Research ContributionArnold Mathematical Journal

Received: 25 November 2014 / Revised: 20 August 2015 / Accepted: 11 November 2015

# Solvability of Linear Differential Systems with Small Exponents in the Liouvillian Sense

R. R. Gontsov Insitute for Information Transmission Problems RAS,
Bolshoy Karetny per. 19 Moscow 127994 Russia,
gontsovrr@gmail.com
I. V. Vyugin National Research University Higher School of Economics,
Vavilova str. 7 Moscow 117312 Russia,
vyugin@gmail.com

### Abstract

The paper is devoted to solvability of linear differential systems by quadratures, one of the classical problems of differential Galois theory. As known, solvability of a system depends entirely on properties of its differential Galois group. However, detecting solvability or non-solvability of a given system is a difficult problem, because the dependence of its differential Galois group on the coefficients of the system remains unknown. We consider systems with regular singular points as well as those with non-resonant irregular ones, whose exponents (respectively, so-called formal exponents in the irregular case) are sufficiently small. It turns out that for systems satisfying such restrictions criteria of solvability can be given in terms of the coefficient matrix.

###### Keywords
Picard–Vessiot extension, Differential Galois groupLiouvillian solution, Stokes matrices
###### Mathematics Subject Classification
34M03, 34M15, 34M35, 12H05

## 1 Introduction

Consider on the Riemann sphere $\overline{\mathbb{C}}$ a linear differential system

 $\displaystyle\frac{dy}{dz}=B(z)\,y,\quad y(z)\in{\mathbb{C}}^{p},$ (1)

of $p$ equations with a meromorphic coefficient matrix $B$ (whose entries are thus rational functions) and singularities at some points $a_{1},\ldots,a_{n}$. Solvability of linear differential equations and systems in the Liouvillian sense (in other words, by quadratures) is a classical problem of differential Galois theory developed by Picard and Vessiot at the beginning of the twentieth century. In analogy to Galois theory of algebraic equations, they connected to the system a group called the differential Galois group and showed that solvability of the system by quadratures depends entirely on properties of its differential Galois group. Later, in the middle of the century, Kolchin completed this theory by considering other types of solvability and their dependence on properties of the differential Galois group.

However, the Picard–Vessiot–Kolchin theory reveals the cause of solvability or non-solvability of linear differential equations by quadratures rather than answers this question addressed to a specific equation, since one does not know how the differential Galois group of an equation depends on its coefficients. In our paper, we are interested in the cases when the answer to the question of solvability of the system (1) by quadratures can be given in terms of its coefficient matrix $B$. For example, in the case of a Fuchsian system

 $\displaystyle\frac{dy}{dz}=\left(\sum_{i=1}^{n}\frac{B_{i}}{z-a_{i}}\right)y, \quad B_{i}\in{\rm Mat}(p,{\mathbb{C}}),$ (2)

with sufficiently small entries of the matrices $B_{i}$, [Ilyashenko and Khovanskii1974] (see also [Khovanskii2008], Ch. 6, §2.3) obtained an explicit criterion of solvability. Namely, the following statement holds:

There exists an $\varepsilon=\varepsilon(n,p)>0$ such that a condition of solvability by quadratures for the Fuchsian system (2) with $\|B_{i}\|<\varepsilon$ takes an explicit form: the system is solvable by quadratures if and only if all the matrices $B_{i}$ are triangular (in some common basis).

Using results of Kolchin, these authors also obtained criteria for other types of solvability of the Fuchsian system (2) with small residue matrices $B_{i}$ in terms of these matrices. Moreover, a topological version of Galois theory developed by Khovanskii at the beginning of 1970’s allowed him to obtain stronger results concerning non-solvability of Fuchsian systems.

Our paper may be considered as an attempt to generalize results of Ilyashenko and Khovanskii to the case of non-Fuchsian systems. It is organized as follows. In the next two sections we recall basic notions of differential Galois theory and of the local theory of linear differential systems, which are used in the paper.

