Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification
We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of the solvability of the Lie algebra of the differential Galois group of the system. However...
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| Цитувати: | Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification / M.A. Barkatou, R.R. Gontsov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 23 назв. — англ. |
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| citation_txt | Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification / M.A. Barkatou, R.R. Gontsov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 23 назв. — англ. |
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| description | We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of the solvability of the Lie algebra of the differential Galois group of the system. However, the dependence of this Lie algebra on the system coefficients remains unknown. We show that for the particular class of systems with non-resonant irregular singular points that have sufficiently small coefficient matrices, the problem is reduced to that of solvability of the explicit Lie algebra generated by the coefficient matrices. This extends the corresponding Ilyashenko-Khovanskii theorem obtained for linear differential systems with Fuchsian singular points. We also give some examples illustrating the practical verification of the presented criteria of solvability by using general procedures implemented in Maple.
|
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 058, 15 pages
Linear Differential Systems with Small Coefficients:
Various Types of Solvability and their Verification
Moulay A. BARKATOU † and Renat R. GONTSOV ‡§
† Laboratoire XLIM (CNRS UMR 72 52), Département Mathématiques-Informatique,
Université de Limoges, Faculté des Sciences et Techniques, 123 avenue Albert Thomas,
F-87060 LIMOGES Cedex, France
E-mail: moulay.barkatou@unilim.fr
‡ Institute for Information Transmission Problems RAS,
Bolshoy Karetny per. 19, build. 1, Moscow 127051, Russia
§ Moscow Power Engineering Institute, Krasnokazarmennaya 14, Moscow 111250, Russia
E-mail: gontsovrr@gmail.com
Received January 30, 2019, in final form July 31, 2019; Published online August 09, 2019
https://doi.org/10.3842/SIGMA.2019.058
Abstract. We study the problem of solvability of linear differential systems with small co-
efficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this
problem is equivalent to that of solvability of the Lie algebra of the differential Galois group
of the system. However, dependence of this Lie algebra on the system coefficients remains
unknown. We show that for the particular class of systems with non-resonant irregular sin-
gular points that have sufficiently small coefficient matrices, the problem is reduced to that
of solvability of the explicit Lie algebra generated by the coefficient matrices. This extends
the corresponding Ilyashenko–Khovanskii theorem obtained for linear differential systems
with Fuchsian singular points. We also give some examples illustrating the practical veri-
fication of the presented criteria of solvability by using general procedures implemented in
Maple.
Key words: linear differential system; non-resonant irregular singularity; formal exponents;
solvability by generalized quadratures; triangularizability of a set of matrices
2010 Mathematics Subject Classification: 34M03; 34M25; 34M35; 34M50
1 Introduction
Solvability of linear differential equations and systems in finite terms is a classical question of
differential Galois theory. It begins with the problem of solvability by generalized quadratures
(or, in the Liouvillian sense, which means the representability of all solutions of an equation
in terms of elementary or algebraic functions and their integrals, speaking informally) in the
1830’s in the works of Liouville on second order equations. Generalized in 1910 by Mordukhai–
Boltovskii for nth order equations, this was independently developed in a quite different way
by Picard and Vessiot who connected to an equation (system) a group, turned out to be a lin-
ear algebraic group, called the differential Galois group. They showed that solvability of the
equation (system) in the Liouvillian sense depends entirely on properties of this group or its
Lie algebra. Namely, solvability holds if and only if the identity component of the differential
Galois group is solvable (which is equivalent to the solvability of the Lie algebra of the group).
Later Kolchin completed this theory by considering other, more particular, types of solvability
and their dependence on properties of the differential Galois group.
This paper is a contribution to the Special Issue on Algebraic Methods in Dynamical Systems. The full
collection is available at https://www.emis.de/journals/SIGMA/AMDS2018.html
mailto:email@address
mailto:email@address
https://doi.org/10.3842/SIGMA.2019.058
https://www.emis.de/journals/SIGMA/AMDS2018.html
2 M.A. Barkatou and R.R. Gontsov
General methods for computing (in theory, at least) the differential Galois group have been
proposed by several authors [5, 6, 9, 18] in the last past decades. However, those methods
have an extremely high computational complexity and are therefore not practical for nontrivial
problems. Moreover it is not clear how the differential Galois group of a given system depends
on the coefficients of the latter. More recently an algorithm for computing the Lie algebra of
the differential Galois group has been proposed in [3]. This algorithm has a rather reasonable
complexity, compared to the previous methods for computing the differential Galois group, but
like the former it does not allow to clarify the relationship between the coefficients of the input
system and its differential Lie algebra.
Though the main difficulty is that the differential Galois group (and its Lie algebra) of
a specific equation depends on its coefficients very implicitly, in some cases it is possible to leave
aside these implicit objects (which are hard to compute) and obtain an answer to the question of
solvability in terms of objects that are determined by the coefficients of the equation explicitly.
Here are some examples:
• the list of hypergeometric equations solvable in the Liouvillian sense is completely known
(Scwharz–Kimura’s list consisting of the fifteen families [14], see also [23, Chapter 12,
Section 1]);
• the Bessel equation is solvable in the Liouvillian sense if and only if its parameter is
a half-integer (see [17, Section 2.8] or [23, Chapter 11, Section 1]);
• the equation y′′ = (az2 + bz + c)y, a 6= 0, is solvable in the Liouvillian sense if and only if
(4ac− b2)/4a3/2 is an odd integer (see [16]);
• more generally, an equation y′′ = P (z)y with a polynomial P , is not solvable in the
Liouvillian sense if degP is odd (see [11]) whereas in the (2n + 1)-dimensional space of
such equations with degP = 2n, equations solvable in the Liouvillian sense form a union
of countable number of algebraic varieties of dimension n+ 1 each (see [19, 22]).
