Factorization of piecewise constant matrix functions and systems of linear differential equations

Torsten Ehrhardt ^{∗} Fakultät für Mathematik

Technische Universität Chemnitz 09107 Chemnitz, Germany

Ilya M. Spitkovsky ^{†} Department of Mathematics College of William and Mary Williamsburg, VA 23187, USA

Abstract

Let G be a piecewise constant n × n matrix function which is defined on a smooth closed curve Γ in the complex sphere and which has m jumps. We consider the problem of determining the partial indices of the factorization of the matrix function G in the space L^{p}(Γ). We show that this problem can be reduced to a certain problem for systems of linear differential equations.

Studying this related problem, we obtain some results for the partial indices for general n and m. A complete answer is given for n = 2, m = 4 and for n = m = 3. One has to distinguish several cases. In some of these cases, the partial indices can be determined explicitly. In the remaining cases, one is led to two possibilities for the partial indices. The problem of deciding which is the correct possibility is equivalent to the description of the monodromy of n-th order linear Fuchsian differential equations with m singular points.

1 Introduction

Let Γ be a smooth closed positively oriented curve in the complex sphere ℂ̇ = ℂ∪{∞} which divides ℂ̇ into two domains D_{+} and D_{−}. Assume without loss of generality that 0 ∈ D_{+} and ∞ ∈ D_{−}. Let L^{p}(Γ) be the Banach space of all Lebesgue measurable functions on Γ for which

∥f∥_{L^{p}(Γ)} = ( ∫_{Γ} |f(τ)|^{p} |dτ|)^{1/p} < ∞.  (1.1)

For 1 < p < ∞, consider the singular integral operator S_{Γ} defined on L^{p}(Γ),

(S_{Γ}f)(t) = \frac{1}{πi} ∫_{Γ} \frac{f(τ)}{τ − t} dτ,  t ∈ Γ.  (1.2)

∗tehrhard@mathematik.tu-chemnitz.de. Research at MSRI is supported in part by NSF grant DMS-9701755. Research has also been supported in part by DAAD Grant 213/402/537/5.

†ilya@math.wm.edu. Research is supported in part by NSF Grant DMS-9800704.

The operator S_{Γ} is bounded on L^{p}(Γ) and satisfies S_{Γ}^{2} = I (see [10]). Introduce the projections

P_{Γ} = (I + S_{Γ})/2,  Q_{Γ} = (I − S_{Γ})/2,  (1.3)

and the Banach spaces

L_{+}^{p}(Γ) = P_{Γ}(L^{p}(Γ)),  L_{−}^{p}(Γ) = Q_{Γ}(L^{p}(Γ)) ∔ ℂ,  (1.4)

i.e., the image of P_{Γ} and the image of Q_{Γ} plus the set of constant functions.

It is well known ([11], see also [15]) that functions f ∈ L_{+}^{p}(Γ) can be identified with functions f̂ which are analytic in D_{+} and for which there exists an expanding sequence of domains D_{k} with rectifiable boundaries Γ_{k} such that D_{k} ∪ Γ_{k} ⊂ D_{+}, ⋃_{k} D_{k} = D_{+}, and sup_{k} ∫_{Γ_{k}} |f(t)|^{p} |dt| < ∞. In fact, D_{k} can always be chosen as the image of { z: |z| < r_{k} } under a conformal mapping of the unit disk 𝔻 = { z: |z| < 1 } onto D_{+}, where r_{k} is an arbitrary increasing sequence of positive numbers converging to 1. The function f is given by the non-tangent limits of the function f̂ at the boundary Γ of D_{+}. Conversely, the function f̂ represents the analytic extension of f into D_{+}. A similar result holds for L_{−}^{p}(Γ).

Let L^{∞}(Γ) denote the set of all Lebesgue measurable and essentially bounded functions on Γ. A factorization of an n × n matrix function G ∈ L^{∞}(Γ)^{n×n} in the space L^{p}(Γ) is a representation of the form

G(t) = G_{+}(t)Λ(t)G_{−}(t),  t ∈ Γ,  (1.5)

where Λ(t) = diag (t^{κ_{1}}, …, t^{κ_{n}}) is a diagonal matrix with κ_{1}, …, κ_{n} ∈ ℤ, and the factors G_{+} and G_{−} satisfy the following conditions:

(i) G_{+} ∈ L_{+}^{p}(Γ)^{n×n}, G_{+}^{−1} ∈ L_{+}^{q}(Γ)^{n×n},

(ii) G_{−} ∈ L_{−}^{q}(Γ)^{n×n}, G_{−}^{−1} ∈ L_{−}^{p}(Γ)^{n×n}.

Here 1/p + 1/q = 1. The integers κ_{1}, …, κ_{n} are called the partial indices of the factorization.

Due to (i)-(ii), the operator G_{−}^{−1}Q_{Γ}G_{+}^{−1} is defined on the (dense in L^{p}(Γ)^{n}) set of rational vector functions with poles off Γ and maps this set into L^{1}(Γ)^{n}. If this mapping is bounded in the L^{p}-norm, then it can be extended by continuity to a bounded operator on the whole space L^{p}(Γ)^{n}. In this case, the representation (1.5) is called a Φ-factorization in the space L^{p}(Γ).

A detailed discussion of the Φ-factorization, its properties and the history of the subject can be found in [7, 15]; the latter monograph touches also on the (not necessarily Φ-) factorization in L^{p}(Γ). The proofs of all the subsequent results stated in this section can be found in [7, 15]. The next theorem explains why the notion of the Φ-factorization is natural and important.

Theorem 1.1 The matrix function G ∈ L^{∞}(Γ)^{n×n} admits a Φ-factorization in the space L^{p}(Γ) if and only if the singular integral operator A = P_{Γ} + GQ_{Γ} is a Fredholm operator on the space L^{p}(Γ)^{n}. In this case, we have

dim ker A = ∑_{κ_{j}>0}κ_{j}, dim ker A^{∗} = − ∑_{κ_{j}<0}κ_{j}.

Hence the partial indices κ_{1}, …, κ_{n} contain important information about the operator A. In particular, A is invertible if and only if the function G admits a socalled canonical Φ-factorization, i.e., a Φ-factorization where all partial indices are zero. If A is a Fredholm operator, then its index, ind A := dim ker A − dim ker A^{∗}, is equal to

κ = ∑_{j=1}^{n} κ_{j},  (1.6)

the total index of the factorization.

The notion of Φ-factorization plays also an important role in the study of more general singular integral operators. One can show that the operator G_{1}P_{Γ} + G_{2}Q_{Γ} with G_{1}, G_{2} ∈ L^{∞}(Γ)^{n×n} is a Fredholm operator if and only if the functions G_{1} and G_{2} are invertible in L^{∞}(Γ)^{n×n} and the function G_{1}^{−1}G_{2} admits a Φ-factorization.

The factorization of a matrix function is not unique. However, the partial indices are always uniquely determined up to the change of order. For this reason, we will assume that

κ_{1} ≥ κ_{2} ≥ … ≥ κ_{n}.  (1.7)

The relation between factors G_{+} and G_{−} corresponding to different factorizations of a function G is described in the following theorem. The characterization involves certain rational block triangular matrix functions whose size is determined by the multiple occurrence of same values for the partial indices. Let I_{l} denote the identity matrix of size l × l, and let Gℂ^{l×l} stand for the group of all invertible l × l matrices.

Theorem 1.2 Assume that G ∈ L^{∞}(Γ)^{n×n} admits two factorizations in L^{p}(Γ),

G(t) = G_{+}(t)Λ(t)G_{−}(t) = G̃_{+}(t)Λ(t)G̃_{−}(t), t ∈ Γ,

where Λ(t) = diag (t^{\overline{κ}^{1}}I_{l_{1}}, t^{\overline{κ}^{2}}I_{l_{2}}, …, t^{\overline{κ}^{k}}I_{l_{k}}) with \overline{κ}_{1}, …, \overline{κ}_{k} ∈ ℤ, \overline{κ}_{1} > \overline{κ}_{2} > … > \overline{κ}_{k}, l_{1}, …, l_{k} ∈ {1, 2, …}, l_{1} + … + l_{k} = n. Then

G_{+}(t) = G̃_{+}(t)V (t),  G̃_{−}(t) = U(t)G_{−}(t),  (1.8)

where U(t) and V (t) are matrix functions of the form

V (t) = ( \begin{matrix}A_{11} & V_{12}(t) & … & V_{1k}(t) \\ 0 & A_{22} & & V_{2k}(t) \\ ⋮ & & ... & ⋮ \\ 0 & 0 & … & A_{kk} \end{matrix} ) , U(t) = ( \begin{matrix}A_{11} & U_{12}(t) & … & U_{1k}(t) \\ 0 & A_{22} & & U_{2k}(t) \\ ⋮ & & ... & ⋮ \\ 0 & 0 & … & A_{kk} \end{matrix} ) ,

with A_{jj} ∈ Gℂ^{l_{j}×l_{j}}, V_{ij}(t) = ∑_{r=0}^{\overline{κ}_{i}−\overline{κ}_{j}} A_{ij}^{(r)}t^{r}, U_{ij}(t) = t^{\overline{κ}_{j}−\overline{κ}_{i}}V_{ij}(t) and A_{ij}^{(r)} ∈ ℂ^{l_{i}×l_{j}}.

In fact, also the converse of the above theorem is true. Suppose we are given a factorization (Φ-factorization) G = G_{+}ΛG_{−}, and we determine G̃_{+} and G̃_{−} by (1.8), then also G = G̃_{+}ΛG̃_{−} is a factorization (Φ-factorization). Using this theorem and its converse, we can conclude the following statement.

Corollary 1.3 Suppose G ∈ L^{∞}(Γ)^{n×n} admits a Φ-factorization and G = G_{+}ΛG_{−} is a factorization. Then G = G_{+}ΛG_{−} is such a Φ-factorization.

Let P C(Γ)^{n×n} stand for the set of all piecewise continuous n×n matrix functions, i.e., functions for which the one-sided limits G(τ +0) and G(τ −0) exist at each τ ∈ Γ. Here G(γ(θ_{0}) ± 0) := lim_{θ→θ_{0}±0} G(γ(θ)), where the periodic function γ : ℝ → Γ is a parameterization of the positively oriented curve Γ. The following result solves the problem of the existence of a Φ-factorization for functions in P C(Γ)^{n×n}.

Theorem 1.4 The matrix function G ∈ P C(Γ)^{n×n} admits a Φ-factorization in the space L^{p}(Γ) if and only if the following is satisfied:

(i) The matrices G(τ + 0) and G(τ − 0) are invertible for each τ ∈ Γ.

(ii) For each j = 1, …, n and each τ ∈ Γ, we have

\frac{1}{2π} arg λ_{j}(τ) + \frac{1}{p} ∉ ℤ,  (1.9)

where λ_{1}(τ), …, λ_{n}(τ) are the eigenvalues of the matrices G(τ − 0)G(τ + 0)^{−1}.

If the assumptions of this theorem are satisfied, then one can define the numbers ζ_{j}(τ) as the (unique) values of −(2π)^{−1} arg λ_{j}(τ) which lie in the interval J_{p} := (1/p − 1, 1/p).

It is also possible to evaluate the total index κ related to the above factorization. For simplicity assume that G has only finitely many jumps at points a_{1}, …, a_{m} ∈ Γ, which lie in this order on the positively oriented curve Γ. Then

κ = ∑_{k=1}^{m} [\frac{1}{2π} arg det G(τ)]_{τ=a_{k}+0}^{a_{k+1}−0} + ∑_{k=1}^{m} ∑_{j=1}^{n} ζ_{j}(a_{k}).  (1.10)

Here a_{m+1} = a_{1}, and […] denotes the total increment along the subarcs on which G is continuous. Note that, in general, the value of κ depends on the space L^{p}(Γ) because so do the numbers ζ_{j}(τ). Only in the case where all λ_{j}(τ) are positive real numbers (hence ζ_{j}(τ) = 0), the value of κ does not depend on the underlying space L^{p}(Γ).

The calculation of the partial indices κ_{1}, …, κ_{n} in the case n > 1 is a more delicate problem and can be solved only in special situations. It is the purpose of this paper to consider the case of piecewise constant matrix functions, i.e., functions which have only a finite number of jump discontinuities and which are constant along the subarcs joining these points. We will show that the problem of determining the partial indices can be reduced to a certain problem for systems of linear differential equations. Elaborating on this related problem, we will obtain some information about the partial indices.

Preparing the following sections, we define certain matrices. Let G ∈ P C(Γ)^{n×n} be a piecewise constant matrix function with m jumps at the points a_{1}, …, a_{m} ∈ Γ, which are arranged on Γ in this order. Assume that G admits a Φ-factorization in the space L^{p}(Γ). Introduce the matrices

M_{k} = G(a_{k} − 0)G(a_{k} + 0)^{−1},  k = 1, …, m.  (1.11)

Because G is constant along the subarcs joining a_{k} and a_{k+1} and takes the values G(a_{k} + 0) = G(a_{k+1} − 0) there, these matrices satisfy the relation

M_{1}M_{2} ⋯ M_{m} = I.  (1.12)

Moreover, let E_{1}, …, E_{m} be matrices such that

M_{k} ∼ exp(−2πiE_{k}),  k = 1, …, m,  (1.13)

and the real parts of the eigenvalues of E_{k} are contained in the interval J_{p}. Here “∼” stands for similarity of matrices. Because G admits a Φ-factorization and J_{p} has length one, the matrices E_{k} exist and are uniquely determined up to similarity. In fact, the numbers ζ_{1}(a_{k}), …, ζ_{n}(a_{k}) defined above are equal to the real parts of the eigenvalues of E_{k}. Formula (1.10) for the total index can also be simplified:

κ = ∑_{k=1}^{m} trace E_{k}.  (1.14)

Note that the matrices E_{1}, …, E_{m} depend on the underlying space L^{p}(Γ). Roughly speaking, they contain the information about the “proper” choice of the logarithm of the eigenvalues of M_{1}, …, M_{m}. For sake of further reference, the m-tuples of n × n matrices

[M_{1}, …, M_{m}] and [E_{1}, …, E_{m}]  (1.15)

will be called the “data” associated with the piecewise constant matrix function G with respect to the space L^{p}(Γ).