In Sect. 4, we describe a weak version of Ramis’s theorem (due to Ilyashenko and Khovanskii) concerning the description of the local differential Galois group at a singular point of a system in terms of local meromorphic invariants of the latter. This theorem is essentially used to prove statements on solvability of non-Fuchsian systems by quadratures. In this section we also recall the notion of exponential torus, a subgroup of the local differential Galois group, which also appears in further proofs.

Section 5 is devoted to an approach that uses holomorphic vector bundles with meromorphic connections on the Riemann sphere in a context of the analytic theory of linear differential equations. This approach was developed and successfully applied by Bolibrukh to solving inverse monodromy problems of this theory, in particular, Hilbert’s 21st (the Riemann–Hilbert) problem. Here we also use this approach, combining it with techniques from differential Galois theory, to obtain statements concerning solvability of linear differential systems by quadratures.

Section 6 is about solvability of Fuchsian systems (more generally, systems with regular singular points) by quadratures. In particular, we refine here the Ilyashenko–Khovanskii criterion in such a way that it is sufficient that the eigenvalues of the residue matrices $B_{i}$ be small (the estimate is given) rather than the matrices themselves. This refinement is naturally extended to other types of solvability, as well as to strong non-solvability of Fuchsian systems.

In Sect. 7, we propose a generalization of the criterion of solvability by quadratures to the case of essentialy non-Fuchsian systems (systems with non-resonant irregular singular points) with small formal exponents, and also discuss other types of solvability, including local solvability by quadratures over the field of meromorphic germs at an irregular singular point.

## 2 Solvability by Quadratures and the Differential Galois Group

In this section, we recall the definitions of some basic notions of differential Galois theory. Besides solvability of a linear differential system by quadratures, we consider other types of solvability and explain how they depend on properties of the differential Galois group of the system.

A Picard–Vessiot extension of the field ${\mathbb{C}}(z)$ of rational functions corresponding to the system (1) is a differential field $F={\mathbb{C}}(z)\langle Y\rangle$ generated as a field extension of ${\mathbb{C}}(z)$ by all the entries of a fundamental matrix $Y(z)$ of the system (1). Let us specify that such a matrix is taken (by Cauchy’s theorem) in the field of germs of meromorphic functions at a non-singular point $z_{0}$ of the system. One says that the system (1) is solvable by quadratures, if there is a fundamental matrix $Y$, whose entries are expressed in elementary or algebraic functions and their integrals, or, more formally, if the field $F$ is contained in some differential field extension of ${\mathbb{C}}(z)$ generated by algebraic functions, integrals and exponentials of integrals:

 $\displaystyle{\mathbb{C}}(z)=F_{1}\subset\cdots\subset F_{m},\quad F\subseteq F _{m},$

where each $F_{i+1}=F_{i}\langle x_{i}\rangle$ is a field extension of $F_{i}$ by an element $x_{i}$, which is either:

• algebraic over $F_{i}$, or

• an integral (that is, an element whose derivative belongs to $F_{i}$), or

• an exponential of an integral (that is, an element whose logarithmic derivative belongs to $F_{i}$).

Such an extension ${\mathbb{C}}(z)\subseteq F_{m}$ is called Liouvillian; thus solvability by quadratures means that the Picard–Vessiot extension $F$ is contained in some Liouvillian extension of the field of rational functions.

In analogy to classical Galois theory, solvability or non-solvability of a linear differential system by quadratures is related to properties of its differential Galois group. The differential Galois group ${\bf G}={\rm Gal}\,(F/{\mathbb{C}}(z))$ of the system (1) (of a Picard–Vessiot extension ${\mathbb{C}}(z)\subseteq F$) is the group of differential automorphisms of the field $F$ (i.e., automorphisms commuting with differentiation) that preserve the elements of the field ${\mathbb{C}}(z)$:

 $\displaystyle{\bf G}=\left\{\sigma\colon F\rightarrow F\Bigg|\sigma\circ\frac{ d}{dz}=\frac{d}{dz}\circ\sigma,\sigma(f)=f\quad\forall f\in{\mathbb{C}}(z) \right\}.$