A rather unexpected and maybe less known subclass of linear differential systems whose
various types of solvability can be checked in terms of explicit input data, is formed by systems
with sufficiently small matrix coefficients. The first result of this kind concerns Fuchsian systems.
It was obtained by Ilyashenko, Khovanskii and published in 1974 in Russian, and is revised
in [12], [13, Chapter 6, Section 2.3]. This claims that for Fuchsian (p× p)-systems
dy
dz
=
(
n∑
i=1
Bi
z − ai
)
y, y(z) ∈ Cp, Bi ∈ Mat(p,C),
with fixed singular points a1, . . . , an ∈ C (and maybe ∞, if
n∑
i=1
Bi 6= 0) there exists an ε =
ε(p, n) > 0 such that a criterion of solvability in the Liouvillian sense for a Fuchsian system with
‖Bi‖ < ε takes the following form: the system is solvable if and only if all the matrices Bi can be
simultaneously reduced to a triangular form. Using Kolchin’s results, Ilyashenko and Khovanskii
also obtained criteria for other types of solvability of a Fuchsian system with small residue
matrices, in terms of these matrices. Though, all these criteria still remain not quite explicit
in the sense that the ε is expressed implicitely. This situation was refined in [20] where it was
proved that for applying the above criterion of solvability it is sufficient for the eigenvalues of the
matrices Bi, called the exponents of the system, to be small enough rather than for the matrices
themselves (this refinement had been conjectured earlier by Andrey Bolibrukh, see remark in [12,
Section 6.2.3]). Moreover, an explicit bound for the exponents was given. Then similar criteria
of solvability were obtained in [8] for a (non-resonant) irregular differential system with small
formal exponents.
Linear Differential Systems with Small Coefficients: Various Types of Solvability 3
In the next section we recall these criteria for various types of solvability of a linear differ-
ential system with small (formal) exponents (Theorems 2.1, 2.3, 2.4 and 2.5), previously giving
necessary definitions. Then, in Section 3, we prove that the formal exponents at a non-resonant
irregular singular point z = a of a system
dy
dz
= B(z)y, B(z) =
1
(z − a)r+1
(
B(0) +B(1)(z − a) + · · ·+B(r)(z − a)r + · · ·
)
(1.1)
are small, if the coefficient matrices B(1), . . . , B(r) of the principal part of B(z) at a are small
enough, while the leading term B(0) belongs to some compact subset of Mat(p,C). This allows
us to formulate criteria of solvability inside an open set of non-resonant irregular differential
systems (Theorem 3.1), looking at the problem theoretically, like Ilyashenko and Khovanskii did
in the Fuchsian case.
From a practical point of view, as soon as we have a system whose (formal) exponents satisfy
some numerical restrictions, which can be checked algorithmically, the question on solvability of
the implicit Lie algebra of the differential Galois group of the system is reduced to the question
on solvability of the explicit Lie algebra generated by the system coefficient matrices. This
practical part of the problem is studied in Section 4, where the corresponding algorithms and
examples are provided.
2 Various types of solvability
We consider a linear differential (p× p)-system
dy
dz
= B(z)y, y(z) ∈ Cp, (2.1)
defined on the whole Riemann sphere C, with the meromorphic coefficient matrix B (whose
entries are thus rational functions).
One says that a solution y of the system (2.1) is Liouvillian if there is a tower of elementary
extensions
C(z) = F0 ⊂ F1 ⊂ · · · ⊂ Fm
of the field C(z) of rational functions such that all components of y belong to Fm. Here each
Fi+1 = Fi〈xi〉 is a field extension of Fi by an element xi, which is either:
– an integral of some element in Fi, or
– an exponential of integral of some element in Fi, or
– algebraic over Fi.
The system is said to be solvable in the Liouvillian sense (or, by generalized quadratures), if all
its solutions are Liouvillian. There are other types of solvability studied by Kolchin [15] from
the point of view of differential Galois theory, which are defined in analogy to solvability by
generalized quadratures, and we leave formal definitions to the reader. These are:
1) solvability by integrals and algebraic functions;
2) solvability by integrals;
3) solvability by exponentials of integrals and algebraic functions;
4) solvability by algebraic functions.
4 M.A. Barkatou and R.R. Gontsov
We leave aside the differential Galois group of the system and the description of the above
types of solvability in terms of this group, since in the case of systems having small (formal)
exponents we will deal with, this description is provided in terms of the coefficient matrix. Let
us recall some basic theory.
Assume that the system (2.1) has singular points a1, . . . , an ∈ C of Poincaré rank r1, . . . , rn
respectively and, for the simplicity of exposition, that the point ∞ is non-singular (though we
will consider some example with a singular point at infinity; see Example 2.6 below). This means
that the coefficient matrix B has the form
B(z) =
n∑
i=1
(
B
(0)
i
(z − ai)ri+1
+ · · ·+
B
(ri)
i
z − ai
)
,
n∑
i=1
B
(ri)
i = 0 (2.2)
(B contains a polynomial part and/or
n∑
i=1
B
(ri)
i 6= 0, if ∞ is a singular point).