Let us recall what is already known about the factorization of piecewise constant n × n matrix functions with m jump discontinuities. Any piecewise constant scalar function (n = 1) can be easily factored explicitly. Namely, G(t) = G_{+}(t)t^{κ}G_{−}(t), where

G_{+}(t) = c ∏_{k=1}^{m}(t − a_{k})^{−ε_{k}},  G_{−}(t) = ∏_{k=1}^{m} (1 − \frac{a_{k}}{t} )^{ε_{k}} ,  (1.16)

ε_{k} = E_{k}, and κ is given by (1.14). The branches of the analytic functions in (1.16) are chosen in such a way that G_{±} is analytic on D_{±}, and c is a suitable nonzero constant.

Hence, the case n = 1 is solvable for any m. Another easy case is m = 2 for any n. Indeed, multiplying G by the constant matrix G(a_{1} − 0)^{−1} on the left, we may suppose without loss of generality that G assumes the values I and M(= M_{2}) only. These values obviously commute with each other. In other words, G is a so-called functionally commuting matrix function, and its factorization can be constructed explicitly (see [15, Section 4.4]).

The simplest non-trivial case (n = 2, m = 3) was first studied by Zverovich and Khvoschinskaya [20] in a slightly different setting. In the setting considered here it was treated by Tashbaev and one of the authors in [18]. Using the appropriate modification of the results from [20], they gave explicit formulas for the factors and the partial indices. Therein they had to distinguish several cases. Apart from trivial cases (where G can be reduced to a functionally commuting matrix function), the factors were constructed by using the hypergeometric function. We will not restate here the complete results of [18], however, the results on the partial indices will be obtained again in a corollary in Section 6 in the setting of the related problem.

Let us describe the content of this paper in more detail. In Section 2, we recall some basic facts from the theory of systems of linear differential equations and from the theory of scalar linear differential equations (of higher order).

In Section 3, we introduce a class of n × n systems Y′(z̃) = A(z)Y (z̃) of linear differential equations, which we call systems of standard form. Their solutions Y (z̃) are characterized by the monodromy, which is given by [M_{1}, …, M_{m}], and by the behavior at the singularities a_{1}, …, a_{m} ∈ ℂ, which is given by [E_{1}, …, E_{m}]. In addition, we allow an apparent singularity at infinity, where the behavior is described by certain integers κ_{1}, …, κ_{n}, called the indices. Then we prove an “existence” and a “uniqueness” theorem for such systems. Moreover, we show that these systems of linear differential equations provide us with a solution to our factorization problem. Regarding the calculation of the partial indices, we are led to the following (wellposed) question: What are the indices of the systems which are characterized by given data [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}] and singularities a_{1}, …, a_{m}? In fact, the indices for the systems correspond to the partial indices of the factorization.

In Section 4, we reformulate the conditions regarding the behavior at the singularities at a_{1}, …, a_{m} and at infinity in terms of the matrix function A(z) rather than in terms of the solution Y (z̃). This is a simple step, but all further considerations are based on it. We note that the above question can now be expressed as follows: What is the monodromy of a system given by A(z), where A(z) has a certain form?

In Section 5, we obtain two “general” results. Firstly, if the monodromy is irreducible, then the indices (which are supposed to be ordered decreasingly) satisfy κ_{k} − κ_{k+1} ≤ m − 2 for each k = 1, …, n − 1. Secondly, we show that this condition is sharp. For this we produce a subclass of systems for which κ_{k} − κ_{k+1} = m − 2 for each k = 1, …, n − 1. It turns out that the monodromy of such systems coincides with the monodromy of certain scalar Fuchsian linear differential equations of n-th order and with m singularities. We remark that Bolibruch [5] already obtained the first result in the setting of vector bundles (and by using a different method) and that there is also a strong connection of the second result with his work.

In Section 6, we consider the case n = 2. One is led naturally to distinguishing three cases relating to the “reducibility type” of [M_{1}, …, M_{m}], i.e., the structure of the invariant subspaces of these matrices. For general m, we can determine the indices in some cases explicitly. In the remaining cases, we obtain the estimate κ_{1} − κ_{2} ≤ m − 2. These remaining cases include irreducible ones (for which this result is already clear from Section 5) as well as certain reducible ones. The essential point in the proof is a theorem which characterizes the data (hence, in particular, the monodromy) of a certain class of 2 × 2 triangular systems.

With these results we are immediately in a position to give an explicit answer for the case n = 2, m = 3, hence restating the results of [18]. However, the results obtained here are slightly more general because the assumptions on the matrices E_{1}, …, E_{m} are less restrictive.

The complete answer is given for the case n = 2 and m = 4 by resorting in addition to the second result of Section 5. However, the answer is explicit only in some cases. In fact, in the other cases, we are led to two possibilities for the indices. The problem of deciding which possibility is the correct one is equivalent to the description of the monodromy of second order linear Fuchsian differential equations with four singular points.

In Section 7, we treat the case n = m = 3. It is technically much more complicated than the previous case as now nine reducibility types for [M_{1}, M_{2}, M_{3}] occur. We also have to prove theorems which characterize the data (monodromy) of triangular and block-triangular systems. Finally, in the main theorem we arrive at several cases, where, similar as before, in some cases the indices can be given explicitly whereas in the other cases they depend on the description of the monodromy of third order linear Fuchsian differential equations with three singular points.

2 Systems of linear differential equations

A system of linear differential equations in the complex domain can be written as

\frac{dy}{dz} = A(z)y,  (2.1)

where A(z) is a given n × n matrix function. We assume that A(z) is analytic on the complex plane ℂ with the exception of a finite number of points, which are called the singularities of the system. The point a = ∞ is by definition a singularity if the function z^{−2}A(z^{−1}) is not analytic at z = 0, which is justified by a change of variables z ↦ 1/z.

Let a_{1}, …, a_{m} ∈ ℂ̇ be the singularities of the system, and let S̃ be the universal covering surface of S := ℂ̇ \ {a_{1}, …, a_{m}}. Then the solutions y of (2.1) are analytic functions defined on S̃ and taking values in ℂ^{n}. Associated with S̃ there is a covering map ρ : S̃ → S. Points on S̃ will usually be denoted by z̃ and points on S by z = ρ(z̃). The set of all solutions of (2.1) forms an n-dimensional linear space.

Taking n linearly independent solutions y_{1}(z̃), …, y_{n}(z̃), one can consider the analytic n × n matrix function Y (z̃) = (y_{1}(z̃), …, y_{n}(z̃)). It satisfies the matrix equation

Y′(z̃) = A(z)Y (z̃),  z̃ ∈ S̃,  (2.2)

and we have det Y (z̃) ≠ 0 for all z̃ ∈ S̃. In fact, for any solution of (2.2) with det Y (z̃_{0}) ≠ 0 for some z̃_{0} ∈ S̃ we have det Y (z̃) ≠ 0 for all z̃ ∈ S̃. We will consider only such solutions (i.e., analytic functions Y : S̃ → Gℂ^{n×n}) because only they contain the full information. Any two solutions Y_{1}(z̃) and Y_{2}(z̃) are related to each other by Y_{2}(z̃) = Y_{1}(z̃)C with C ∈ Gℂ^{n×n}.

Associated with the universal covering surface S̃, there is a group Δ of deck transformations. A deck transformation is an analytic bijection σ : S̃ → S̃ which satisfies ρ ∘ σ = ρ.

Given a solution Y (z̃) of (2.2) and σ ∈ Δ, then Y (σ(z̃)) is also a solution of (2.2). Hence there exists a unique matrix χ(σ) ∈ Gℂ^{n×n} such that

Y (z̃) = Y (σ(z̃))χ(σ).  (2.3)

In fact, the mapping χ : Δ → Gℂ^{n×n} is a representation (group homomorphism), i.e., χ(στ) = χ(σ)χ(τ). This mapping is called the monodromy representation of Y (z̃).

Now let Ŷ(z̃) be another solution of (2.2), and let χ̂ : Δ → Gℂ^{n×n} be the monodromy representation of Ŷ(z̃). If Y and Ŷ are related by Ŷ(z̃) = Y (z̃)C with C ∈ Gℂ^{n×n}, then

χ̂(σ) = C^{−1}χ(σ)C  (2.4)

for all σ ∈ Δ. This means that to the system (2.2) there corresponds a class of mutually conjugate representations. This class is called the monodromy of the system (2.2).

It is interesting to note that the matrix function A(z) can be reconstructed from a solution Y (z̃). Assume that Y : S̃ → Gℂ^{n×n} is an analytic function which satisfies (2.3) with some representation χ : Δ → Gℂ^{n×n}. Then one can define

A(z) := Y′(z̃)Y (z̃)^{−1},  z̃ ∈ S̃.  (2.5)

This definition is correct since the right hand side turns out to be a single-valued analytic function (i.e., it is invariant under any deck transformation).

We will occasionally have to modify a solution of a system Y′(z̃) = A(z)Y (z̃) with a (single-valued) analytic matrix function V : S → Gℂ^{n×n} as follows: Ŷ(z̃) = V (z)Y (z̃). Then formula (2.5) tells us that Ŷ(z̃) is a solution of the system Ŷ′(z̃) = Â(z)Ŷ(z̃) with

Â(z) = V (z)A(z)V^{−1}(z) + V′(z)V^{−1}(z),  z ∈ S.  (2.6)

Moreover, the modified system has the same monodromy as the original one. Conversely, suppose that A(z) and Â(z) are related to each other by (2.6) with some single-valued matrix function V (z). Then the solutions of the corresponding systems are connected with each other by Ŷ(z̃) = V (z)Y (z̃)C with C ∈ Gℂ^{n×n}.

Because the group Δ is finitely generated, by elements σ_{1}, …, σ_{m} say, a complete characterization of the monodromy is given by the matrices χ(σ_{1}), …, χ(σ_{m}). Although there is no distinguished choice for the generators, we will choose them in a specific way.

Assume that a_{1}, …, a_{m} ∈ ℂ lie in this order on a smooth closed positively oriented curve Γ ⊂ ℂ. Let z̃_{0} ∈ S̃ be a point such that ρ(z̃_{0}) lies inside of Γ. For k = 1, …, m, determine the (unique) point z̃_{k} ∈ S̃ satisfying ρ(z̃_{k}) = ρ(z̃_{0}) and the following condition: there exists a simple path γ̃_{k} : [0, 1] → S̃ with γ̃_{k}(0) = z̃_{0} and γ̃_{k}(1) = z̃_{k} such that the curve ρ(γ̃_{k}) intersects Γ exactly twice, first on the subarc joining a_{k−1} and a_{k}, and second on the subarc joining a_{k} and a_{k+1}. (That means, the curve ρ(γ̃_{k}) goes once around a_{k} in positive direction.) Having determined z̃_{k}, there exists a unique deck transformation σ_{k} ∈ Δ for which σ_{k}(z̃_{0}) = z̃_{k}. These deck transformations σ_{1}, …, σ_{m} ∈ Δ satisfy the condition

σ_{1} ∘ σ_{2} ∘ … ∘ σ_{m} = id.  (2.7)

The construction of σ_{1}, …, σ_{m} depends primarily on the points a_{1}, …, a_{m} and the curve Γ. It also depends on z̃_{0}, but for a different point z̃_{0}, we get deck transformations σ′_{k} = τ^{−1} ∘ σ_{k} ∘ τ with some τ ∈ Δ.

The monodromy representation of a solution of (2.2) can now be described by the m-tuple [χ(σ_{1}), …, χ(σ_{m})], which (because of (2.7)) satisfies the relation

χ(σ_{1})χ(σ_{2}) ⋯ χ(σ_{m}) = I.  (2.8)

As we are interested in monodromy, let us introduce the following equivalence relation:

[M_{1}, …, M_{m}] ∼ [M′_{1}, …, M′_{m}]  (2.9)

if and only if there exists a C ∈ Gℂ^{n×n} such that M′_{k} = C^{−1}M_{k}C for all k = 1, …, m. The equivalence classes are denoted by [M_{1}, …, M_{m}]_{∼} and will be called the simultaneous similarity class. Hence the monodromy of the system (2.2) is given by class [χ(σ_{1}), …, χ(σ_{m})]_{∼}.

In order to study the behavior of the solutions of the system (2.2) in a neighborhood of the singularities, we introduce the punctured neighborhoods D_{a} = {z ∈ ℂ : 0 < |z − a| < ε} for a ∈ ℂ and D_{∞} = {z ∈ ℂ : |z| > 1/ε} for a = ∞, where ε is sufficiently small. Let D̃_{a} be the universal covering surface of D_{a}. The associated group of deck transformations is singly generated by a transformation σ_{a}, which sends a point z̃ ∈ D̃_{a} into its correspondent after going once around the point a in positive direction.