(The differential Galois group can be defined by any Picard–Vessiot extension, since these are all isomorphic as differential fields.) As follows from the definition, the image $\sigma(Y)$ of the fundamental matrix $Y$ of the system (1) under any element $\sigma$ of the Galois group is a fundamental matrix of this system again. Hence, $\sigma(Y)=YC,\;C\in{\rm GL}(p,{\mathbb{C}})$. As every element of the differential Galois group is determined uniquely by its action on a fundamental matrix of the system, the group $\bf G$ can be regarded as a subgroup of the matrix group ${\rm GL}(p,{\mathbb{C}})$ for any such $Y$. Moreover, this subgroup ${\bf G}\subseteq{\rm GL}(p,{\mathbb{C}})$ is algebraic, i.e., closed in the Zariski topology of the space ${\rm GL}(p,{\mathbb{C}})$ (the topology, whose closed sets are those determined by systems of polynomial equations), see ([Kaplansky1957], Th. 5.5). A very good reference for the basics of differential Galois theory, especially the algebraic geometric aspects, is the book by [Crespo and Hajto2011].

The differential Galois group $\bf G$ is a union of a finite number of disjoint connected sets that are open and closed simultaneously (in the Zariski topology), and the set containing the identity matrix is called the identity component. The identity component ${\bf G}^{0}\subseteq{\bf G}$ is a normal subgroup of finite index ([Kaplansky1957], Lemma 4.5). According to the Picard–Vessiot theorem, solvability of the system (1) by quadratures is equivalent to solvability of the subgroup ${\bf G}^{0}$ (see [Kaplansky1957], Th. 5.12; [Khovanskii2008], Ch. 3, Th. 5.1). Recall that a group $H$ is said to be solvable, if there exist intermediate subgroups $\{e\}=H_{0}\subset H_{1}\subset\cdots\subset H_{m}=H$, such that, for every $i=1,\dots,m$, $H_{i-1}$ is normal in $H_{i}$ and the factor group $H_{i}/H_{i-1}$ is Abelian.

Alongside the differential Galois group, one considers the monodromy group $\bf M$ of the system (1) generated by the monodromy matrices $M_{1},\ldots,M_{n}$ corresponding to analytic continuation of the fundamental matrix $Y$ around the singular points $a_{1},\ldots,a_{n}$. Each $M_{i}$ is defined as follows: since the operation of analytic continuation commutes with differentiation, the matrix $Y$ considered in a neighbourhood of a non-singular point $z_{0}$ goes to another fundamental matrix, $YM_{i}$, under an analytic continuation along a simple loop $\gamma_{i}$ encircling a point $a_{i}$ and no other $a_{j}$. As analytic continuation also preserves elements of the field ${\mathbb{C}}(z)$ (since they are single-valued functions), one has ${\bf M}\subseteq\bf G$. Furthermore, the differential Galois group of a Fuchsian system coincides with the closure of its monodromy group in the Zariski topology (see [Khovanskii2008], Ch. 6, Cor. 1.3). Hence, a Fuchsian system is solvable by quadratures if and only if the identity component ${\bf M}\cap{\bf G}^{0}$ of its monodromy group is solvable.

Now we will discuss other types of solvability. These are defined in analogy to solvability by quadratures, and we leave formal definitions to the reader. [Kolchin1948] gave criteria for a linear differential system to be solvable with respect to each of these types in terms of its differential Galois group. Further we will need his results only for a system, whose differential Galois group is triangular, meaning that all matrices of the group are triangular in some common basis.

###### Kolchin’s criteria.

(See also [Khovanskii2008], Ch. 3, §8). Let the differential Galois group $\bf G$ of the system (1) be triangular. Then the system is

1. 1.

solvable by integrals and algebraic functions if and only if the eigenvalues of all elements of $\bf G$ are roots of unity;

2. 2.

solvable by integrals if and only if the eigenvalues of all elements of $\bf G$ are equal to unity;

3. 3.

solvable by exponentials of integrals and algebraic functions if and only if $\bf G$ is diagonal;

4. 4.

solvable by algebraic functions if and only if $\bf G$ is diagonal and the eigenvalues of all its elements are roots of unity.