Genericity assumption. We also restrict ourselves to the generic case, when each singular
point ai is either Fuchsian or irregular non-resonant. In the first case the Poincaré rank ri
equals zero, while in the second case it is positive and the eigenvalues of the leading term B
(0)
i
are pairwise distinct. Such a system will be called generic.
Near a Fuchsian singular point z = ai, the system possesses a fundamental matrix Y of the
form
Y (z) = U(z)(z − ai)A(z − ai)E ,
where the matrix U is holomorphically invertible at ai (that is, detU(ai) 6= 0), A is a diagonal
integer matrix, E is a triangular matrix. The eigenvalues λ1
i , . . . , λ
p
i of the triangular matrix
A+ E are called the exponents of the system at a Fuchsian singular point ai and they coincide
with the eigenvalues of the residue matrix B
(ri)
i = B
(0)
i .
Near a non-resonant irregular singular point z = ai, the system possesses a formal funda-
mental matrix Ŷ of the form
Ŷ (z) = F̂ (z)(z − ai)ΛeQ(1/(z−ai)), (2.3)
where Λ = diag
(
λ1
i , . . . , λ
p
i
)
is a diagonal matrix, F̂ is a matrix formal Taylor series in (z − ai),
with det F̂ (ai) 6= 0, and Q is a polynomial. The numbers λ1
i , . . . , λ
p
i are called the formal
exponents of the system at the non-resonant irregular singular point ai, and the algorithm of
their calculation will be recalled in the next section.
Now we give some theorems which follow from [7, 8] and which we will be based on further.
Theorem 2.1. Let the exponents of the generic system (2.1), (2.2) at all singular points satisfy
the condition
Reλji > −1/n(p− 1), i = 1, . . . , n, j = 1, . . . , p, (2.4)
furthermore, for every Fuchsian singular point ai each difference λji−λki 6∈ Q\Z. Then the system
is solvable by generalized quadratures if and only if there exists a constant matrix C ∈ GL(p,C)
such that all the matrices CB
(l)
i C−1 are upper triangular.
Remark 2.2. By the genericity assumption, each irregular singular point of the system is
non-resonant, whereas Fuchsian singular points are not assumed to be non-resonant: integer
differences λji − λki of the exponents at a Fuchsian singular point are not forbidden.
Linear Differential Systems with Small Coefficients: Various Types of Solvability 5
Theorem 2.3. The assertion of Theorem 2.1 remains valid if the condition (2.4) is replaced by∣∣Reλji − Reλki
∣∣ < 1/n(p− 1), i = 1, . . . , n, j, k = 1, . . . , p.
According to the behaviour of solutions of a linear differential system near its irregular singu-
lar point, the system (2.1) with at least one irregular singular point is not solvable by integrals
and algebraic functions. For solvability by exponentials of integrals and algebraic functions, the
following criterion holds.
Theorem 2.4. Under the assumptions of Theorem 2.1, the generic system (2.1), (2.2) is solvable
by exponentials of integrals and algebraic functions1 if and only if there exists a constant matrix
C ∈ GL(p,C) such that all the matrices CB
(l)
i C−1 are diagonal.
For a Fuchsian system, we additionally have the following criteria of solvability.
Theorem 2.5. If all the singular points of the system (2.1), (2.2) are Fuchsian then, under the
assumptions of Theorem 2.1, this system is
1) solvable by integrals and algebraic functions2 if and only if there exists C ∈ GL(p,C) such
that all the matrices CB
(0)
i C−1 are triangular and the eigenvalues of each B
(0)
i are rational
numbers differing by an integer;
2) solvable by integrals if and only if there exists C ∈ GL(p,C) such that all the matrices
CB
(0)
i C−1 are triangular and their eigenvalues are equal to zero;
3) solvable by algebraic functions3 if and only if there exists C ∈ GL(p,C) such that all the
matrices CB
(0)
i C−1 are diagonal and the eigenvalues of each B
(0)
i are rational numbers
differing by an integer.
Example 2.6. Let us give an example which illustrates that to expect the equivalence of solv-
ability in the Liouvillian sense to the triangularizability of a system via a constant gauge trans-
formation, one indeed need to put some restrictions on the (formal) system exponents.
Consider an equation
d2u
dz2
=
(
z2 + c
)
u,
where c is a constant. Written in the form of a (2 × 2)-system with respect to the vector of
unknowns y = (u, du/dz)>, this becomes
dy
dz
=
(
0 1
z2 + c 0
)
y,
or, in the variable t = 1/z,
dy
dt
=
(
0 −1/t2
−1/t4 − c/t2 0
)
y.
1Under the assumptions of the theorem, this type of solvability is equivalent to solvability by exponentials of
integrals.
2Under the assumptions of the theorem, this type of solvability is equivalent to solvability by integrals and
radicals.
3Under the assumptions of the theorem, this type of solvability is equivalent to solvability by radicals.
6 M.A. Barkatou and R.R. Gontsov
This is a system with the unique singular point t = 0 of Poincaré rank 3 which is resonant though,
since the leading term of the coefficient matrix is nilpotent. Thus we make the transformation
ỹ = tdiag(0,1)y, under which the coefficient matrix B(t) of the system is changed as follows:
A(t) = tdiag(0,1)B(t)t−diag(0,1) +
1
t
diag(0, 1) =
(
0 −1/t3
−1/t3 − c/t 0
)
+
(
0 0
0 1/t
)
=
(
0 −1/t3
−1/t3 0
)
+
(
0 0
−c/t 1/t
)
=
1
t3
(
A(0) +A(2)t2
)
.