Let us restrict the solutions of the system (2.2) onto D̃_{a_{k}} and assume for simplicity that a_{k} ≠ ∞. Then the function Y : D̃_{a_{k}} → Gℂ^{n×n} has the property Y (z̃) = Y (σ_{a_{k}}(z̃))M_{k}, where M_{k} is similar to χ(σ_{k}). Let E_{k} be any matrix satisfying M_{k} = exp(−2πiE_{k}), and introduce

(z̃ − a_{k})^{E_{k}} := exp(E_{k} ln(z̃ − a_{k})),  z̃ ∈ D̃_{a_{k}},  (2.10)

noting that the logarithm is well defined on D̃_{a_{k}}. Then Y (z̃) can be written as

Y (z̃) = Z_{k}(z)(z̃ − a_{k})^{E_{k}},  z̃ ∈ D̃_{a_{k}},  (2.11)

where Z_{k} : D_{a_{k}} → Gℂ^{n×n} is a (single-valued) analytic function, which may have (along with its inverse) an isolated singularity at a_{k}. The description of the local behavior of the solution is now reduced to characterizing the matrix E_{k} and the behavior of Z_{k}(z) and its inverse.

When speaking of a restriction of a solution, which is defined on S̃, onto D̃_{a_{k}}, we have to be a little bit careful. Namely, if ρ : S̃ → S is the covering map, then the preimage ρ^{−1}(D_{a_{k}}) can be identified with D̃_{a_{k}} × Ω where Ω is an index set, which is countably infinite for m ≥ 3. For a solution Y : S̃ → Gℂ^{n×n}, the relation (2.11) should therefore correctly be written as

Y (z̃_{∗}) = Z_{k}(z)(z̃ − a_{k})^{E_{k}}C_{ω},  z̃_{∗} = (z̃, ω) ∈ D̃_{a_{k}} × Ω,  (2.12)

where C_{ω} ∈ Gℂ^{n×n} depends on the connected component Ω of the preimage ρ^{−1}(D_{a_{k}}).

A singularity a_{k} of a system is called apparent if the solutions have no branching behavior at a_{k}, i.e., if χ(σ_{k}) = I. In this case it is not necessary to take this singularity into account when defining the universal covering surface.

The following fundamental theorem says that there exist systems with prescribed singularities and prescribed monodromy. A first proof was given by Plemelj [16], who employed the theory of singular integral equations. Another proof [17, 1] can be given by using the theory of vector bundles [9, 19].

Theorem 2.1 For any distinct points a_{1}, …, a_{m} ∈ ℂ̇ and matrices M_{1}, …, M_{m} ∈ ℂ^{n×n} satisfying M_{1} ⋯ M_{m} = I, there exists a system with singularities only at a_{1}, …, a_{m} and with monodromy given by [M_{1}, …, M_{m}]_{∼}.

This theorem (as it is stated) does not say anything about the type of the singularities. A singularity a_{k} of a system is called Fuchsian if A(z) has only a simple pole at z = a_{k}. A singularity a_{k} is called regular if the function Z_{k}(z) in (2.12) has not an essential singularity at z = a_{k}. A Fuchsian singularity is always regular.

The result of Plemelj is actually stronger than the statement of the previous theorem. One can construct a system which has only Fuchsian singularities with the possible exception of one singularity, which is (in general) merely regular.

The question whether there exists a system with prescribed singularities and monodromy such that all singularities are Fuchsian is known as the Riemann–Hilbert problem or Hilbert’s 21st problem. Surprisingly, Bolibruch [3, 1, 5] showed that the answer may be negative. Notice, however, that Plemelj’s results still implies that the answer is always positive if one allows one additional apparent singularity.

A generalized version of the Riemann–Hilbert problem, where one asks for systems with prescribed Fuchsian singularities, prescribed monodromy and prescribed local behavior (in a certain specified sense) was also studied by Bolibruch [6].

The Riemann–Hilbert problem and its modifications have an intimate relation to the problem which will be proposed in Section 3. In fact, all these problems can be formulated by using the language of vector bundles. We will not elaborate on this direction, but merely confine ourselves to some comments later on.

Finally, let us recall some facts about (scalar) linear differential equations of higher order [13, 14, 8]. A linear differential equation of n-th order can be written as

y^{(n)} + q_{1}(z)y^{(n−1)} + … + q_{n}(z)y = 0,  (2.13)

where y^{(k)} stands for the k-th derivative of the complex valued function y. We assume that q_{1}(z), …, q(z) are analytic functions on ℂ with the exception of a finite number of points, which represent the singularities of the equation. The point a = ∞ is a singularity if so is the point z = 0 for the equation resulting from a change of variables z ↦ 1/(z − a). (This definition does not depend on the choice of a ∈ ℂ.)

The solutions of the above equation with singularities a_{1}, …, a_{m} ∈ ℂ̇ are analytic functions y : S̃ → ℂ defined on the corresponding universal covering surface. The set of all solutions forms an n-dimensional linear space. Given n linearly independent solutions y_{1}(z̃), …, y_{n}(z̃), the monodromy representation χ : Δ → Gℂ^{n×n} is defined by

(y_{1}(z̃), …, y_{n}(z̃)) = (y_{1}(σ(z̃)), …, y_{n}(σ(z̃)))χ(σ),  (2.14)

where χ(σ) ∈ Gℂ^{n×n} is a uniquely determined matrix and σ ∈ Δ.

A singularity a ∈ ℂ of equation (2.13) is called Fuchsian if one can write q_{k}(z) = r_{k}(z)/(z − a)^{k} where r_{k}(z) is analytic at z = a for each k = 1, …, n. The local exponents ρ_{1}, …, ρ_{n} of this singularity are by definition the roots of the indical equation

ρ(ρ − 1) ⋯ (ρ − n + 1) + ∑_{k=1}^{n} r_{k}(a)ρ(ρ − 1) ⋯ (ρ − n + 1 + k) = 0. (2.15)

The singularity at infinity is Fuchsian if q_{k}(z) = O(1/z^{k}) as z → ∞ for each k = 1, …, n, and the local exponents are defined similarly.

Now suppose that equation (2.13) has exactly m Fuchsian singularities a_{1}, …, a_{m} with local exponents ρ_{1}^{(k)}, …, ρ_{n}^{(k)} for a_{k}. Then the well known Fuchs’ relation says that

∑_{k=1}^{m}(ρ_{1}^{(k)} + … + ρ_{n}^{(k)}) = (m − 2)\frac{n(n − 1)}{2}.  (2.16)

A couple of more facts about linear differential equations will be stated in Section 5 when they are needed.

3 Systems of standard form and their relation to the factorization problem

In this section, we introduce a class of systems which are characterized by their singularities, their monodromy and their local behavior near the singularities. In addition, we allow an apparent singularity at infinity the behavior of which is described by certain integers.

We prove that such systems exist for given singularities, monodromy and local data. We also discuss the “uniqueness”, i.e., we examine the relation between two systems having the same singularities and the same data. Finally, we show that the factors appearing in the factorization of a piecewise constant matrix function can be constructed by using the solutions of such systems. The problem of determining the partial indices reduces to a corresponding problem for systems of linear differential equations.

Let a_{1}, …, a_{m} ∈ ℂ be distinct points which lie in this order on a smooth closed positively oriented curve Γ ⊂ ℂ. Denote by S̃ the universal covering surface of S := ℂ̇ \ {a_{1}, …, a_{m}}. Define deck transformations σ_{1}, …, σ_{m} for S̃ in the way explained in the previous section.

A matrix E is called non-resonant if the difference of any two eigenvalues of E is not a nonzero integer. A collection [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}] of m-tuples of n × n matrices will be called admissible data if the following conditions are satisfied:

(a) M_{1}M_{2} ⋯ M_{m} = I.

(b) M_{k} ∼ exp(−2πiE_{k}) for each k = 1, …, m.

(c) The matrices E_{1}, …, E_{m} are non-resonant.

Definition 3.1 We say that the system Y′(z̃) = A(z)Y (z̃) is of standard form with respect to admissible data [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}], with singularities a_{1}, …, a_{m} ∈ ℂ and with indices κ_{1}, …, κ_{n} ∈ ℤ, where κ_{1} ≥ κ_{2} ≥ … ≥ κ_{n}, if the following conditions are satisfied:

(i) The system has singularities only at a_{1}, …, a_{m} and at infinity, where the singularity at infinity is apparent.

(ii) The monodromy of the system is given by [χ(σ_{1}), …, χ(σ_{m})]_{∼} = [M_{1}, …, M_{m}]_{∼}.

(iii) For each k = 1, …, m, the solutions of the system defined on D̃_{a_{k}} can be written

as

Y (z̃) = Z_{k}(z)(z̃ − a_{k})^{E_{k}}C,  z̃ ∈ D̃_{a_{k}},

where the function Z_{k} : D_{a_{k}} ∪ {a_{k}} → Gℂ^{n×n} is analytic and C ∈ Gℂ^{n×n}.

(iv) The solutions of the system defined on D_{∞} can be written as

Y (z) = diag (z^{κ_{1}}, …, z^{κ_{n}})Z_{∞}(z)C, z ∈ D_{∞},

where the function Z_{∞} : D_{∞} ∪ {∞} → Gℂ^{n×n} is analytic and C ∈ Gℂ^{n×n}.

The necessity of the assumption (a) and (b) on the data is obvious. Assumption (c) is not necessary for this definition, but it will be important later on. Note that the solutions Y (z̃) are analytic and invertible functions on S̃^{∗} = S̃ \ ρ^{−1}(∞).

A simple observation is in order. The data [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}] of a given system of standard form can be replaced by data [CM_{1}C^{−1}, …, CM_{m}C^{−1}] and [C_{1}E_{1}C_{1}^{−1}, …, C_{m}E_{m}C_{m}^{−1}], where C, C_{1}, …, C_{m} ∈ Gℂ^{n×n}. Conversely, to a given system of standard form one can uniquely associate the monodromy data [M_{1}, …, M_{m}]_{∼} and the local data [(E_{1})_{∼}, …, (E_{m})_{∼}]. Indeed, in order to see that the matrices E_{k} are unique up to similarity, one can use (iii) and consider the residue of A(z) = Y′(z̃)Y^{−1}(z̃) at z = a_{k}.

Finally, also the indices of a system of standard form are unique. Here one can use (iv) and consider the residue of Y^{−1}(z)Y′(z) of any solution defined on D_{∞}.

Before discussing the existence and uniqueness of systems of standard form for given singularities and data, we will illustrate how these systems provide us with the solution of the original factorization problem for piecewise constant matrix functions.

In the following theorem we assume that the curve Γ divides the complex sphere ℂ̇ into two domains D_{+} and D_{−} such that 0 ∈ D_{+} and ∞ ∈ D_{−}. As usual, we identify functions which are analytic on the domains D_{+} or D_{−} with their boundary values, considered as functions on Γ.

Theorem 3.2 Let G ∈ P C(Γ)^{n×n} be a piecewise constant function with jumps only at a_{1}, …, a_{m}. Assume that G admits a Φ-factorization in the space L^{p}(Γ), 1 < p < ∞, and let [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}] be the data associated with G with respect to L^{p}(Γ).

Assume that there exists a system of standard form with these data, with singularities a_{1}, …, a_{m} and with indices κ_{1}, …, κ_{n}. Let Y_{1} and Y_{2} be solutions of this system defined on D_{+} and D_{−} \ {∞}, respectively. Then there exist C_{1}, C_{2} ∈ Gℂ^{n×n} such that the representation

G(t) = G_{+}(t)Λ(t)G_{−}(t),  t ∈ Γ,

is a Φ-factorization of G in the space L^{p}(Γ), where Λ(t) = diag (t^{κ_{1}}, …, t^{κ_{n}}) and

G_{+}(z) = C_{1}^{−1}Y_{1}^{−1}(z), z ∈ D_{+}, G_{−}(z) = Λ^{−1}(z)Y_{2}(z)C_{2}, z ∈ D_{−} \ {∞}.

Proof. Note first that Y_{1} and Y_{2} exist because D_{+} and D_{−} are simply connected and infinity is an apparent singularity of the system. Let G̃_{+}(z) = Y_{1}^{−1}(z), z ∈ D_{+}, and let G̃_{−}(z) = Λ^{−1}(z)Y_{2}(z), z ∈ D_{−} \ {∞}. From the condition (iv) of Definition 3.1, it follows that G̃_{−} is also analytic and invertible at ∞. Hence G_{+} and G_{−} are analytic and invertible on all of D_{+} and D_{−}, respectively. It follows from (iii) that in a neighborhood of the singularity a_{k}, the entries of the functions G̃_{+}^{−1} and G̃_{−} behave as (z − a_{k})^{λ}(ln(z − a_{k}))^{σ} and the entries of the functions G̃_{+} and G̃_{−}^{−1} behave as (z − a_{k})^{−λ}(ln(z − a_{k}))^{σ}, where λ is taken from the eigenvalues of E_{k} and σ ∈ {0, 1, …} refers to their multiplicity. By assumption we have −1/q = 1/p − 1 < Re λ < 1/p. This implies that G̃_{+} ∈ L_{+}^{p}(Γ)^{n×n}, G̃_{−}^{−1} ∈ L_{−}^{q}(Γ)^{n×n}, G̃_{+}^{−1} ∈ L_{+}^{p}(Γ)^{n×n} and G̃_{−} ∈ L_{−}^{q}(Γ)^{n×n}.

The functions Y_{1} and Y_{2} can be extended analytically onto the boundary Γ \ {a_{1}, …, a_{m}}. The values of the functions defined by G̃_{+}(t) = Y_{1}^{−1}(t) and G̃_{−}(t) = Λ^{−1}(t)Y_{2}(t) with t ∈ Γ \ {a_{1}, …, a_{m}} coincide with the boundary values of the above functions G̃_{+} and G̃_{−}, which are defined on D_{+} and D_{−}.