## 3 A Local Form of Solutions Near a Singular Point

In this section, we recall the definitions of regular and irregular singular points of a linear differential system and describe the structure of solutions near a singular point of each type.

A singular point $z=a_{i}$ of the system (1) is said to be regular, if any solution of the system has at most polynomial growth in any sector with the vertex at this point of sufficiently small radius and an opening less than $2\pi$. Otherwise, the point $z=a_{i}$ is said to be irregular.

A singular point $z=a_{i}$ of the system (1) is said to be Fuchsian, if the coefficient matrix $B(z)$ has a simple pole at this point. Due to Sauvage’s theorem, a Fuchsian singular point is always regular (see [Hartman1964], Th. 11.1). However, the coefficient matrix of a system at a regular singular point may in general have a pole of order greater than one. Let us write the Laurent expansion of the coefficient matrix $B$ of the system (1) near its singular point $z=a$ in the form

 $\displaystyle B(z)=\frac{B_{-r-1}}{(z-a)^{r+1}}+\cdots+\frac{B_{-1}}{z-a}+B_{0 }+\cdots,\quad B_{-r-1}\neq 0.$ (3)

The number $r$ is called the Poincaré rank of the system (1) at this point (or the Poincaré rank of the singular point $z=a$). For example, the Poincaré rank of a Fuchsian singular point is equal to zero.

The system (1) is said to be Fuchsian, if all its singular points are Fuchsian (then it can be written in the form (2)). A system, whose singular points are all regular, will be called regular singular.

### 3.1 A Regular Singular Point

According to Levelt’s theorem [Levelt1961], in a neighbourhood of each regular singular point $a_{i}$ of the system (1), there exists a fundamental matrix of the form

 $\displaystyle Y_{i}(z)=U_{i}(z)(z-a_{i})^{A_{i}}(z-a_{i})^{{\widetilde{E}}_{i}},$ (4)

where $U_{i}(z)$ is a holomorphic matrix at the point $a_{i}$, $A_{i}={\rm diag}(\varphi_{i}^{1},\ldots,\varphi_{i}^{p})$ is a diagonal matrix, whose entries $\varphi_{i}^{j}$ are integers organized in a non-increasing sequence, and ${\widetilde{E}}_{i}=(1/2\pi{\bf i})\ln{\widetilde{M}}_{i}$ is an upper triangular matrix (the normalized logarithm of the corresponding monodromy matrix ${\widetilde{M}}_{i}$), whose eigenvalues $\rho_{i}^{j}$ satisfy the condition

 $\displaystyle 0\leqslant{\rm Re}\,\rho_{i}^{j}<1.$

Such a fundamental matrix is called a Levelt matrix, and one also says that its columns form a Levelt basis in the solution space of the system (in a neighborhood of the regular singular point $a_{i}$). The complex numbers $\beta_{i}^{j}=\varphi_{i}^{j}+\rho_{i}^{j}$ are called the (Levelt) exponents of the system at the regular singular point $a_{i}$.

A singular point $a_{i}$ is Fuchsian if and only if the corresponding matrix $U_{i}$ in the decomposition (4) is holomorphically invertible at this point, that is, $\det U_{i}(a_{i})\neq 0$. It is not difficult to check that in this case the exponents of the system at the point $a_{i}$ coincide with the eigenvalues of the residue matrix $B_{i}$. In the general case of a regular singularity $a_{i}$, estimates for the order of the function $\det U_{i}$ at this point were obtained by [Corel2001] (see also [Gontsov2004]):

 $\displaystyle r_{i}\leqslant{\rm ord}_{a_{i}}\det U_{i}\leqslant\frac{p(p-1)}{ 2}\,r_{i},$

where $r_{i}$ is the Poincaré rank of the regular singular point $a_{i}$. These estimates imply the inequalities for the sum of exponents of a regular singular system over all its singular points, which are called the Fuchs inequalities:

 $\displaystyle-\frac{p(p-1)}{2}\sum_{i=1}^{n}r_{i}\leqslant\sum_{i=1}^{n}\sum_{ j=1}^{p}\beta_{i}^{j}\leqslant-\sum_{i=1}^{n}r_{i}$ (5)

(the sum of exponents is an integer).