The transformed system has two singular points: non-resonant irregular t = 0 of Poincaré
rank 2 (the eigenvalues of the leading term A(0) are ±1) and Fuchsian t =∞. The eigenvectors
of the A(0) are e1 ∓ e2, e1,2 being the standard basic vectors (1, 0)>, (0, 1)> of C2, whereas
A(2)(e1 − e2) = (−c− 1)e2, A(2)(e1 + e2) = (−c+ 1)e2.
Hence, the matrices A(0), A(2) are simultaneously triangularizable (have a common eigenvector)
if and only if c = 1 or c = −1. However, due to [16] (see the third example in our Introduction)
the initial scalar equation and thus the transformed system, are solvable in the Liouvillian sense
not only when c = ±1 but also in the case of any other odd integer c. It remains to note that
the exponents of the system at the Fuchsian point t = ∞ are 0 and −1 (the eigenvalues of the
matrix −A(2)), that is, one of them does not satisfy the condition (2.4) of Theorem 2.1 which
in this case (p = 2, n = 2) looks like
Reλji > −1/2, i, j = 1, 2.
3 On the smallness of formal exponents
Let us denote by W ⊂ Mat(p,C) a (Zariski open) subset of (p × p)-matrices having pairwise
distinct eigenvalues. Assuming all the Poincaré ranks of the system (2.1), (2.2) to be positive
we regard it as a point in the set
Sr1,...,rna1,...,an =
{(
B
(0)
1 , . . . , B
(r1)
1
)
, . . . ,
(
B(0)
n , . . . , B(rn)
n
)
|B(0)
1 , . . . , B(0)
n ∈W
}
.
Here we prove the following statement generalizing the Ilyashenko–Khovanskii criterion of
solvability to the non-resonant irregular case.
Theorem 3.1. For any open disc D bW there exists δ = δ(D, p, n) > 0 such that in a set
Sr1,...,rna1,...,an(D, δ) =
{(
B
(0)
i , . . . , B
(ri)
i
)n
i=1
∈ Sr1,...,rna1,...,an |B
(0)
i ∈ D,
∥∥B(1)
i
∥∥ < δ, . . . ,
∥∥B(ri)
i
∥∥ < δ
}
of systems with fixed singular points of fixed positive Poincaré ranks and sufficiently small non-
leading coefficient matrices, systems solvable by generalized quadratures are those and only those
whose matrices B
(0)
i , . . . , B
(ri)
i , i = 1, . . . , n, are simultaneously conjugate to triangular ones.
This theorem will clearly follow from Theorem 3.2 below and Theorem 2.1.
Theorem 3.2. For any open disc D bW and any ε > 0 there exists δ = δ(D, ε) > 0 such that
a local system (1.1) with
B(0) ∈ D,
∥∥B(1)
∥∥ < δ, . . . ,
∥∥B(r)
∥∥ < δ, r > 0,
has the formal exponents λ1, . . . , λp at the non-resonant irregular singular point z = a satisfying
the condition |λj | < ε, j = 1, . . . , p.
Linear Differential Systems with Small Coefficients: Various Types of Solvability 7
Proof. To prove the statement of the theorem, let us first recall the procedure of a formal
transformation of the system (1.1) to a diagonal form, the splitting lemma (from [21, Section 11],
and we assume that a = 0 for simplicity).
Under a transformation y = T (z)ỹ, the system is changed as follows
dỹ
dz
= A(z)ỹ, A(z) = T−1B(z)T − T−1 dT
dz
,
and a new coefficient matrix A has the Laurent expansion
A(z) =
1
zr+1
(
A(0) +A(1)z + · · ·
)
.
One may always assume that T (z) = I + T (1)z + · · · and B(0) = A(0) = diag
(
α1, . . . , αp
)
. Then
gathering the coefficients at each power zk from the relation
T (z)zr+1A(z)− zr+1B(z)T (z) + zr+1 dT
dz
= 0,
we obtain
T (k)B(0) −B(0)T (k) +A(k) −B(k) +
k−1∑
l=1
(
T (l)A(k−l) −B(k−l)T (l)
)
+ (k − r)T (k−r) = 0
(the last summand equals zero for k 6 r). There are two sets of unknowns in this system of
matrix equations: A(k) =
(
A
(k)
ij
)
and T (k) =
(
T
(k)
ij
)
. Requiring all the A(k)’s to be diagonal and
assuming A(0), . . . , A(k−1) and T (0), . . . , T (k−1) to be already found, one first obtains
A
(k)
jj = B
(k)
jj −H
(k)
jj ,
where H(k) =
k−1∑
l=1
(
T (l)A(k−l) −B(k−l)T (l)
)
+ (k − r)T (k−r), and then
T
(k)
ij =
1
αj − αi
(
B
(k)
ij −H
(k)
ij
)
, i 6= j, T
(k)
jj = 0.
Since a diagonal system is integrated explicitly, one can see that the factors of the formal
fundamental matrix Ŷ (z) = F̂ (z)zΛeQ(1/z) (see (2.3)) of the local system (1.1) (we are especially
interested in Λ) are
F̂ (z) = T (z)eA
(r+1)z+A(r+2) z2
2
+···, Λ = A(r),
Q(1/z) = −A
(0)
rzr
− A(1)
(r − 1)zr−1
− · · · − A(r−1)
z
.
Therefore, the formal exponents of the system (the eigenvalues of Λ) are
λj = A
(r)
jj = B
(r)
jj −H
(r)
jj .
