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

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

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

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.

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:

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 ${\mathbb{C}}(z)$:

(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.

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

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.

Alongside the differential Galois group $\bf G$ of the system (1), one also defines the local differential Galois groups ${\bf G}_{a}$ corresponding to each singular point $z=a$ of the system. Let ${\cal M}_{a}={\mathbb{C}}(\{z-a\})$ be the field of meromorphic functions at the point $a$. Consider a fundamental matrix $Y_{S}(z)$ of the system (1) on some open sector $S$ with vertex $a$ (its entries are elements of the field of meromorphic functions in $S$) and the Picard–Vessiot extension ${\cal M}_{a}\subseteq F_{a}={\cal M}_{a}\langle Y_{S}\rangle$ of ${\cal M}_{a}$ generated by all the entries of $Y_{S}$. The corresponding differential Galois group ${\bf G}_{a}={\rm Gal}\,(F_{a}/{\cal M}_{a})$ is the group of differential automorphisms of the field $F_{a}$ that preserve the elements of the field ${\cal M}_{a}$. Like the global differential Galois group $\bf G$, this group is regarded as an algebraic subgroup of the matrix group ${\rm GL}(p,{\mathbb{C}})$, where this representation depends of course on $a$ and on $Y_{S}$.

We have the following natural inclusion of each ${\bf G}_{a}$ into $\bf G$. Consider a fundamental matrix $Y$ of the system (1) in a neighbourhood $U_{0}$ of a non-singular point $z_{0}$ with respect to which the group $\bf G$ is determined. The entries of $Y$ are elements of the field of meromorphic functions in $U_{0}$. We analytically continue the matrix $Y$ to the sector $S$ along some path $\gamma$ and obtain a matrix ${\widetilde{Y}}_{S}$, whose entries are elements of the field $F_{a}$. Then we apply to ${\widetilde{Y}}_{S}$ some element $\sigma$ of the group ${\mathbf{G}}_{a}$ and return $\sigma({\widetilde{Y}}_{S})$ back to $U_{0}$ along the path $\gamma^{-1}$. Since ${\mathbb{C}}(z)\subset{\cal M}_{a}$, the automorphism $\sigma$ preserves rational functions, and thus the transformation $\tau^{-1}_{\gamma}\circ\sigma\circ\tau_{\gamma}$, where $\tau_{\gamma}$ is the operator of an analytic continuation along the path $\gamma$ (we mean that $\tau_{\gamma}$ is an operator from the solution space of (1) in $U_{0}$ to that in $S$), is an element of the group $\bf G$. On the level of matrices, this looks as follows:

where $C_{\gamma}\in{\rm GL}(p,{\mathbb{C}})$ is a matrix connecting $Y_{S}$ and the analytic continuation ${\widetilde{Y}}_{S}$ of $Y$ into $S$ along $\gamma$, that is, ${\widetilde{Y}}_{S}(z)=Y_{S}(z)C_{\gamma}$.

Let us recall some notions concerning holomorphic vector bundles and meromorphic connections in a context of linear differential equations. We mainly follow [Bolibruch et al.2006] or [Ilyashenko and Yakovenko2008], Ch. 3 (see also [Gontsov and Poberezhnyi2008]).

In an analytic interpretation, a holomorphic bundle $E$ of rank $p$ over the Riemann sphere is defined by a cocycle $\{g_{\alpha\beta}(z)\}$, that is, a collection of holomorphic matrix functions corresponding to a covering $\{U_{\alpha}\}$ of the Riemann sphere:

and satisfying the conditions

Two holomorphically equivalent cocycles $\{g_{\alpha\beta}(z)\}$ and $\{g^{\prime}_{\alpha\beta}(z)\}$ define the same bundle. Equivalence of cocycles means that there exists a set $\{h_{\alpha}(z)\}$ of holomorphic matrix functions $h_{\alpha}\colon U_{\alpha}\longrightarrow{\rm GL}(p,{\mathbb{C}})$ such that

A section $s$ of the bundle $E$ is determined by a set $\{s_{\alpha}(z)\}$ of vector functions $s_{\alpha}\colon U_{\alpha}\longrightarrow{\mathbb{C}}^{p}$ that satisfy the conditions $s_{\alpha}(z)=g_{\alpha\beta}(z)s_{\beta}(z)$ in non-empty intersections $U_{\alpha}\cap U_{\beta}$.