We introduce the function G̃(t) = G̃_{+}(t)Λ(t)G̃_{−}(t), t ∈ Γ, and observe that this representation is a factorization of G̃ in the space L^{p}(Γ). On the other hand, we have G̃(t) = Y_{1}^{−1}(t)Y_{2}(t), and because Y_{1} and Y_{2} are solutions of the system, it follows that G̃ is a piecewise constant function on Γ with jumps only at a_{1}, …, a_{m}.

Now let us express the monodromy of the system in terms of G̃. We start from the solution Y_{1}(z) on D_{+} and continue analytically by going once around the point a_{k} (see the construction of the deck transformations σ_{k}). When we first cross Γ, the proper analytic continuation is the function Y_{2}(z)G̃(a_{k} − 0)^{−1} on D_{−}. After the second crossing of Γ, we obtain the function Y_{1}(z)G̃(a_{k} + 0)G̃(a_{k} − 0)^{−1} on D_{+}. Hence, up to simultaneous similarity, we have χ(σ_{k}) = G̃(a_{k} − 0)G̃(a_{k} + 0)^{−1}. On the other hand, by condition (ii), the monodromy is given by [M_{1}, …, M_{m}]. Hence χ(σ_{k}) = C_{1}M_{k}C_{1}^{−1} = C_{1}G(a_{k} − 0)G(a_{k} + 0)^{−1}C_{1}^{−1} with some C_{1} ∈ Gℂ^{n×n}. This implies that G̃(a_{k} − 0)G̃(a_{k} + 0)^{−1} = C_{1}G(a_{k} − 0)G(a_{k} + 0)^{−1}C_{1}^{−1}, hence G̃(a_{k} + 0)^{−1}C_{1}G(a_{k} + 0) = G̃(a_{k} − 0)^{−1}C_{1}G(a_{k} − 0) for each k = 1, …, m. Because both G and G̃ are piecewise constant, there is a C_{2} ∈ Gℂ^{n×n} such that C_{2} = G̃(t)^{−1}C_{1}G(t). Hence G(t) = C_{1}G̃(t)C_{2}, which implies that G(t) = G_{+}(t)Λ(t)G_{−}(t) is a factorization in the space L^{p}(Γ). By Corollary 1.3, this is even a Φ-factorization. □

Note that the data associated with a piecewise constant matrix function is necessarily admissible data. Actually, assumption (c) is even weaker than the condition that the real parts of the eigenvalues of the matrices E_{k} lie in the open interval J_{p}.

We need the following factorization-type result about analytic matrix functions defined on a punctured disk D_{a}. It is the analytic version of the Birkhoff-Grothendieck theorem [2]. A quite simple proof has also been given by Leiterer [12] (see also [1]).

Lemma 3.3 For a ∈ ℂ, let Φ : D_{a} → Gℂ^{n×n} be an analytic function. Then there exist κ_{1}, …, κ_{n} ∈ ℤ such that Φ can be written in the form

Φ(z) = U(z)diag ((z − a)^{κ_{1}}, …, (z − a)^{κ_{n}})V (z),  z ∈ D_{a},

where U : ℂ̇ \ {a} → Gℂ^{n×n} and V : D_{a} ∪ {a} → Gℂ^{n×n} are analytic functions.

Using a well known modification idea [1], we prove the existence of systems of standard form with prescribed data and singularities, which have certain (a priori not known) indices.

Theorem 3.4 Let the n × n matrices [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}] be admissible data, and let a_{1}, …, a_{m} ∈ ℂ be distinct points. Then there exists a system of standard form with these data and singularities and with certain indices κ_{1}, …, κ_{n} ∈ ℤ.

Proof. From Theorem 2.1 it follows that there exists a system with singularities only at a_{1}, …, a_{m} whose monodromy is given by [M_{1}, …, M_{m}]_{∼}. Let Y_{0} be a solution of this system, i.e., Y_{0} : S̃^{∗} → Gℂ^{n×n} is analytic and has the given monodromy representation.

We are going to modify Y_{0} step by step in order to obtain an analytic matrix function Y : S̃^{∗} → Gℂ^{n×n} which also satisfies (iii) and (iv) of Definition 3.1. Let us consider Y_{0} near the point a_{1}. Because the monodromy is given as above, the solution changes by a matrix similar to M_{1} after going once around a_{1} in positive direction. Hence, restricted to D̃_{a_{1}}, the function Y_{0} can be written in the form Y_{0}(z̃) = Φ_{1}(z)(z̃ − a_{1})^{E_{1}}C, where Φ_{1} : D_{a_{1}} → Gℂ^{n×n} is a certain analytic function. We apply Lemma 3.3 with Φ = Φ_{1} and a = a_{1}. We obtain that Φ_{1}(z) = Ũ_{1}(z)V_{1}(z), z ∈ D_{a_{1}}, where Ũ_{1} is analytic and invertible on ℂ \ {a_{1}} and V_{1} is analytic and invertible on D_{a_{1}} ∪ {a_{1}}. Hence Ũ_{1}^{−1}(z)Y_{0}(z̃) = V_{1}(z)(z̃ − a_{1})^{E_{1}}C for z̃ ∈ D̃_{a_{1}}. We define the function Y_{1}(z̃) := Ũ_{1}^{−1}(z)Y_{0}(z̃) for z̃ ∈ S̃^{∗}. This function satisfies the conditions (i), (ii), and also the condition (iii) at the point a_{1}.

The next steps consist in doing the same for a_{2}, …, a_{m}. The essential point is that, at the k-step, we define the function Y_{k}(z̃) = Ũ_{k}^{−1}(z)Y_{k−1}(z̃), where Ũ_{k}(z) is analytic and invertible on ℂ\{a_{k}}. This modification does not produce additional singularities (except at infinity), it does not change the monodromy, and it changes the local behavior only at the a_{k} (in the desired way), but not at the other singularities. So we obtain a function Y_{m}(z̃) which satisfies the conditions (i)–(iii).

Finally, we consider the point infinity. Restricting Y_{m}(z̃) to D_{∞} we can write Y_{m}(z) = Φ(1/z)C, where Φ(1/z) is analytic and invertible on D_{0}. We apply Lemma 3.3 with a = 0 and factor Φ(z) = V (z)Λ(z)U(z), z ∈ D_{0}, where V (z) is analytic and invertible on ℂ̇ \ {0}, Λ(z) is the diagonal matrix, and U(z) is analytic and invertible on D_{0} ∪ {0}. It follows that V^{−1}(1/z)Y_{m}(z) = Λ(1/z)U(1/z)C. We define Y (z̃) = V^{−1}(1/z)Y_{m}(z̃), z̃ ∈ S̃^{∗}. Because V (1/z) is analytic and invertible on ℂ, the function Y (z̃) satisfies also the required properties (i)–(ii). Since Y (z) = Λ(1/z)U(1/z)C on D_{∞}, this function also satisfies condition (iv) after an appropriate permutation of the entries of Λ(1/z).

Finally, we define the corresponding matrix function A(z) by A(z) = Y′(z̃)Y^{−1}(z̃). □

Next we consider the question about the uniqueness of systems of standard form with prescribed data and singularities. First of all, the indices are uniquely determined. The systems themselves are not unique, but they are related to each other in a simple way. Because of the relation to factorization (described in Theorem 3.2) it should not be surprising that the description is analogous to that given in Theorem 1.2.

Theorem 3.5 Let Y′_{1}(z̃) = A_{1}(z)Y_{1}(z̃) and Y′_{2}(z̃) = A_{2}(z)Y_{2}(z̃) be two systems of standard form with the same admissible data and singularities a_{1}, …, a_{m} ∈ ℂ. Then the indices corresponding to both systems coincide. Moreover, if the indices are given by

\overline{κ}_{1}, …, \overline{κ}_{1}, \overline{κ}_{2}, … \overline{κ}_{2}, \overline{κ}_{3}, …, \overline{κ}_{k−1}, \overline{κ}_{k}, …, \overline{κ}_{k},

where \overline{κ}_{j} occurs exactly l_{j} times, \overline{κ}_{1} > \overline{κ}_{2} > … > \overline{κ}_{k}, l_{1}, …, l_{k} ∈ \{1, 2, …\}, l_{1} + … + l_{k} = n, then both systems are related to each other by

A_{2}(z) = V′(z)V^{−1}(z) + V (z)A_{1}(z)V^{−1}(z),  z ∈ S \ {∞},  (3.1)

or, equivalently, by

Y_{2}(z̃) = V (z)Y_{1}(z̃)C,  z̃ ∈ S̃^{∗},  (3.2)

where C ∈ Gℂ^{n×n} and V : ℂ → Gℂ^{n×n} is a rational matrix function of the form

V (z) = ( \begin{matrix}V_{11} & V_{12}(z) & … & V_{1k}(z) \\ 0 & V_{22} & & V_{2k}(z) \\ ⋮ & & ... & ⋮ \\ 0 & 0 & … & V_{kk} \end{matrix} )  (3.3)

with V_{jj} ∈ Gℂ^{l_{j}×l_{j}}, V_{ij}(z) = ∑_{r=0}^{\overline{κ}_{i}−\overline{κ}_{j}} V_{ij}^{(r)}z^{r}, where V_{ij}^{(r)} ∈ ℂ^{l_{i}×l_{j}} for i < j.

Proof. Because the systems have the same singularities and the same monodromy, there exist an analytic function V : S \ {∞} → Gℂ^{n×n} and a C ∈ Gℂ^{n×n} such that Y_{2}(z̃) = V (z)Y_{1}(z̃)C. We are first going to show that V (z) is analytic and invertible on all of ℂ.

Let us restrict the solutions Y_{1} and Y_{2} onto a neighborhood D̃_{a_{k}} of the point a_{k}. From (iii) of Definition 3.1 it follows that there are C_{1}, C_{2} ∈ Gℂ^{n×n} such that

Y_{1}(z̃) = Z_{k}^{(1)}(z)(z̃ − a)^{E_{k}}C_{1},  Y_{2}(z̃) = Z_{k}^{(2)}(z)(z̃ − a)^{E_{k}}C_{2},  z̃ ∈ D̃_{a_{k}},

where Z_{k}^{(1)} and Z_{k}^{(2)} are analytic and invertible at D_{a_{k}} ∪ {a_{k}}. Hence

Z_{k}^{(2)}(z)(z̃ − a)^{E_{k}} C̃ = V (z)Z_{k}^{(1)}(z)(z̃ − a)^{E_{k}},  (3.4)

where C̃ = C_{2}C^{−1}C_{1}^{−1}. By going once around the point a_{k} in positive direction it follows that

Z_{k}^{(2)}(z)(z̃ − a)^{E_{k}}N_{k}C̃ = V (z)Z_{k}^{(1)}(z)(z̃ − a)^{E_{k}}N_{k},

where N_{k} := exp(2πiE_{k}). Combining both equations it follows that N_{k} = C̃^{−1}N_{k}C̃, i.e., N_{k} and C̃ commute with each other. Now we use the fact that the matrix E_{k} is non-resonant. Because N_{k} = exp(2πiE_{k}), this implies that E_{k} can be written as a polynomial in the matrix M_{k}. Hence also E_{k} and C̃ commute. Using this in connection with (3.4) we obtain

Z_{k}^{(2)}(z)C̃(z̃ − a)^{E_{k}} = V (z)Z_{k}^{(1)}(z)(z̃ − a)^{E_{k}}.

Hence V (z) = Z_{k}^{(2)}(z)C̃(Z_{k}^{(1)}(z))^{−1}, which implies that V (z) is analytic and invertible at a_{k}.

So far we have shown that V (z) and V^{−1}(z) are entire analytic functions. We analyze their behavior at infinity by restricting the solutions Y_{1} and Y_{2} onto D_{∞}. From (iv) of Definition 3.1, it follows that

Y_{1}(z) = Λ_{1}(z)Z_{∞}^{(1)}C_{1},  Y_{2}(z) = Λ_{2}(z)Z_{∞}^{(2)}C_{2},  z ∈ D_{∞},

where C_{1}, C_{2} ∈ Gℂ^{n×n} and Z_{∞}^{(1)} and Z_{∞}^{(2)} are analytic and invertible on a neighborhood of infinity. Here we assume that Λ_{1} and Λ_{2} are of the form

Λ_{s}(z) = diag (z^{\overline{κ}^{1}}I_{l_{1}^{(s)}}, z^{\overline{κ}^{2}}I_{l_{2}^{(s)}}, …, z^{\overline{κ}^{k}}I_{l_{k}^{(s)}}), s = 1, 2,

where \overline{κ}_{1}, …, \overline{κ}_{k} ∈ ℤ, \overline{κ}_{1} > \overline{κ}_{2} > … > \overline{κ}_{k}, l_{1}^{(s)}, …, l_{k}^{(s)} ∈ \{0, 1, …\}, l_{1}^{(s)} + … + l_{k}^{(s)} = n. Here I_{l} stands for the identity matrix of size l × l. Now the above equations give

Λ_{2}(z)Z_{∞}^{(2)}C_{2} = V (z)Λ_{1}(z)Z_{∞}^{(1)}C_{1}C, z ∈ D_{∞}.