This implies the estimate∣∣λj∣∣ 6 ∥∥A(r)
∥∥ 6 ∥∥B(r)
∥∥+
∥∥H(r)
∥∥
(we will use, for example, the matrix 1-norm ‖ · ‖1 here).
8 M.A. Barkatou and R.R. Gontsov
Now we are ready to prove the smallness of the formal exponents λj ’s in the case of small
coefficients B(1), . . . , B(r). For this, we should prove the smallness of H(1), . . . ,H(r). Denoting
ρ = min
i 6=j
∣∣αj − αi
∣∣ > 0 we will have
∣∣T (k)
ij
∣∣ 6 1
ρ
(∣∣B(k)
ij
∣∣+
∣∣H(k)
ij
∣∣) =⇒
∥∥T (k)
∥∥ 6 1
ρ
(∥∥B(k)
∥∥+
∥∥H(k)
∥∥),
hence∥∥H(k)
∥∥ 6 k−1∑
l=1
∥∥T (l)
∥∥(∥∥A(k−l)∥∥+
∥∥B(k−l)∥∥)
6
k−1∑
l=1
1
ρ
(∥∥B(l)
∥∥+
∥∥H(l)
∥∥)(2∥∥B(k−l)∥∥+
∥∥H(k−l)∥∥).
Assume that all
∥∥B(k)
∥∥ < δ, k = 1, . . . , r. Then
∥∥H(k)
∥∥ 6 k−1∑
l=1
1
ρ
(
2δ2 + δ
(
2
∥∥H(l)
∥∥+
∥∥H(k−l)∥∥)+
∥∥H(l)
∥∥∥∥H(k−l)∥∥)
=
1
ρ
(
2(k − 1)δ2 + 3δ
k−1∑
l=1
∥∥H(l)
∥∥+
k−1∑
l=1
∥∥H(l)
∥∥∥∥H(k−l)∥∥) . (3.1)
Thus having H(1) = 0,
∥∥H(2)
∥∥ 6 2δ2/ρ, one obtains∥∥H(r)
∥∥ 6 δPr−1(δ/ρ) and
∣∣λj∣∣ 6 δ(1 + Pr−1(δ/ρ)),
where Pr−1 is a polynomial of degree r − 1. Indeed, since one already has∥∥H(2)
∥∥ 6 δP1(δ/ρ), P1(x) = 2x,
making an inductive assumption∥∥H(k)
∥∥ 6 δPk−1(δ/ρ) for k = 2, 3, . . . , r − 1
(where Pk−1 is a polynomial of degree k − 1) we deduce from (3.1) the required estimate
for
∥∥H(r)
∥∥:
∥∥H(r)
∥∥ 6 1
ρ
(
2(r − 1)δ2 + 3δ
r−1∑
l=2
δPl−1(δ/ρ) +
r−2∑
l=2
δ2Pl−1(δ/ρ)Pr−l−1(δ/ρ)
)
= δ
(
2(r − 1)
δ
ρ
+
r−1∑
l=2
3
δ
ρ
Pl−1(δ/ρ) +
r−2∑
l=2
δ
ρ
Pl−1(δ/ρ)Pr−l−1(δ/ρ)
)
= δPr−1(δ/ρ). �
Remark 3.3. As follows from the above, the polynomials Pk in the proof of Theorem 3.2 are
defined by the recurrence relations
Pk(x) = 2kx+
k∑
l=2
3xPl−1(x) +
k−1∑
l=2
xPl−1(x)Pk−l(x), k = 2, . . . , r − 1, P1(x) = 2x,
in particular,
P2(x) = 4x+ 6x2, P3(x) = 6x+ 18x2 + 22x3, . . . ,
Pr−1(x) = 2(r − 1)x+ 3(r − 1)(r − 2)x2 + · · ·+ cr−1x
r−1.
Therefore,
Linear Differential Systems with Small Coefficients: Various Types of Solvability 9
• when ε → 0 and ρ is fixed (or, more generally, ε/ρ is bounded), then the leading term of
the polynomial δ
(
1 + Pr−1(δ/ρ)
)
is δ, and the condition
δ
(
1 + Pr−1(δ/ρ)
)
< ε (3.2)
implies an asymptotic estimate δ(ε) = O(ε);
• when ε→ 0 and ρ→ 0 such that ε/ρ is unbounded, then the leading term of the polynomial
δ
(
1 + Pr−1(δ/ρ)
)
satisfying (3.2) is cr−1δ
r/ρr−1, and hence asympotically
δ(ε, ρ) = O
(
ε1/rρ(r−1)/r
)
.
There are two simple corollaries of Theorem 3.1.
Corollary 3.4 (for systems with singular points of Poincaré rank 1). In a set
S 1,...,1
a1,...,an
(
1/n(p− 1)
)
=
{(
B
(0)
i , B
(1)
i
)n
i=1
∈ S 1,...,1
a1,...,an |
∥∥B(1)
i
∥∥ < 1/n(p− 1)
}
,
systems solvable by generalized quadratures are those and only those whose matrices B
(0)
i , B
(1)
i ,
i = 1, . . . , n, are simultaneously conjugate to triangular ones.
Proof. If all the Poincaré ranks ri = 1, then
∣∣λji ∣∣ 6 ∥∥B(1)
i
∥∥ and thus one has no necessity to
restrict the leading terms B
(0)
i on some disc D bW and may take any B
(0)
i ∈W . �
Corollary 3.5. System (2.1) with the coefficient matrix B of the form
B(z) =
n∑
i=1
B
(0)
i
(z − ai)ri+1
, B
(0)
i ∈W, ri > 0,
is solvable by generalized quadratures if and only if all the matrices B
(0)
1 , . . . , B
(0)
n are simulta-
neously conjugate to triangular ones.
