55–112.
  * [3] Aparicio-Monforte A., Compoint E., Weil J.-A., A characterization of reduced forms of linear differential systems, J. Pure Appl. Algebra 217 (2013), 1504–1516, arXiv:1206.6661.
  * [4] Barkatou M., Cluzeau T., Weil J.-A., Di Vizio L., Computing the Lie algebra of the differential Galois group of a linear differential system, in Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2016, 63–70.
  * [5] Bertrand D., Groupes algébriques et équations différentielles linéaires, Astérisque 206 (1992), Exp. No. 750, 4, 183–204.
  * [6] Borel A., Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, New York, 1991.
  * [7] Combot T., Sanabria C., A symplectic Kovacic’s algorithm in dimension 4, in ISSAC’18 – Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2018, 143–150, arXiv:1802.01023.
  * [8] Compoint E., Singer M.F., Computing Galois groups of completely reducible differential equations, J. Symbolic Comput. 28 (1999), 473–494.
  * [9] Dreyfus T., Weil J.-A., Computing the Lie algebra of the differential Galois group: the reducible case, arXiv:1904.07925.
  * [10] Feng R., Hrushovski’s algorithm for computing the Galois group of a linear differential equation, Adv. in Appl. Math. 65 (2015), 1–37, arXiv:1312.5029.
  * [11] Hessinger S.A., Computing the Galois group of a linear differential equation of order four, Appl. Algebra Engrg. Comm. Comput. 11 (2001), 489–536.
  * [12] Hrushovski E., Computing the Galois group of a linear differential equation, in Differential Galois Theory (Bȩdlewo, 2001), Banach Center Publ., Vol. 58, Polish Acad. Sci. Inst. Math., Warsaw, 2002, 97–138.
  * [13] Katz N.M., A conjecture in the arithmetic theory of differential equations, Bull. Soc. Math. France 110 (1982), 203–239.
  * [14] Kovacic J.J., An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput. 2 (1986), 3–43.
  * [15] Nguyen K.A., van der Put M., Solving linear differential equations, Pure Appl. Math. Q. 6 (2010), 173–208.
  * [16] Person A.C., Solving homogeneous linear differential equations of order 4 in terms of equations of smaller order, Ph.D. Thesis, North Carolina State University, 2002.
  * [17] Seidenberg A., Some basic theorems in differential algebra (characteristic $p$, arbitrary), Trans. Amer. Math. Soc. 73 (1952), 174–190.
  * [18] Seidenberg A., Contribution to the Picard–Vessiot theory of homogeneous linear differential equations, Amer. J. Math. 78 (1956), 808–818.
  * [19] Singer M.F., Ulmer F., Galois groups of second and third order linear differential equations, J. Symbolic Comput. 16 (1993), 9–36.
  * [20] van der Hoeven J., Around the numeric-symbolic computation of differential Galois groups, J. Symbolic Comput. 42 (2007), 236–264.
  * [21] van der Put M., Singer M.F., Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, Vol. 328, Springer-Verlag, Berlin, 2003.
  * [22] van Hoeij M., Decomposing a 4th order linear differential equation as a symmetric product, in Differential Galois Theory (Bȩdlewo, 2001), Banach Center Publ., Vol. 58, Polish Acad. Sci. Inst. Math., Warsaw, 2002, 89–96.
  * [23] van Hoeij M., Ragot J.F., Ulmer F., Weil J.-A., Liouvillian solutions of linear differential equations of order three and higher, J. Symbolic Comput. 28 (1999), 589–609.
  * [24] Wei J., Norman E., On global representations of the solutions of linear differential equations as a product of exponentials, Proc. Amer. Math. Soc. 15 (1964), 327–334.