### 3.2 An Irregular Singular Point

Let us now describe the structure of solutions of the system (1) near one of its irregular singular points. We assume that the irregular singularity $z=a$ of Poincaré rank $r$ is non-resonant, that is, the eigenvalues $b_{1},\ldots,b_{p}$ of the leading term $B_{-r-1}$ of the matrix $B(z)$ in the expansion (3) are pairwise distinct. Let us fix a matrix $T$, which reduces the leading term $B_{-r-1}$ to the diagonal form:

 $\displaystyle T^{-1}B_{-r-1}T={\rm diag}(b_{1},\ldots,b_{p}).$

The system possesses a uniquely determined formal fundamental matrix $\widehat{Y}$ of the form (see [Wasow1965], §§10, 11)

 $\displaystyle\widehat{Y}(z)=\widehat{F}(z)(z-a)^{\Lambda}e^{Q(z)},$ (6)

where

1. (a)

$\widehat{F}(z)$ is a matrix formal Taylor series in $z-a$, and $\widehat{F}(a)=T$;

2. (b)

$\Lambda$ is a constant diagonal matrix, whose diagonal entries are called the formal exponents of the system (1) at the irregular singular point $z=a$;

3. (c)

$Q(z)={\rm diag}(q_{1}(z),\ldots,q_{p}(z))$ is a diagonal matrix, whose entries $q_{j}(z)$ are polynomials in $(z-a)^{-1}$ of degree $r$ without a constant term,

 $\displaystyle q_{j}(z)=-\frac{b_{j}}{r}\,(z-a)^{-r}+o((z-a)^{-r}).$

For each pair $(b_{j},b_{k})$ of eigenvalues of the matrix $B_{-r-1}$, one has $2r$ rays starting at the point $a$, which are called Stokes lines of the system (1) at this point:

 $\displaystyle\left\{z\in{\mathbb{C}}\Bigg|{\rm Re}\,\frac{b_{j}-b_{k}}{(z-a)^{ r}}=0\right\}$ $\displaystyle=$ $\displaystyle\left\{z\in{\mathbb{C}}\Bigg|{\rm arg}\,(z-a)=\frac{1}{r}\left({ \rm arg}(b_{j}-b_{k})+\frac{\pi}{2}+\pi m\right),\right.$ $\displaystyle\quad\left.m=0,1,\ldots,2r-1\vphantom{\left\{z\in{\mathbb{C}} \Bigg|{\rm arg}\,(z-a)=\frac{1}{r}\left({\rm arg}(b_{j}-b_{k})+\frac{\pi}{2}+ \pi m\right),\right.}\right\}.$

These rays are asymptotic to the corresponding curves $\{{\rm Re}\,(q_{j}-q_{k})=0\}$, which divide a neighbourhood of $a$ into domains, in which the function $e^{q_{j}-q_{k}}$ has an exponential growth or decay.

Consider a covering of a punctured neighbourhood of $a$ by $2r$ congruent sectors

###### Acknowledgements
The authors are very thankful to professor Claude Mitschi for reading the text accurately and giving many improving remarks as well as for kindly providing her (yet unpublished) notes on differential Galois theory from a forthcoming volume of the CIMPA lecture notes (which we essentially used in preparing Sect. 4.2 concerning the exponential torus). We thank professor Askold Khovanskii who has drawn our attention to other types of solvability and has given an advice concerning a presentation of the paper. We also thank the referee for nice and relevant comments and for improving our English. The work is supported by the Russian Foundation for Basic Research (Grant No. 14-01-00346), IUM-Simons Fellowship and Dynasty Foundation.

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