Proof. In the case B
(1)
i = · · · = B
(ri)
i = 0, i = 1, . . . , n, all the formal exponents equal zero,
and there is again no need to restrict the matrices B
(0)
i on some disc D bW . �
4 Practical remarks and examples
We conclude by noting that the results presented in this paper can be turned into an algo-
rithm which allows to check the solvability of a given concrete system. The main steps can be
summarized as follows.
1. Compute the formal exponents at each singular point. The practical calculation of the
formal exponents presented in the previous section, is already implemented in Maple as
a part of the general program ISOLDE [4] based on algorithms from [1].
2. Check the required smallness of the formal exponents of the system under consideration.
3. The simultaneous triangularizability of a set of matrices is equivalent to the solvability of
the Lie algebra generated by them. This can be checked using the package LieAlgebras of
Maple.4
4Note that checking the simultaneous triangularizability of a given set of matrices can be done using more
direct algorithms (that is, without recourse to Lie algebra computations). Those algorithms consist in trying first
to find a common eigenvector of the input matrices and then to proceed recursively (this is a work in progress [2]).
10 M.A. Barkatou and R.R. Gontsov
In order to illustrate the various steps of our algorithm we give two examples of computations
performed by using our implementation in Maple.
Example 4.1. We consider a (3× 3)-system (2.1) with the coefficient matrix
B(z) = M1/z
3 +M4/z +M2/(z − 1)2 +M5/(z − 1) +M3/(z + 1)2 +M6/(z + 1),
where the matrices Mi are given by
M1 =
−5 −4 −4
17 14 13
−10 −8 −7
, M2 =
−6 −5 −5
23 17 15
−14 −9 −7
, M3 =
1 1 1
−11 −7 −6
8 4 3
,
M4 =
2a− c a− c a− c
−3− 6a+ 5c −2− 3a+ 5c −2− 3a+ 5c
3 + 4a− 3c 2 + 2a− 3c 2 + 2a− 3c
,
M5 =
0 0 0
b+ 2 −b+ 1 −2b+ 1
−b− 2 b− 1 2b− 1
,
M6 =
−2a+ c −a+ c −a+ c
−b+ 6a+ 1− 5c b+ 3a+ 1− 5c 2b+ 3a+ 1− 5c
b− 4a− 1 + 3c −b− 2a− 1 + 3c −2b− 2a− 1 + 3c
and a, b and c are parameters. It has three singular points a1 = 0, a2 = 1, a3 = −1 of
Poincaré rank r1 = 2, r2 = 1, r3 = 1, respectively. The point z = ∞ is non-singular, since
M4 +M5 +M6 = 0.
Our implementation allows to check that the three singularities are non-resonant and gives
the formal exponents for each of them.
> read "/Users/barkatou/Desktop/Spliting/split":
We apply the splitting lemma at z = a1 up to k = 2.
> splitlemma(B, z=0, 2, T);
We get the equivalent matrix
− 1
z3
+
a
z
0 0
0
1
z3
0
0 0
2
z3
+
c
z
+ regular part.
The formal exponents at z = a1 are then λ1
1 = 0, λ2
1 = a, λ3
1 = c.
We apply now the splitting lemma at z = a2 up to k = 1.
> splitlemma(B, z=1, 1, T);
We get the equivalent matrix
− 1
(z − 1)2
0 0
0
2
(z − 1)2
+
b
z − 1
0
0 0
3
(z − 1)2
+ regular part.
The formal exponents at z = a2 are then λ1
2 = 0, λ2
2 = 0, λ3
2 = b.
Finally, we apply the splitting lemma at z = a3 up to k = 1.
Linear Differential Systems with Small Coefficients: Various Types of Solvability 11
> splitlemma(B, z=-1, 1, T);
We get the equivalent matrix
− a
z + 1
0 0
0 − 2
(z + 1)2
− c
z + 1
0
0 0 − 1
(z + 1)2
− b
z + 1
+ regular part.
The formal exponents at z = a3 are then λ1
3 = −a, λ2
3 = −c, λ3
3 = −b.
Note that in our case (p = 3, n = 3) the condition (2.4) of Theorem 2.1 is as follows:
Reλji > −1/6, i, j = 1, 2, 3.
This is equivalent here to the condition
|Reλ| < 1/6, for λ ∈ {a, b, c}.
Now let us check in which case the matrices Mi, i = 1, . . . , 6, are simultaneously triangular-
izable. For this, we can first use the LieAlgebras package of Maple.
1. First one computes the Lie algebra L generated by the set S = {M1, . . . ,M6}.
2. Then one can use the Query function with the argument “Solvable” to check the solvability
of L.
> with(DifferentialGeometry):
> with(LieAlgebras):
> S:=[seq(M_i, i = 1 .. 6)]:
> L := MatrixLieAlgebra(S);
−5 −4 −4
17 14 13
−10 −8 −7
,
−6 −5 −5
23 17 15
−14 −9 −7
,
1 1 1
−11 −7 −6
8 4 3
,
−1 −1 −1
14 10 10
−13 −9 −9
,
2a− c a− c a− c
−3− 6a+ 5c −2− 3a+ 5c −2− 3a+ 5c
3 + 4a− 3c 2 + 2a− 3c 2 + 2a− 3c
,
0 0 0
b+ 2 −b+ 1 −2b+ 1
−b− 2 b− 1 2b− 1
.