A meromorphic connection $\nabla$ on the holomorphic vector bundle $E$ is determined by a set $\{\omega_{\alpha}\}$ of matrix meromorphic differential 1-forms that are defined in the corresponding neighbourhoods $U_{\alpha}$ and satisfy gluing conditions

Under a transition to an equivalent cocycle $\{g^{\prime}_{\alpha\beta}\}$ connected with the initial one by the relations (11), the 1-forms $\omega_{\alpha}$ of the connection $\nabla$ are transformed into the corresponding 1-forms

Conversely, the existence of holomorphic matrix functions $h_{\alpha}\colon U_{\alpha}\longrightarrow{\rm GL}(p,{\mathbb{C}})$ such that the matrix 1-forms $\omega_{\alpha}$ and $\omega^{\prime}_{\alpha}$ (satisfying the conditions (12) for $g_{\alpha\beta}$ and $g^{\prime}_{\alpha\beta}$ respectively) are connected by the relation (13) in $U_{\alpha}$ implies the equivalence of the cocycles $\{g_{\alpha\beta}\}$ and $\{g^{\prime}_{\alpha\beta}\}$ (one may assume that the intersections $U_{\alpha}\cap U_{\beta}$ do not contain singular points of the connection).

Vector functions $s_{\alpha}(z)$ satisfying linear differential equations $ds_{\alpha}=\omega_{\alpha}s_{\alpha}$ in the corresponding $U_{\alpha}$ can be chosen, by virtue of the conditions (12), in such a way that the set $\{s_{\alpha}(z)\}$ determines a section of the bundle $E$, which is called horizontal with respect to the connection $\nabla$. Thus, horizontal sections of a holomorphic vector bundle with a meromorphic connection are determined by solutions of local linear differential systems. The monodromy of a connection (the monodromy group) characterizes the ramification of horizontal sections under their analytic continuation along loops in $\overline{\mathbb{C}}$ not containing singular points of the connection 1-forms and is defined similarly to the monodromy group of the system (1). A connection will be called Fuchsian (logarithmic), regular, or irregular, depending on the type of the singular points of its 1-forms (as singular points of the corresponding linear differential systems).

If a bundle is holomorphically trivial (all matrices of the cocycle can be taken as the identity matrices), then, by virtue of the conditions (12), the matrix 1-forms of a connection coincide on non-empty intersections $U_{\alpha}\cap U_{\beta}$. Hence, horizontal sections of such a bundle are solutions of a global linear differential system defined on the whole Riemann sphere. Conversely, the linear system (1) determines a meromorphic connection on the holomorphically trivial vector bundle of rank $p$ over $\overline{\mathbb{C}}$. Such a bundle has a standard definition by the cocycle that consists of the identity matrices, while the connection is defined by the matrix 1-form $B(z)dz$ of coefficients of the system. To our purpose, it will be more convenient to use the following equivalent coordinate description [this construction appears already in [Bolibruch et al.2006]].

At first, we consider a covering $\{U_{\alpha}\}$ of the punctured Riemann sphere $\overline{\mathbb{C}}{\setminus}\{a_{1},\ldots,{}a_{n}\}$ by simply connected neighbourhoods. Then, on the corresponding non-empty intersections $U_{\alpha}\cap U_{\beta}$, one defines the matrix functions of a cocycle, $g^{\prime}_{\alpha\beta}(z)\equiv{\rm const}$, which are expressed in terms of the monodromy matrices $M_{1},\ldots,M_{n}$ of the system (1) via the operations of multiplication and taking the inverse (see [Gontsov and Poberezhnyi2008]). In this case, the matrix differential 1-forms $\omega^{\prime}_{\alpha}$ defining a connection are equal to zero. Further, the covering $\{U_{\alpha}\}$ is complemented by small neighbourhoods $O_{i}$ of the singular points $a_{i}$ of the system; thus, we obtain the covering of the Riemann sphere $\overline{\mathbb{C}}$. To non-empty intersections $O_{i}\cap U_{\alpha}$, there correspond matrix functions $g^{\prime}_{i\alpha}(z)=Y_{i}(z)$ of the cocycle, where $Y_{i}(z)$ is a germ of a fundamental matrix of the system, whose monodromy matrix at the point $a_{i}$ is equal to $M_{i}$ (so, for analytic continuations of the chosen germ to non-empty intersections $O_{i}\cap U_{\alpha}\cap U_{\beta}$, the cocycle relations $g_{i\alpha}g_{\alpha\beta}=g_{i\beta}$ hold). The matrix differential 1-forms $\omega^{\prime}_{i}$ determining the connection in the neighbourhoods $O_{i}$ coincide with the 1-form $B(z)dz$ of coefficients of the system. To prove the holomorphic equivalence of the cocycle $\{g^{\prime}_{\alpha\beta},g^{\prime}_{i\alpha}\}$ to the identity cocycle, it is sufficient to check the existence of holomorphic matrix functions