This means that

V (z) = Λ_{2}(z)Z_{∞}(z)Λ_{1}^{−1}(z), V^{−1}(z) = Λ_{1}(z)Z_{∞}^{−1}(z)Λ_{2}^{−1}(z),

where Z_{∞} := Z_{∞}^{(2)}C_{2}C^{−1}C_{1}^{−1}(Z_{∞}^{(1)})^{−1} is analytic and invertible in a neighborhood of infinity. Now we introduce block partitions of the matrices V (z) and V^{−1}(z):

V (z) = (V_{ij}(z))_{i,j=1}^{k}, V^{−1}(z) = (U_{ij}(z))_{i,j=1}^{k},

where V_{ij}(z) is of size l_{i}^{(2)} × l_{j}^{(1)} and U_{ij}(z) is of size l_{i}^{(1)} × l_{j}^{(2)}. Using the structure of Λ_{1}(z) and Λ_{2}(z), it follows that ∥V_{ij}(z)∥ = O(z^{\overline{κ}_{i}−\overline{κ}_{j}}) and ∥U_{ij}(z)∥ = O(z^{\overline{κ}_{i}−\overline{κ}_{j}}) as z → ∞. Because V_{ij}(z) and U_{ij}(z) are entire analytic functions, we obtain that V_{ij} = 0 and U_{ij} = 0 for i > j, and that V_{ij}(z) and U_{ij}(z) are matrix polynomials of degree at most \overline{κ}_{i} − \overline{κ}_{j} if i ≤ j. In particular, the diagonal blocks are constant matrices. The block triangular structure of V (z) and V^{−1}(z) implies that V_{ii}U_{ii} = I_{l_{i}^{(2)}}

and U_{ii}V_{ii} = I_{l_{i}^{(1)}}. Hence V_{ii} and U_{ii} are invertible square matrices, i.e., l_{i}^{(1)} = l_{i}^{(2)}. Hence the indices corresponding to both systems are the same. Thereby, we have also proved that V (z) is of the desired form. □

It is easy to see that also the converse of the above theorem holds. This suggests the following definition of an equivalence relation for systems of standard form. Two systems Y′_{1}(z̃) = A_{1}(z)Y_{1}(z̃) and Y′_{2}(z̃) = A_{2}(z)Y_{2}(z̃) of standard form will be called equivalent if their indices are the same and if there exists a matrix V (z) of the form (3.3) and the properties stated there such that (3.1) holds (or, equivalently, (3.2) holds).

The next corollary resumes the results about existence a uniqueness in a very brief way. In fact, the uniqueness results will be employed implicitly in Section 6.

Corollary 3.6 Let a_{1}, …, a_{m} ∈ ℂ be distinct points. Then Definition 3.1 establishes a one-to-one correspondence between the equivalence classes of admissible data,

[M_{1}, …, M_{m}]_{∼} and [(E_{1})_{∼}, …, (E_{m})_{∼}],

and the equivalence classes of systems of standard form with above singularities.

The previous considerations show that to each class of admissible data and given singularities one can associate a class of systems of standard form, and hence certain (uniquely determined) indices. For the problem of determining the partial indices for a piecewise constant matrix function, one has now to solve the following question: what are the indices associated to given admissible data and singularities? As we will see, this question can be answered explicitly only in some cases. In general, it leads to monodromy problems for which no explicit solution is known.

In order to indicate the importance of the non-resonance assumption for the matrices E_{1}, …, E_{m}, we show that Theorem 3.5 breaks down if this assumption is dropped. Consider the following matrix functions, which are the solutions of certain 2 × 2 systems with three singularities at a_{1}, a_{2}, a_{3},

Y_{1}(z) = ( \begin{matrix}(z − a_{1})(z − a_{2})(z − a_{3}) & 0 \\ 0 & 1 \end{matrix} ) ,  (3.5)
Y_{2}(z) = ( \begin{matrix}(z − a_{1})(z − a_{2}) & 0 \\ 0 & (z − a_{3}) \end{matrix} ) ,  (3.6)
Y_{3}(z) = ( \begin{matrix}(z − a_{1})(z − a_{2}) & 0 \\ 1 & (z − a_{3}) \end{matrix} ) .  (3.7)

The corresponding systems have trivial monodromy (M_{1} = M_{2} = M_{3} = I), the same local behavior (with E_{1} = E_{2} = E_{3} = diag (1, 0)), but their indices are (3, 0) for Y_{1} and (2, 1) for Y_{2} and Y_{3}. Moreover, one can show (which requires a little effort) that Y_{2} and Y_{3} are not equivalent in the above sense, although they have the same indices. Obviously, Corollary 3.6 also breaks down, and the question “what are the indices for given data?” does not make sense. One might think of replacing this question by asking for all possible indices for given data, but we will not pursue this direction in this paper.

Finally, let us mention the connection to vector bundles (we refer to [1, Sect. 5.1] for details). For given data and singularities one constructs in a certain way a vector bundle over ℂ̇. Due to the fact that the matrices E_{k} are non-resonant, this construction is essentially unique. Any such vector bundle is characterized by certain integers κ_{1}, …, κ_{n}, which are called the splitting type of the vector bundle. Using the construction of the vector bundle and the splitting type, one can easily construct the corresponding system of standard form, which has indices κ_{1}, …, κ_{n}. So the above question takes the following form: what is the splitting type of the vector bundles constructed for given data and singularities?

4 Equivalent characterizations of systems of standard form

The definition of systems of standard form has been given entirely in terms of the solutions Y (z̃) of the system. In this section, we show by using standard arguments that the conditions (i), (iii) and (iv) of Definition 3.1 (i.e., except the monodromy condition) can be replaced by equivalent characterizations given entirely in terms of A(z).

We prepare with the following well known lemma. Note that the assumption that E is non-resonant is essential for the validity of the lemma.

Lemma 4.1 Let A(z) be an analytic matrix function on D_{a}, a ∈ ℂ, and assume that A(z) has only a simple pole at z = a whose residue is a non-resonant matrix E,

A(z) = \frac{E}{z − a} + “analytic term at z = a”,  z ∈ D_{a}.

Then the solutions Y : D̃_{a} → Gℂ^{n×n} of Y′(z̃) = A(z)Y (z̃) can be written as

Y (z̃) = Z(z)(z̃ − a)^{E}C,  z̃ ∈ D̃_{a},

where the function Z : D_{a} ∪ {a} → Gℂ^{n×n} is analytic and C ∈ Gℂ^{n×n}.

Proposition 4.2 The system Y′(z̃) = A(z)Y (z̃) is of standard form with (admissible) data [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}], with singularities a_{1}, …, a_{m} ∈ ℂ, and with indices κ_{1}, …, κ_{n} ∈ ℤ if and only if the following conditions are satisfied:

(i) The matrix function A(z) is analytic on ℂ \ {a_{1}, …, a_{m}}.

(ii) The monodromy of the system is given by [χ(σ_{1}), …, χ(σ_{m})]_{∼} = [M_{1}, …, M_{m}]_{∼}.

(iii) For each k = 1, …, m, the function A(z) has at most a simple pole at z = a_{k} whose residue is a matrix similar to E_{k}:

A(z) = \frac{S_{k}E_{k}S_{k}^{−1}}{z − a_{k}} + “analytic term at z = a_{k}”,  z ∈ D_{a_{k}}.

(iv) At infinity, the function A(z) is of the form

A(z) = \frac{diag (κ_{1}, …, κ_{n})}{z} + Λ(z)R(z)Λ^{−1}(z),  z ∈ D_{∞},

where Λ(z) = diag (z^{κ_{1}}, …, z^{κ_{n}}) and R : D_{∞} → ℂ^{n×n} is an analytic function satisfying ∥R(z)∥ = O(1/z^{2}) as z → ∞.

Proof. Assume (i)–(iv) of Definition 3.1 holds. Then also conditions (i) and (ii) of this proposition are fulfilled. As to (iii), remark that Y (z̃) = Z_{k}(z)(z̃ − a_{k})^{E_{k}}C implies

A(z) = Y′(z̃)Y^{−1}(z̃) = Z′_{k}(z)Z_{k}^{−1}(z) + Z_{k}(z)\frac{E_{k}}{z − a_{k}} Z_{k}^{−1}(z)
= \frac{Z_{k}(a_{k})E_{k}Z_{k}^{−1}(a_{k})}{z − a_{k}} + “analytic term at z = a_{k}”

because Z_{k}(z) is analytic and invertible at z = a_{k}. At infinity we have Y (z) = Λ(z)Z_{∞}(z)C. Hence, similarly,

A(z) = Y′(z̃)Y^{−1}(z̃) = Λ′(z)Λ^{−1}(z) + Λ(z)Z′_{∞}(z)Z_{∞}^{−1}(z)Λ^{−1}(z)
= \frac{diag (κ_{1}, …, κ_{n})}{z} + Λ(z)R(z)Λ^{−1}(z),

where R(z) = Z′_{∞}(z)Z_{∞}^{−1}(z). Since Z_{∞}(z) is analytic and invertible, ∥R(z)∥ = O(1/z^{2}) as z → ∞. This implies (iv).

In order to prove the converse, we remark first that (i) and (ii) of this proposition imply (i) and (ii) of Definition 3.1 with the exception of the apparentness of the singularity at infinity, which will follow from (iv).

Condition (iii) of this proposition in connection with Lemma 4.1 immediately implies (iii) of Definition 3.1. Note that the mere similarity of the residue to E_{k} does not cause problems because (z̃ − a_{k})^{S_{k}E_{k}S_{k}^{−1}} = S_{k}(z̃ − a_{k})^{E_{k}}S_{k}^{−1}.

Finally, let us consider the singularity at infinity. Because of the asymptotics of R(z) at infinity, there exists an analytic solution Z_{∞} : D_{∞} ∪ {∞} → Gℂ^{n×n} of the system Z′_{∞}(z) = R(z)Z_{∞}(z) near infinity. Now define Y_{∞}(z) = Λ(z)Z_{∞}(z), z ∈ D_{∞}. Then,

Y′_{∞}(z)Y_{∞}^{−1}(z) = Λ′(z)Λ^{−1}(z) + Λ(z)Z′_{∞}(z)Z_{∞}^{−1}(z)Λ^{−1}(z)
= \frac{diag (κ_{1}, …, κ_{n})}{z} + Λ(z)R(z)Λ^{−1}(z) = A(z).

Hence Y_{∞}(z) is a solution of our original system considered on the domain D_{∞}. However, the relation of such a “restricted” solution to any solution of the system is given by Y (z) = Y_{∞}(z)C with some C. This is exactly condition (iv) of Definition 3.1. Obviously, this also shows that infinity is an apparent singularity. □

Now we present the desired description in terms of the matrix function A(z). For k ∈ ℤ, k ≥ 0, let P_{k} stand for the set of all polynomials of degree less than or equal to k. For k ∈ ℤ, k < 0, let P_{k} stand for the set containing only the function identically equal to zero.

Theorem 4.3 The system Y′(z̃) = A(z)Y (z̃) is of standard form with (admissible) data [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}], with singularities a_{1}, …, a_{m} ∈ ℂ and indices κ_{1}, …, κ_{n} ∈ ℤ if and only if the following conditions are satisfied:

(i) A(z) = [A_{ij}(z)]_{i,j=1}^{n}, where

A_{ij}(z) = \frac{κ_{i}δ_{ij}z^{m−1} + p_{ij}(z)}{(z − a_{1})(z − a_{2}) ⋯ (z − a_{m})},  (4.1)

where p_{ij} ∈ P_{m−2+κ_{i}−κ_{j}} and δ_{ij} is the Kronecker symbol.

(ii) For each k = 1, …, m, the residue of A(z) at z = a_{k} is similar to the matrix E_{k}.

(iii) The monodromy of the system is given by [χ(σ_{1}), …, χ(σ_{m})]_{∼} = [M_{1}, …, M_{m}]_{∼}.

Proof. In regard to the previous proposition, we have to show that the characterization given by (4.1) is equivalent to the property that A(z) is analytic on ℂ \ {a_{1}, …, a_{m}}, has simple poles at a_{1}, …, a_{m} and behaves at infinity as described in (iv) of Proposition 4.2.

The fact that A(z) is analytic on ℂ \ {a_{1}, …, a_{m}} and has simple poles at a_{1}, …, a_{m} is equivalent to writing the entries of A(z) as A_{ij}(z) = q_{ij}(z)(z − a_{1})^{−1}(z − a_{2})^{−1} ⋯ (z − a_{m})^{−1}, where q_{ij} is an entire analytic function. The characterization of q_{ij} as a certain polynomial is now equivalent to (iv) of Proposition 4.2. □

Considering the residues of A(z) at a_{1}, …, a_{m} and at infinity and taking traces, we obtain the following simple relation,

κ_{1} + … + κ_{n} = trace E_{1} + … + trace E_{m}.  (4.2)

Hence the sum of the indices is determined by the the eigenvalues of E_{k}. This relation is the counterpart to formula (1.14) for the total index of the factorization of a piecewise constant matrix function.

5 Some general results in the irreducible case

In this section we present some necessary conditions for the indices of systems of standard form with irreducible monodromy data [M_{1}, …, M_{m}]. By establishing a relationship to scalar linear Fuchsian differential equations of n-th order, we show that, in general, these conditions cannot be improved.

A subspace X ⊆ ℂ^{n} is called an invariant subspace of a collection of n×n matrices {M_{ω}}_{ω∈Ω} if the image M_{ω}X is contained in X for each ω ∈ Ω. We say that a matrix function F (z) has the invariant subspace X if F (z)X ⊆ X for each z.

A collection of matrices {M_{ω}}_{ω∈Ω} is called reducible if there exists a non-trivial invariant subspace X (i.e., X ≠ {0} and X ≠ ℂ^{n}). Otherwise, it is called irreducible. The monodromy of a system of linear differential equations is called reducible (resp., irreducible) if the collection {χ(σ)}_{σ∈Δ} (or, equivalently, the collection [χ(σ_{1}), …, χ(σ_{m})]) is reducible (resp., irreducible) for some (hence each) monodromy representation χ of the system.