> LD1:=LieAlgebraData(L,Alg1):
> DGsetup(LD1):
We check the solvability of L
> Query("Solvable");
true
This tells us that the Lie algebra L is sovable (apparently for all a, b, c) but it does not give
a transformation P that simultaneously triangularizes the matrices Mi. Such a transformation
can be directly obtained using the implementation from [2].
> read "/Users/barkatou/Desktop/Simul-Triang/utilitaires_reduction.mpl":
> read "/Users/barkatou/Desktop/Simul-Triang/SimultTriang.mpl":
> P := SimultaneousTriangularization([seq(M[i], i = 1 .. 6)]);
12 M.A. Barkatou and R.R. Gontsov
P :=
0 −1 1
1 1 0
−1 0 0
.
One can check that indeed the matrices P−1MiP are triangular:
> seq(1/P . M[i] . P, i = 1 .. 6);
1 −2 10
0 −1 7
0 0 2
,
2 −5 14
0 −1 9
0 0 3
,
−1 4 −8
0 0 −3
0 0 −2
,
0 1 + 2a −3− 4a+ 3c
0 a −2a+ 2c
0 0 c
,
b −2b− 1 b+ 2
0 0 0
0 0 0
,
−b 2b− 2a −b+ 4a+ 1− 3c
0 −a 2a− 2c
0 0 −c
.
In conclusion, the system in this example is solvable in the Liouvillian sense without any
restriction on the parameters a, b, c. On the other hand, the matrices Mi, i = 1, . . . , 6, are never
simultaneously diagonalizable since, for example, M1, M2 do not commute:
[M1,M2] =
−1 −1 −1
14 10 10
−13 −9 −9
.
Hence, due to Theorem 2.4 we can assert that the system is not solvable by exponentials of
integrals and algebraic functions whenever |Reλ| < 1/6, for λ ∈ {a, b, c}.
Example 4.2. We consider a (3× 3)-system (2.1) with the coefficient matrix
B(z) = M1/(z − a1)1+r1 +M2/(z − a2)1+r2 +M3/(z − a3)1+r3 ,
where the matrices Mi are given by
M1 =
1 0 0
0 −1 0
0 0 2
, M2 =
0 0 0
3a 3 + b 1
−3ab −b2 − 5b −2− b
, M3 =
−1 0 0
0 4 0
−2 0 1
,
ri are positive integers and a, b are parameters. The matrix M2 has three distinct eigenvalues:
0, 3, −2. Hence the system has three non-resonant singularities and is of the form assumed in
Corollary 3.5.
> with(DifferentialGeometry):
> with(LieAlgebras):
> S:=[seq(M_i, i = 1 .. 3)]:
> L := MatrixLieAlgebra(S);
1 0 0
0 −1 0
0 0 2
,
0 0 0
3a 3 + b 1
−3ab −b2 − 5b −2− b
,
−1 0 0
0 4 0
−2 0 1
,
0 0 0
−6a 0 −3
−3ab −3b2 − 15b 0
,
0 0 0
0 0 0
−2 0 0
,
0 0 0
−15a− 2 0 −3
6ab+ 2b+ 4 −3b2 − 15b 0
,
0 0 0
12a 0 9
−3ab −9b2 − 45b 0
.
Linear Differential Systems with Small Coefficients: Various Types of Solvability 13
> LD1:=LieAlgebraData(L,Alg1):
> DGsetup(LD1):
We check the solvability of L
> Query("Solvable");
false
This should be interpreted as that the Lie algebra L is not solvable (and hence our three
matrices Mi are not simultaneously triangularizable) for generic values of a, b. Therefore, the
system is not solvable by generalized quadratures for generic values of a, b.
Our implementation [2] gives the same answer:
> read "/Users/barkatou/Desktop/Simul-Triang/utilitaires_reduction.mpl":
> read "/Users/barkatou/Desktop/Simul-Triang/SimultTriang.mpl":
> P := SimultaneousTriangularization([seq(M[i], i = 1 .. 3)]);
"The matrices are not simultaneously triangularizable!"
Now one can check directly that for b = −5 or b = 0, the matrices Mi are simultaneously
triangularizable. Indeed, if we consider the following matrix
P :=
0 0 1
−9 0 0
0 1 0
and compute the matrices P−1MiP , we find that
> seq(1/P . M[i] . P, i = 1 .. 3);
−1 0 0
0 2 0
0 0 1
,
3 + b −1/9 −a/3
9b(b+ 5) −2− b −3ab
0 0 0
,
4 0 0
0 1 −2
0 0 −1
,
from which we see that the matrices Mi are simultaneously triangularizable when b(b+ 5) = 0.
This means that for b = −5 or b = 0, the system is solvable by generalized quadratures and, on
the other hand, is not solvable by exponentials of integrals and algebraic functions (since the
matrices Mi are not simultaneously diagonalizable: [M1,M2] 6= 0).
Note that for this kind of examples where the matrices Mi depend on some parameters, one
question naturally arises, namely: is it possible to find the equations for the solvability locus in
the parameter space?
The answer to this question is positive. Indeed, one can use Cartan’s criterion for solvabili-
ty [10] which states that a matrix Lie algebra g is solvable (in characteristic zero) if and only if
K(g, [g, g]) = 0, where K designates the Killing form5 of g. This is also equivalent to Tr(uv) = 0
for all u ∈ g and all v ∈ [g, g], the derived Lie algebra of g.