such that

Since we have $\omega_{\alpha}=B(z)dz$ and $\omega^{\prime}_{\alpha}=0$ for all $\alpha$, the first equation in (14) can be rewritten as a linear system

which has a holomorphic solution $h^{-1}_{\alpha}\colon U_{\alpha}\longrightarrow{\rm GL}(p,{\mathbb{C}})$, since the 1-form $B(z)dz$ is holomorphic in a simply connected neighbourhood $U_{\alpha}$. The second equation in (14) has a holomorphic solution $h_{i}(z)\equiv I$, since $\omega_{i}=\omega^{\prime}_{i}=B(z)dz$.

One says that the bundle $E$ has a subbundle $E^{\prime}\subset E$ of rank $k<p$ that is stabilized by the connection $\nabla$, if the pair $(E,\nabla)$ admits a coordinate description $\{g_{\alpha\beta}\}$, $\{\omega_{\alpha}\}$ of the following block upper triangular form:

where $g_{\alpha\beta}^{1}$ and $\omega_{\alpha}^{1}$ are $k\times k$ blocks (then the cocycle $\{g_{\alpha\beta}^{1}\}$ defines the subbundle $E^{\prime}$, and the 1-forms $\omega_{\alpha}^{1}$ define the restriction $\nabla^{\prime}$ of the connection $\nabla$ to the subbundle $E^{\prime}$).

We use the above coordinate descriptions of the bundle and the connection with the cocycle $\{g^{\prime}_{\alpha\beta},g^{\prime}_{i\alpha}\}$ and set $\{\omega^{\prime}_{\alpha},\omega^{\prime}_{i}\}$ of matrix 1-forms. The matrices $g^{\prime}_{\alpha\beta}$ are already block upper triangular, since the monodromy matrices $M_{1},\ldots,M_{n}$ are (and $\omega^{\prime}_{\alpha}=0$), while the matrices $g^{\prime}_{i\alpha}=Y_{i}$ can be transformed to such a form, $\Gamma_{i}\,g^{\prime}_{i\alpha}=\Gamma_{i}\,Y_{i}$. Thus, changing the matrices $g^{\prime}_{i\alpha}$ to $\Gamma_{i}\,g^{\prime}_{i\alpha}$ and matrix 1-forms $\omega^{\prime}_{i}$ to $\Gamma_{i}\,\omega^{\prime}_{i}\,\Gamma^{-1}_{i}+(d\Gamma_{i})\Gamma^{-1}_{i},$ we obtain a holomorphically equivalent coordinate description with the cocycle matrices and connection matrix 1-forms, which have the same block upper triangular form.

The following auxiliary lemma points to a certain block structure of a linear differential system in the case when the corresponding holomorphically trivial vector bundle with the meromorphic connection has a holomorphically trivial subbundle that is stabilized by the connection.

The degree $\deg E$ (which is an integer) of a holomorphic vector bundle $E$ endowed with a meromorphic connection $\nabla$ may be defined as the sum

of the residues of local differential 1-forms ${\rm tr}\,\omega_{i}$ over all singular points of the connection $\nabla$ (the notation “tr” stays for the trace), where $\omega_{i}$ is the local matrix differential 1-form of the connection $\nabla$ in a neighbourhood of its singular point $a_{i}$. Later, when calculating the degree of a bundle, we apply the Liouville formula ${\rm tr}\,\omega_{i}=d\ln\det Y_{i}$, where $Y_{i}$ is a fundamental matrix of the local linear differential system $dy=\omega_{i}\,y$.