The following auxiliary result relates invariant subspaces of a matrix function A(z) with invariant subspaces for the solution and the monodromy of the corresponding system.

Lemma 5.1 Let A(z) be an analytic matrix function which has invariant subspaces {X_{r}}_{r∈R}. Then there exists a solution of the system Y′(z̃) = A(z)Y (z̃) such that this solution and its corresponding monodromy representation also have the invariant subspaces {X_{r}}_{r∈R}.

Proof. Let P_{r} ∈ ℂ^{n×n} be projections such that the image of P_{r} is the subspace X_{r}. Then (I − P_{r})A(z)P_{r} = 0, and we obtain

(I − P_{r})Y′(z̃)P_{r} = (I − P_{r})A(z)(I − P_{r})Y (z̃)P_{r}.

Choose any point z̃_{0} ∈ S̃ and consider the solution Y (z̃) for which Y (z̃_{0}) = I. Hence (I − P_{r})Y (z̃_{0})P_{r} = 0. Because the matrix function (I − P_{r})Y (z̃)P_{r} is a solution of the above first order system, it follows that (I −P_{r})Y (z̃)P_{r} = 0 for all z̃ ∈ S̃. This means that X_{r} is an invariant subspace of Y (z̃). Hence X_{r} is also an invariant subspace of the inverse Y^{−1}(z̃). Let χ be the monodromy representation of Y (z̃) and σ ∈ Δ. Then χ(σ) = Y^{−1}(σ(z̃))Y (z̃). Hence X_{r} is an invariant subspace of χ(σ). □

If we choose another solution of the system, Ŷ(z̃) = Y (z̃)C, then the corresponding monodromy representation has the invariant subspaces {C^{−1}X_{r}}_{r∈R} (see formula (2.4)).

Now we state the conditions on the indices in the case of irreducible monodromy. These conditions (formulated in the setting of vectors bundles and with a different proof) have already been obtained by Bolibruch [4]. Recall that the indices are ordered decreasingly, κ_{1} ≥ κ_{2} ≥ … ≥ κ_{n}.

Theorem 5.2 Let Y′(z̃) = A(z)Y (z̃) be a system of standard form with m singularities a_{1}, …, a_{m} ∈ ℂ. If the monodromy of this system is irreducible, then the indices κ_{1}, …, κ_{n} ∈ ℤ satisfy, for each k = 1, …, n − 1, the condition

κ_{k} − κ_{k+1} ≤ m − 2.  (5.1)

Proof. Suppose that this condition is not fulfilled for some k, i.e., κ_{k} −κ_{k+1} > m−2. Because the indices are ordered decreasingly, it follows that κ_{j} − κ_{i} > m − 2 for each j = 1, …, k and each i = k + 1, …, n. Theorem 4.3 implies that for those i and j, the entries A_{ij}(z) of the matrix A(z) vanish identically. This means that the matrix A(z) is of block triangular form, i.e., it possesses the non-trivial invariant subspace X = ℂ^{k} ⊕ {0}^{n−k}. By the previous lemma, the monodromy representation of some solution has also this invariant subspace. Hence the monodromy of the system is reducible, which contradicts the assumption. □

We want to emphasize that the above condition holds regardless of the location of the singularities and the values of [E_{1}, …, E_{m}]. Reinterpreted in terms of the factorization problem, this implies the independence of this condition from the underlying space L^{p}(Γ).

It is also easy to see that this condition (even combined with the knowledge of the total index κ) is in general not sufficient to determine the values of the partial indices uniquely. In this connection, we are going to show that (5.1) cannot be improved.

For this purpose we introduce systems of standard form for which the above relation holds with equality, i.e., κ_{k}−κ_{k+1} = m−2 for each k = 1, …, n−1. We single out a subclass of such systems which have – generically – irreducible monodromy.

Suppose we are given indices κ_{1}, …, κ_{n} ∈ ℤ satisfying κ_{k} − κ_{k+1} = m − 2. It follows from Theorem 4.3 that systems of standard form Y′(z̃) = A(z)Y (z̃) with singularities a_{1}, …, a_{m} and above indices have a matrix function A(z) which is of the form

\frac{1}{p} ( \begin{matrix}κ_{1}z^{m−1} + p_{11} & p_{12} & p_{13} & … & p_{1n} \\ α_{1} & κ_{2}z^{m−1} + p_{22} & p_{23} & & p_{2n} \\ 0 & α_{2} & κ_{3}z^{m−1} + p_{33} & ... & ⋮ \\ ⋮ & & ... & ... & p_{n−1,n} \\ 0 & 0 & … & α_{n−1} & κ_{n}z^{m−1} + p_{nn} \end{matrix} ) , (5.2)

where p(z) = (z − a_{1}) ⋯ (z − a_{m}), α_{k} ∈ ℂ and p_{ij} ∈ P_{(m−2)(1+j−i)}. The condition on the polynomials p_{ij} follows from κ_{i} − κ_{j} = (m − 2)(j − i). Conversely, any system with matrix A(z) given by (5.2) such that the residues of A(z) at a_{1}, …, a_{m} are nonresonant matrices is a system of standard form with above indices and singularities and with certain admissible data.

If α_{k} = 0 for some k = 1, …, n − 1, then the monodromy of the above system is reducible because A(z) is of block triangular form. Hence, as we are interested in irreducible monodromy, we will focus on systems with α_{1} ⋯ α_{n−1} ≠ 0. A preliminary characterization of the data of such systems is given next.

For S ∈ ℂ^{n×n}, let min(S) stand for the degree of the minimal polynomial of S, i.e., the polynomial p of smallest degree for which p(S) = 0.

Proposition 5.3 Assume that Y′(z̃) = A(z)Y (z̃) is a system of standard form with admissible data [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}] where A(z) is given by (5.2) with α_{1} ⋯ α_{n−1} ≠ 0. Then min(E_{k}) = min(M_{k}) = n for each k = 1, …, m.

Proof. Because α_{1} ⋯ α_{n−1} ≠ 0 the residue Ê_{k} of A(z) at z = a_{k} has nonzero entries on the diagonal below the main diagonal. All entries below this diagonal are zero. By considering the powers of E_{k}, it follows that min(Ê_{K}) = n. By Proposition 4.2(iii) we have Ê_{k} ∼ E_{k}, which implies min(E_{k}) = n. Because M_{k} ∼ exp(−2πiE_{k}) and E_{k} is non-resonant, we obtain min(M_{k}) = n by considering the Jordan normal forms. □

The next result relates the systems singled out above to scalar linear Fuchsian differential equations. Therein, the local exponents of the linear Fuchsian differential equations at the points a_{1}, …, a_{m} correspond to the eigenvalues of the matrices E_{1}, …, E_{m} of the given data.

This result has a strong relationship to a result obtained by Bolibruch [4]. He examined the minimal number of additional apparent singularities which a scalar Fuchsian differential equation must have in order to realize given irreducible monodromy. The answer he obtained involves the notion of the maximal Fuchsian weight, which is defined in terms of the splitting type for a class of vector bundles.

Theorem 5.4 Let [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}] be admissible data of n × n matrices, and let a_{1}, …, a_{m} ∈ ℂ be distinct points. For k = 1, …, m, denote by ε_{k}^{(1)}, …, ε_{k}^{(n)} the eigenvalues of the matrices E_{k}. Then the following two statements are equivalent:

(i) [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}] is the data associated to some system with A(z) given by (5.2) and α_{1} ⋯ α_{n−1} ≠ 0.

(ii) [M_{1}, …, M_{m}] is the monodromy of some n-th order linear differential equation with Fuchsian singularities at a_{1}, …, a_{m} and local exponents given by {ε_{k}^{(1)}, …, ε_{k}^{(n)}} for z = a_{k}, k = 1, …, m − 1, and {ε_{m}^{(1)} − κ_{n}, …, ε_{m}^{(n)} − κ_{n}} for z = a_{m}.

Proof. (ii)⇒(i): Suppose we are given an n-th order linear differential equation (2.13) with the above properties. Let y_{1}(z̃), …, y_{n}(z̃) be n linear independent solutions. We introduce

W (z̃) = ( \begin{matrix}y_{1}^{(n−1)}(z̃) & y_{2}^{(n−1)}(z̃) & … & y_{n}^{(n−1)}(z̃) \\ ⋮ & ⋮ & & ⋮ \\ y′_{1}(z̃) & y′_{2}(z̃) & … & y′_{n}(z̃) \\ y_{1}(z̃) & y_{2}(z̃) & … & y_{n}(z̃) \end{matrix} ) ,  z̃ ∈ S̃^{∗},  (5.3)

Then W (z̃) is analytic on S̃^{∗}, and it is well known that the Wronskian det W (z̃) does not vanish on all of S̃^{∗}. In particular, W (z̃) is a solution of the n × n system W′(z̃) = A_{0}(z)W (z̃), where

A_{0}(z) = ( \begin{matrix}−q_{1}(z) & −q_{2}(z) & … & −q_{n}(z) \\ 1 & 0 & … & 0 \\ ⋮ & ... & ... & ⋮ \\ 0 & … & 1 & 0 \end{matrix} ) ,  z ∈ S \ \{∞\},  (5.4)

and q_{1}(z), …, q_{n}(z) are the coefficients of the scalar differential equation. Obviously, the system for A_{0}(z) has the same singularities a_{1}, …, a_{m} (and possibly an apparent singularity at infinity) and the same monodromy as the scalar differential equation.

Below we will replace the system W′(z̃) = A_{0}(z)W (z̃) by a modified system Y′(z̃) = A(z)Y (z̃) which will have the desired properties. However, in order to analyze the behavior at the singular points a_{k}, we will first consider differently modified systems Y_{k}(z̃) = A_{k}(z)Y_{k}(z̃) near these singularities.

For k = 1, …, m, we introduce

P_{k}(z) = diag ((z − a_{k})^{n−1}, (z − a_{k})^{n−2}, …, (z − a_{k}), 1),  (5.5)

and write q_{j}(z) = r_{jk}(z)/(z − a_{k})^{j} for each j = 1, …, n. Because the singularities of the scalar equation are assumed to be Fuchsian, the functions r_{jk}(z) are analytic at z = a_{k}.

For k = 1, …, m − 1, we define Y_{k}(z̃) = P_{k}(z)W (z̃). Then Y_{k}(z̃) is a solution of the system Y′_{k}(z̃) = A_{k}(z)Y_{k}(z̃) with A_{k}(z) = P_{k}(z)A_{0}(z)P_{k}^{−1}(z) + P′_{k}(z)P_{k}^{−1}(z). From this we obtain that A_{k}(z) = Ê_{k}/(z − a_{k})+“analytic term at z = a_{k}” with

Ê_{k} = ( \begin{matrix}−r_{1k}(a_{k}) & −r_{2k}(a_{k}) & … & −r_{nk}(a_{k}) \\ 1 & 0 & … & 0 \\ ⋮ & ... & ... & ⋮ \\ 0 & … & 1 & 0 \end{matrix} ) + diag (n − 1, …, 1, 0). (5.6)

A straightforward calculation shows that the characteristic equation det(Ê_{k} −ρI) = 0 for the matrix Ê_{k} coincides with the indical equation (2.15) for the local exponents of the scalar equation with respect to the singularity a_{k}. Hence the eigenvalues of Ê_{k} are just ε_{k}^{(1)}, …, ε_{k}^{(n)}.

For k = m we make a modification. We define Y_{m}(z̃) = (z − a_{m})^{κ_{n}}P_{m}(z)W (z̃), which is a solution of Y′_{m}(z̃) = A_{m}(z)Y_{m}(z̃). Again A_{m}(z) = Ê_{m}/(z − a_{m})+“analytic term at z = a_{m}”, but now Ê_{m} contains an additional term κ_{n}I. Consequently, det(Ê_{m} − κ_{n}I − ρI) = 0 coincides with the indical equation for the singularity a_{m}. Because the assumption on the local exponents is also different, we obtain nevertheless that the eigenvalues of Ê_{m} are ε_{m}^{(1)}, …, ε_{m}^{(n)}.

In fact, the converse is also true. The scalar differential equation is Fuchsian at a_{k} if and only if the system Y′_{k}(z̃) = A_{k}(z)Y_{k}(z̃) obtained in the above way is Fuchsian at a_{k}.

Next we are going to analyze the singularity at infinity. We make a change of variables z ↦ ξ = (z −a_{k})^{−1} and introduce ŷ_{1}(ξ), …, ŷ_{n}(ξ) by ŷ_{j}((z −a_{m})^{−1}) = y_{j}(z) for j = 1, …, n. These new functions are solutions of a modified scalar linear differential equation. The point ξ = 0 corresponds to the point z = ∞. By assumption they are not singular points. Hence

Ŵ(ξ) = ( \begin{matrix}ŷ_{1}^{(n−1)}(ξ) & ŷ_{2}^{(n−1)}(ξ) & … & ŷ_{n}^{(n−1)}(ξ) \\ ⋮ & ⋮ & & ⋮ \\ ŷ′_{1}(ξ) & ŷ′_{2}(ξ) & … & ŷ′_{n}(ξ) \\ ŷ_{1}(ξ) & ŷ_{2}(ξ) & … & ŷ_{n}(ξ) \end{matrix} ) ,  ξ ∈ D_{0},  (5.7)

is analytic in a neighborhood of ξ = 0 and det Ŵ(0) ≠ 0. We remark that the identity

W (z) = P_{m}^{−1}(z)SP_{m}^{−1}(z)Ŵ((z − a_{m})^{−1}),  z ∈ D_{∞},  (5.8)

holds, which can be verified by a direct calculation. Here S is a certain invertible upper triangular n × n matrix. This matrix does not depend on any parameters and can (in principle) be evaluated explicitly.