In order to check in practice that a given Lie algebra g is solvable, one could proceed as
follows. Let {u1, . . . , ur}, resp. {v1, . . . , vs}, be a basis of g, resp. of [g, g]. Using the package
LieAlgebras of Maple one can compute fij := K(ui, vj) for i = 1, . . . , r, j = 1, . . . , s, and check
whether all the fij ’s are zero or not. In the case when the elements of g are matrices, say, with
coefficients in C(p1, . . . , pt) for some parameters p1, . . . , pt, the fij ’s are then rational functions
of p1, . . . , pt and the solvability locus in the parameter space is given by the zero set of the
5K(u, v) = Tr(ad(u)ad(v)).
14 M.A. Barkatou and R.R. Gontsov
numerators of the fij ’s. In our example, the Maple computation gives the solvability locus
determined by b(b+ 5) = 0.
Concluding remark. As we have already mentioned in Introduction, in practice, to check
the solvability by generalized quadratures of a given linear differential system dy/dz = B(z)y,
it is sufficient (but might be very costly) to compute the Lie algebra of the corresponding
differential Galois group (for example, by the algorithm in [3]) and this works, in theory at
least, for a general system. Now for the class of systems with small coefficients it is much better
(from the computational point of view) to apply the method of the present paper, namely
computing the Lie algebra defined by the coefficients of the matrix B(z). Indeed, let us denote
by g ⊂ gln(C) the Lie algebra of the differential Galois group of our system and by h the
Lie algebra generated by the coefficient matrix. The latter is defined as follows: if B(z) =
f1M1 + · · · + fsMs, where the Mi’s are constant matrices and the fi’s are rational functions
in C(z) that form a basis of the C-vector space generated by the entries of B(z), then h is the
matrix Lie algebra generated by the Mi’s. One has that g is always a Lie subalgebra of h. In
general this inclusion is strict, and when the equality holds the differential system is called in
reduced form (see [3] and references therein). Now, due to the Kolchin–Kovacić theorem, there
exists a gauge transformation y = P ỹ, with P having its entries in the algebraic closure of C(z),
that takes the given system dy/dz = B(z)y into an equivalent one dỹ/dz = B̃(z)ỹ that is in
reduced form. The algorithm in [3] computes such a transformation and hence gets g as the
Lie algebra defined by B̃(z). However computing a reduced form is not an easy task at all,
since this is equivalent to finding the differential Lie algebra g. Now, for systems with small
coefficients what we propose is considering the set of the matrices B
(j)
i involved in the partial
fraction decomposition of B(z) and testing whether they are simultaneously triangularizable
over C or not. For this (even if it is not necessary) one can form the Lie algebra h′ generated
by those matrices and test its solvability. It is not difficult to see that in fact h′ = h (as defined
above). Hence for systems with small coefficients there is no need to compute the differential
Lie algebra g, in other words there is no need to compute a reduced form of the system, one has
only to check the solvability of h′ which is obtained directly from the coefficients of the system.
Acknowledgements
The authors are grateful to Thomas Cluzeau for helpful discussions about the problem of si-
multaneous triangularizability of a set of matrices. They also thank the referee for his/her
nice suggestions which have refined the text. The work of R.G. was partially supported by the
Russian Foundation for Basic Research (projects 16-51-1500005 and 17-01-00515).
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https://doi.org/10.1007/3-7643-7536-1
1 Introduction
2 Various types of solvability
3 On the smallness of formal exponents
4 Practical remarks and examples
References
|
| id | nasplib_isofts_kiev_ua-123456789-210237 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:52Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Barkatou, M.A. Gontsov, R.R. 2025-12-04T13:07:22Z 2019 Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification / M.A. Barkatou, R.R. Gontsov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34M03; 34M25; 34M35; 34M50 arXiv: 1901.09951 https://nasplib.isofts.kiev.ua/handle/123456789/210237 https://doi.org/10.3842/SIGMA.2019.058 We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of the solvability of the Lie algebra of the differential Galois group of the system. However, the dependence of this Lie algebra on the system coefficients remains unknown. We show that for the particular class of systems with non-resonant irregular singular points that have sufficiently small coefficient matrices, the problem is reduced to that of solvability of the explicit Lie algebra generated by the coefficient matrices. This extends the corresponding Ilyashenko-Khovanskii theorem obtained for linear differential systems with Fuchsian singular points. We also give some examples illustrating the practical verification of the presented criteria of solvability by using general procedures implemented in Maple. The authors are grateful to Thomas Cluzeau for helpful discussions about the problem of simultaneous triangularizability of a set of matrices. They also thank the referee for his/her nice suggestions, which have refined the text. The work of R.G. was partially supported by the Russian Foundation for Basic Research (projects 16-51-1500005 and 17-01-00515). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification Article published earlier |
| spellingShingle | Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification Barkatou, M.A. Gontsov, R.R. |
| title | Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification |
| title_full | Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification |
| title_fullStr | Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification |
| title_full_unstemmed | Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification |
| title_short | Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification |
| title_sort | linear differential systems with small coefficients: various types of solvability and their verification |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210237 |
| work_keys_str_mv | AT barkatouma lineardifferentialsystemswithsmallcoefficientsvarioustypesofsolvabilityandtheirverification AT gontsovrr lineardifferentialsystemswithsmallcoefficientsvarioustypesofsolvabilityandtheirverification |