Now we are prepared to define the system which will satisfy the conditions of (i). Let

U(z) = (z − a_{m})^{κ_{n}}P_{1}(z) ⋯ P_{m−1}(z)S^{−1}P_{m}(z),  (5.9)

and define Y (z̃) = U(z)W (z̃). Because the function U(z) is analytic and invertible on S \ {∞}, it follows from the afore-mentioned properties of the system W′(z̃) = A_{0}(z)W (z̃) that the new system Y′(z̃) = A(z)Y (z̃) has singularities only at a_{1}, …, a_{m} (and possibly an apparent singularity at infinity) and has the same monodromy as the scalar differential equation.

Next we analyze the local behavior of the new system at infinity. Combining (5.8) and (5.9), it follows that

Y (z) = (z − a_{m})^{κ_{n}}P_{1}(z) ⋯ P_{m−1}(z)P_{m}^{−1}(z)Ŵ((z − a_{m})^{−1})
= diag (z^{κ_{n}+(m−2)(n−1)}, z^{κ_{n}+(m−2)(n−2)}, …, z^{κ_{n}+m−2}, z^{κ_{n}})Z_{∞}(z) (5.10)

for z ∈ D_{∞}, where Z_{∞} is analytic and invertible on D_{∞}. Hence the behavior at infinity is as required for systems of standard form with indices κ_{1}, …, κ_{n}. (Recall that κ_{k} − κ_{k+1} = m − 2 is assumed.)

In order to analyze the behavior at the singularity a_{k}, we make the connection with the systems Y′_{k}(z̃) = A_{k}(z)Y_{k}(z̃). First write U(z) = (z − a_{m})^{κ_{n}} Û(z)P_{1}(z) ⋯ P_{m}(z), where Û(z) = P_{1}(z) ⋯ P_{m−1}(z)S^{−1}P_{m−1}^{−1}(z) ⋯ P_{1}^{−1}(z) can be shown to be analytic and invertible on all of ℂ. In fact, one uses that S is an upper triangular matrix and that P_{1}(z)·P_{m}(z) is a diagonal matrix function with particular entries. Now, recalling how Y_{k}(z̃) and Y (z̃) were obtained from W (z̃), it follows that Y (z̃) = U_{k}(z)Y_{k}(z̃) where U_{k}(z) is a certain matrix function analytic and invertible on D_{a_{k}}. Hence the local properties of Y (z̃) and Y_{k}(z̃) are essentially the same. In particular, A(z) has a simple pole at z = a_{k} as so has A_{k}(z), and the residue of A(z) at z = a_{k} is similar to Ê_{k}.

Using the characterizations given in Proposition 4.2 and Theorem 4.3, it follows that Y′(z̃) = A(z)Y (z̃) is a system of standard form with “local” data [Ê_{1}, …, Ê_{m}] and the properties mentioned above. In particular, we obtain that A(z) is of the form (5.2).

The assertion that α_{1} ⋯ α_{n−1} ≠ 0 is a consequence of the special structure of A_{0}(z) given in (5.4) and the fact that U(z) is an upper triangular matrix function. Observe that

A(z) = U(z)A_{0}(z)U^{−1}(z) + U′(z)U^{−1}(z).  (5.11)

Finally, we need to show that the “local” data is given also by [E_{1}, …, E_{m}], i.e., E_{k} ∼ Ê_{k}. In fact, Proposition 5.3 implies that min(M_{k}) = min(Ê_{k}) = n. Because by assumption M_{k} ∼ exp(−2πiE_{k}), we obtain min(E_{k}) = n. Again by assumption, the eigenvalues of E_{k} are ε_{k}^{(1)}, …, ε_{k}^{(n)}, hence they are the same as those of Ê_{k}. Now we can conclude E_{k} ∼ Ê_{k}.

(i)⇒(ii): The essence of the above construction was the passage from the matrix function A_{0}(z) to the matrix function A(z) by means of the transformation (5.11) with U(z). The fact that the entries below the first row of A_{0}(z) are of a special form is reason for the connection of A_{0}(z) to the scalar equation.

Now consider the singularities a_{1}, …, a_{m} and the indices κ_{1}, …, κ_{n} as fixed and the functions q_{1}(z), …, q_{n}(z) as arbitrary. It follows from the above argumentation that the matrices A(z) obtained by (5.11) are necessarily of the form (5.2). Moreover, a straightforward computation (using the special structure of U(z) and A_{0}(z)) shows that the entries below the first row of A(z) are also of a special form, i.e., they only depend on a_{1}, …, a_{m} and κ_{n} but not on q_{1}(z), …, q_{n}(z). Moreover, one obtains that α_{1} ⋯ α_{n−1} ≠ 0. In this sense, these entries of A(z) have to be considered as “fixed”. In principle, they can be computed.

We want to make the converse transformation with U^{−1}(z), i.e., to pass from A(z) to A_{0}(z) by means of

A_{0}(z) = U^{−1}(z)A(z)U(z) − U^{−1}(z)U′(z).  (5.12)

Obviously, the matrix A_{0}(z) will be only of the desired form (5.4) if the entries below the first row of A(z) are chosen “properly”. The crucial point is that if these entries of A(z) are given indeed in the appropriate way, then the matrix A_{0}(z) will be of the form (5.4) with certain q_{1}(z), …, q_{n}(z). This can be shown again by a straightforward computation.

Hence we cannot start with just any matrix A(z) of the form (5.2). Fortunately, the following result is true. Any system Y′(z̃) = A(z)Y (z̃) with A(z) given by (5.2) and α_{1} ⋯ α_{n−1} ≠ 0 is equivalent to some system with another matrix Â(z) which is of the same form and whose entries below the first row are prescribed arbitrarily.

In order to prove this statement we need only recall Theorem 3.5. We have to prove that there exists a matrix V (z) of the form (3.3) and the properties stated there such that a transformation with V (z) takes the matrix A(z) to the matrix Â(z). Note that the entries of V (z) are polynomials of a certain degree. We can split this transformation into several steps. First, let V (z) be a diagonal matrix with suitable diagonal entries such that A(z) is taken into a matrix with entries α_{1}, …, α_{n−1}. At the k-step, k = 1, …, n, we then choose the matrix V (z) to have ones on the main diagonal, suitable polynomials on the k-th diagonal above the main diagonal and zero entries everywhere else. In fact, a straightforward computation (analyzing A ↦ V AV^{−1} + V′V^{−1}) shows that these polynomials can be chosen in such a way that the entries of the (k − 1)-th diagonal above the main diagonal of A(z) (except the entry in the first row) can be modified in any desired way. At the k-th step, all entries of A(z) below this diagonal remain unaltered. Hence after the n-th step we arrive at a matrix Â(z) of the same form and with desired entries below the first row.

After this, we may assume that the matrix A(z) of the system described in (i) is of such a form that the transformation (5.12) gives a matrix A_{0}(z) which is of the form (5.4) with certain q_{1}(z), …, q_{n}(z). Hence this new system gives raise to an n-th order linear differential equation. We have to show that this equation has the properties stated in (ii).

In fact, the argumentation given above can be reversed. Using the properties of U(z) (and employing the systems Y′_{k}(z̃) = A_{k}(z)Y_{k}(z̃) at an intermediate step) it follows that q_{1}(z), …, q_{n}(z) are analytic on S \ {∞} and that q_{j}(z) has at most a j-th order pole at z = a_{k}. Hence the scalar equation is a Fuchsian equation and it follows also from above that the local exponents are equal to the eigenvalues of the residue of A(z), which is a matrix similar to E_{k}. Obviously, the monodromy is the same as for the given system.

The assertion that the scalar equation has no singularity at infinity is only slightly more difficult. We know that the solution Y (z̃) of the system can be written as (5.10). Introducing a modified scalar equation by change of variables as above, we see that (5.8) holds with W (z̃) being the solution for the system with A_{0}(z), or, equivalently, the matrix (5.13) corresponding to the solution of the scalar equation. Combining this with (5.9) and the fact that Y (z̃) = U(z)W (z̃), it is possible to conclude that Ŵ(ξ) is analytic and invertible near ξ = 0. Hence ξ = 0 is no singular point for the modified scalar equation, and neither is z = ∞ a singular point for the desired scalar equation. □

We remark in this connection that the identity

trace E_{1} + … + trace E_{m} = κ_{1} + … + κ_{n} = \frac{n(n − 1)}{2}(m − 2) + nκ_{n}, (5.13)

which follows from (4.2) and κ_{k} − κ_{k+1} = m − 2, corresponds to

∑_{j=1}^{n}(ε_{1}^{(j)} + ε_{2}^{(j)} + … + ε_{m}^{(j)} − κ_{n}) = \frac{n(n − 1)}{2}(m − 2),  (5.14)

which is Fuchs’ relation (2.16) for the above linear differential equation.

The previous theorem provides us with an implicit description of a class of data which has indices satisfying κ_{k} − κ_{k+1} = m − 2. (It does not describe all such data because of the assumption α_{1} ⋯ α_{n−1} ≠ 0.) Namely, we may choose indices satisfying κ_{k} − κ_{k+1} = m − 2 and prescribe the eigenvalues of E_{1}, …, E_{m} under the restrictions (5.13) or (5.14). Then the corresponding linear differential equations, which contain (m − 2)n(n + 1)/2 − (m − 1)n + 1 free parameters, give rise to a class of monodromy [M_{1}, …, M_{m}]. Although in general there is no explicit description for this monodromy, one knows that this data has the above indices.

The monodromy of linear Fuchsian differential equations is generically irreducible. In fact, if the numbers ε_{k}^{(j)} are chosen such that

∑_{k=1}^{m} ∑_{r=1}^{R} ε_{k}^{(j_{r,k})} ∉ ℤ  (5.15)

for all possibilities of 1 ≤ R < n and 1 ≤ j_{1,k} < j_{2,k} < … < j_{R,k} ≤ n, then the monodromy of the corresponding differential equation is always irreducible. Hence the condition (5.2) cannot be improved in general. We note, however, that the (nongeneric) case of reducible monodromy is also covered by the previous theorem.

The analogue of the condition min(M_{k}) = n stated in Proposition 5.3 for scalar linear differential equations is well known. If the difference of any two local exponents of a Fuchsian singularity is not a nonzero integer, then the corresponding monodromy matrix M_{k} satisfies min(M_{k}) = n.

6 Some results for the 2 × 2 matrix case

In this section we consider the 2 × 2 matrix case with m singularities. For general m we obtain some information about the indices corresponding to given data. For m = 3, this information provides us with a complete explicit answer. For m = 4, the answer can be given either explicitly or in terms of the monodromy of second order linear Fuchsian differential equations with four singularities.

In order to formulate our results we have to make some preparations related to the “reducibility type” of the data. Let [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}] be admissible data of 2 × 2 matrices. Then the following (exclusive) cases can occur:

(A) [M_{1}, …, M_{m}] is irreducible;

(B) [M_{1}, …, M_{m}] possesses exactly one non-trivial invariant subspace;

(C) [M_{1}, …, M_{m}] possesses at least two non-trivial invariant subspaces.

In all these cases we define

κ = trace E_{1} + … + trace E_{m}.  (6.1)

In case (B), the m-tuple [M_{1}, …, M_{m}] is equivalent to

[( \begin{matrix}μ_{1}^{(1)} & α_{1} \\ 0 & μ_{1}^{(2)} \end{matrix} ) , …, ( \begin{matrix}μ_{m}^{(1)} & α_{m} \\ 0 & μ_{m}^{(2)} \end{matrix} )] ,  (6.2)

where μ_{1}^{(1)} ⋯ μ_{m}^{(1)} = μ_{1}^{(2)} ⋯ μ_{m}^{(2)} = 1, and α_{1}, …, α_{m} ∈ ℂ satisfy the linear equation

α_{1}μ_{2}^{(2)} ⋯ μ_{m}^{(2)} + μ_{1}^{(1)}α_{2}μ_{3}^{(2)} ⋯ μ_{m}^{(2)} + … + μ_{1}^{(1)} ⋯ μ_{m−1}^{(1)}α_{m} = 0. (6.3)

In the representation (6.2), the values of μ_{k}^{(j)} are uniquely determined. In fact, μ_{k}^{(1)} is the eigenvalue of M_{k} corresponding to the non-trivial invariant subspace and μ_{k}^{(2)} is the other eigenvalue. Let ε_{k}^{(1)} and ε_{k}^{(2)} be the eigenvalues of E_{k} numbered in such a way that μ_{k}^{(j)} = exp(−2πiε_{k}^{(j)}). Because of the non-resonance of E_{k}, their values are again uniquely determined, and so are the following integers,

n_{1} = ε_{1}^{(1)} + … + ε_{m}^{(1)},  n_{2} = ε_{1}^{(2)} + … + ε_{m}^{(2)}.  (6.4)

In case (C), the m-tuple [M_{1}, …, M_{m}] is diagonalizable, i.e., it is equivalent to (6.2) with α_{1} = … = α_{m} = 0. Here, in contrast, the values of μ_{k}^{(1)} and μ_{k}^{(2)} can be interchanged. Consequently, the integers n_{1} and n_{2} are only defined up to change of order. In the description of the main results later on, we will therefore assume without loss of generality that n_{1} ≥ n_{2}.

The following proposition provides some information about the equivalence classes of m-tuples of the form (6.2)

Proposition 6.1 Let [M_{1}, …, M_{m}] and [M̃_{1}, …, M̃_{m}] be two m-tuples of the form (6.2). Assume that both m-tuples have the same values μ_{1}^{(1)}, …, μ_{m}^{(1)} and μ_{1}^{(2)}, …, μ_{m}^{(2)}, but possibly different values α_{1}, …, α_{m} and α̃_{1}, …, α̃_{m}, respectively. Then these two m-tuples are equivalent if and only if there exist λ ∈ ℂ \ {0} and ρ ∈ ℂ such that

α_{k} = λα̃_{k} + ρ(μ_{k}^{(1)} − μ_{k}^{(2)})  (6.5)

for each k = 1, …, m.

Proof. The m-tuples [M_{1}, …, M_{m}] and [M̃_{1}, …, M̃_{m}] are equivalent if and only if there exists a matrix C ∈ Gℂ^{2×2} such that M̃_{k} = CM_{k}C^{−1}. If all matrices M_{1}, …, M_{m} are scalar matrices, then the assertion is trivial. Otherwise, if at least one M_{k} is not scalar, one rewrites M̃_{k}C = CM_{k}, and now a straightforward computation shows that C is an upper triangular matrix. Using this, the relation M̃_{k} = CM_{k}C^{−1} is easily seen to be equivalent to (6.5). □

As a conclusion, an m-tuple of the form (6.2) is diagonalizable if and only if the vector α = (α_{1}, …, α_{m}) is a multiple of the vector (μ_{1}^{(1)} − μ_{1}^{(2)}, …, μ_{m}^{(1)} − μ_{m}^{(2)}).

All m-tuples of the form (6.2) are parameterized by the vector α. Because of the linear condition (6.3), all “admissible” vectors α are taken from an (m − 1)- dimensional subspace of ℂ^{m}. The relation (6.5) establishes an equivalence relation between such vectors. The statement of this proposition is that there is a one-to-one correspondence between equivalence classes of such vectors and equivalence classes of such m-tuples. Elaborating on (6.5), it is easy to see that these equivalence classes can be identified with ℙ^{m−2} ∪ {0} in case μ_{k}^{(1)} = μ_{k}^{(2)} for all k = 1, …, m and with ℙ^{m−3} ∪ {0} in case μ_{k}^{(1)} ≠ μ_{k}^{(2)} for some k = 1, …, m. Here ℙ^{n} stands for the n-dimensional complex projective space, and {0} symbolizes the single equivalence class containing the vector α = 0. We remark that the singleton {0} corresponds exactly to the class of matrices [M_{1}, …, M_{m}] which are diagonalizable.

In the following theorem, we consider all systems of standard form Y′(z̃) = A(z)Y (z̃) with indices κ_{1}, κ_{2} ∈ ℤ, κ_{1} ≥ κ_{2}, for which A(z) can be written as

A(z) = \frac{1}{p(z)} ( \begin{matrix}κ_{1}z^{m−1} + p_{1}(z) & p_{12}(z) \\ 0 & κ_{2}z^{m−1} + p_{2}(z) \end{matrix} ) ,  (6.6)

where p(z) = (z − a_{1}) ⋯ (z − a_{m}), p_{1}, p_{2} ∈ P_{m−2}, p_{12} ∈ P_{m−2+κ_{1}−κ_{2}}. We will completely characterize the data which is associated to such systems.

Theorem 6.2 Let κ_{1}, κ_{2} ∈ ℤ, κ_{1} ≥ κ_{2}, and let [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}] be admissible data of 2 × 2 matrices. Then the following two statements are equivalent:

(i) [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}] is the data associated to some system of the form (6.6).

(ii) [M_{1}, …, M_{m}] is equivalent to some m-tuple of the form (6.2) with n_{1} = κ_{1}, n_{2} = κ_{2}, where n_{1} and n_{2} are defined by (6.4).

Proof. (i)⇒(ii): Because A(z) is of triangular form we obtain from Lemma 5.1 that there exists a solution Y (z̃) which is also of triangular form. So let us write Y′(z̃) = A(z)Y (z̃) as

( \begin{matrix}y′_{1}(z̃) & y′_{12}(z̃) \\ 0 & y′_{2}(z̃) \end{matrix} ) = ( \begin{matrix}a_{1}(z) & a_{12}(z) \\ 0 & a_{2}(z) \end{matrix} ) ( \begin{matrix}y_{1}(z̃) & y_{12}(z̃) \\ 0 & y_{2}(z̃) \end{matrix} ) .  (6.7)

For j = 1, 2, we have

a_{j}(z) = \frac{κ_{j}z^{m−1} + p_{j}(z)}{p(z)} = ∑_{k=1}^{m} \frac{ε_{k}^{(j)}}{z − a_{k}} ,  (6.8)

where the last equality defines the numbers ε_{1}^{(j)}, …, ε_{m}^{(j)}. They satisfy κ_{j} = ε_{1}^{(j)} + … + ε_{m}^{(j)}. Because E_{k} is similar to the residue of A(z) at z = a_{k}, the numbers ε_{k}^{(1)} and ε_{k}^{(2)} are the eigenvalues of E_{k}. Once we have shown that μ_{k}^{(j)} = exp(−2πiε_{k}^{(j)}), it follows that κ_{1} = n_{1} and κ_{2} = n_{2}.

In fact, from (6.8) we obtain that the solutions of y′_{j}(z̃) = a_{j}(z)y_{j}(z̃) are given by y_{j}(z̃) = ∏_{k=1}^{m}(z̃ − a_{k})^{ε_{k}^{(j)}} up to a redundant constant. Regardless of the precise expression for y_{12}(z̃), it follows that the monodromy representation for Y (z̃) is given by (6.2) with μ_{k}^{(j)} = exp(−2πiε_{k}^{(j)}) and with certain α_{k}. Hence [M_{1}, …, M_{m}] is equivalent to (6.2) and the μ_{k}^{(j)} and ε_{k}^{(j)} are indeed properly ordered.

(ii)⇒(i): We consider all m-tuples of the form (6.2) with fixed μ_{k}^{(j)}, and we fix also the numbers ε_{k}^{(j)} (i.e., the eigenvalues of E_{k}). Observing that κ_{1} = n_{1} and κ_{2} = n_{2}, we introduce the functions a_{1}(z) and a_{2}(z) by (6.8).

In what follows we consider all systems of the form (6.7) with those a_{1}(z) and a_{2}(z) and with a_{12}(z) = p_{12}(z)/p(z) where p_{12}(z) runs through all polynomials in P_{m−2+κ_{1}−κ_{2}}. These systems are indeed of standard form because κ_{1} ≥ κ_{2} and because E_{1}, …, E_{m} are assumed to be non-resonant. From the first part of this proof we already know that the monodromy representation of the corresponding solutions are of the form (6.2) with certain vectors α = (α_{1}, …, α_{m}). It remains to show that if p_{12} ranges through all of P_{m−2+κ_{1}−κ_{2}}, then the corresponding vectors α take values in all equivalence classes (defined by the equivalence relation (6.5)).

We first examine the question when two systems with A(z) and Ã(z) given by

A(z) = ( \begin{matrix}a_{1}(z) & p_{12}(z)/p(z) \\ 0 & a_{2}(z) \end{matrix} ) ,  Ã(z) = ( \begin{matrix}a_{1}(z) & p̃_{12}(z)/p(z) \\ 0 & a_{2}(z) \end{matrix} )

with p_{12}, p̃_{12} ∈ P_{m−2+κ_{1}−κ_{2}} are equivalent. By definition this is the case if and only if there exists a V such that Ã = V AV^{−1} + V′V^{−1}, where V is of the form

V (z) = ( \begin{matrix}λ_{1} & v(z) \\ 0 & λ_{2} \end{matrix} )

with λ_{1}, λ_{2} ∈ ℂ \ {0} and v ∈ P_{κ_{1}−κ_{2}} if κ_{1} > κ_{2}, and V (z) = V ∈ Gℂ^{2×2} if κ_{1} = κ_{2}. If κ_{1} = κ_{2}, then ÃV = V A implies that V has to be an upper triangular matrix (except for the case where A(z) is given with a_{1}(z) = a_{2}(z) and p_{12}(z) = 0, which can easily be dealt with separately). Using this information about V (z), it follows that the above systems are equivalent if and only if p̃_{12} = λp_{12} + v(a_{2} − a_{1})p + v′p for some λ ∈ ℂ \ {0} and v ∈ P_{κ_{1}−κ_{2}}.

This, in turn, can be rephrased by saying that p̃_{12} − λp_{12} ∈ im Ξ for some λ ∈ ℂ\{0}, where im Ξ is the image of the linear mapping Ξ : P_{κ_{1}−κ_{2}} → P_{m−2+κ_{1}−κ_{2}}, v ↦ v(a_{2}−a_{1})p+v′p. We conclude that dim im Ξ ≤ κ_{1}−κ_{2}+1. If a_{1} = a_{2} (hence κ_{1} = κ_{2}), then Ξ is even the zero mapping. Now decompose P_{m−2+κ_{1}−κ_{2}} = im Ξ⊕X as a direct sum, and remark that dim X ≥ m − 2 and even dim X ≥ m − 1 if a_{1} = a_{2}. Finally, we arrive at the following statement: if p_{12}, p̃_{12} ∈ X, then the above systems are equivalent if and only if p̃_{12} = λp_{12} for some λ ∈ ℂ \ {0}.

Now choose any basis p_{12}^{(1)}, …, p_{12}^{(d)} in X where d = dim X. Assume that a solution of the system (6.7) with a_{12} = p_{12}^{(r)}/p is given by some y_{12} = y_{12}^{(r)}, r = 1, …, d. The corresponding monodromy representation is supposed to be given by (6.2) with some vector α = α^{(r)} ∈ ℂ^{m}. Now consider an arbitrary polynomial p_{12} ∈ X and write p_{12} = ∑ γ_{r}p_{12}^{(r)}. A straightforward computation shows that a solution of (6.7) with a_{12} = p_{12}/p is given with y_{12} = ∑ γ_{r}y_{12}^{(r)} (notice that there are certainly further solutions). Moreover, the monodromy representation of this solution is given by (6.2) with α = ∑ γ_{r}α^{(r)}. In this way, we have defined a linear mapping Λ : X → ℂ^{m}, p_{12} ↦ α from X into the set of all admissible vectors α, which characterize the reducible monodromy.

Now we invoke the fact that there is a one-to-one correspondence between equivalence classes of systems of standard form and equivalence classes of given data (see Corollary 3.6). This fact specialized to the mapping Λ means the following: for p_{12}, p̃_{12} ∈ X, the vector Λ(p_{12}) is equivalent to the vector Λ(p̃_{12}) in the sense of (6.5) if and only if p̃_{12} = λp_{12} for some λ ∈ ℂ \ {0}. This allows us easily to conclude that the kernel of Λ is trivial. Hence, in case a_{1} = a_{2}, we have dim im Λ = d ≥ m − 1, and we are done because the set of all admissible vectors α is an (m−1)-dimensional subspace of ℂ^{m}. On the other hand, a_{1} ≠ a_{2} means that ε_{k}^{(1)} ≠ ε_{k}^{(2)} for some k = 1, …, m. Hence (by the non-resonance assumption) the vector μ = (μ_{1}^{(1)} − μ_{1}^{(2)}, …, μ_{m}^{(1)} − μ_{m}^{(2)}), occurring in (6.5), is not the zero vector. Because dim im Λ = d ≥ m−2, the assertion is proved if we have shown that μ ∉ im Λ. Indeed, if Λ(p_{12}) = μ, then the equivalence of μ to the zero vector implies that p_{12} = 0. Hence μ = 0, which is a contradiction. □

The reader should realize the meaning of the condition n_{1} = κ_{1} and n_{2} = κ_{2} in the previous theorem. Because of the necessary assumption κ_{1} ≥ κ_{2}, not all reducible data is covered by (ii) of the theorem, but only those for which n_{1} ≥ n_{2}. However, the case of diagonalizable data is completely covered.

The next corollary resumes this, and gives in addition some (but in general not a complete) information about the indices in the remaining cases.

Corollary 6.3 Let [M_{1}, …, M_{m}] and [E_{1}, …, E_{m}] be admissible data of 2 × 2 matrices.

(i) If [M_{1}, …, M_{m}] is diagonalizable, or if [M_{1}, …, M_{m}] possesses exactly one nontrivial invariant subspace and n_{1} ≥ n_{2} holds, then the indices are κ_{1} = n_{1} and κ_{2} = n_{2}.

(ii) If [M_{1}, …, M_{m}] is irreducible, or if [M_{1}, …, M_{m}] possesses exactly one nontrivial invariant subspace and n_{1} < n_{2} holds, then the indices fulfill the conditions:

κ_{1} − κ_{2} ≤ m − 2 and κ_{1} + κ_{2} = κ  (6.9)

Proof. Part (i) is a conclusion of the implication (ii)⇒(i) of Theorem 6.2. The second condition of part (ii) is just relation (4.2).

Now assume that κ_{1} − κ_{2} > m − 2. Then Theorem 4.3 implies that the matrix A(z) of the corresponding system is the form (6.6). Using the implication (i)⇒(ii) of Theorem 6.2, it follows that [M_{1}, …, M_{m}] is of the form (6.2) with n_{1} ≥ n_{2}, contradicting the assumption. □

Hence, for data satisfying the assumption in (i), we have a complete knowledge of the indices. Note that they do not depend on the location of the singularities.

For data satisfying the assumption in (ii), we will give precise information for the cases m = 3 and m = 4. In case m = 3, we need only invoke (6.9).

Corollary 6.4 Let [M_{1}, M_{2}, M_{3}] and [E_{1}, E_{2}, E_{3}] be admissible data of 2×2 matrices. Assume that [M_{1}, M_{2}, M_{3}] is irreducible, or that [M_{1}, M_{2}, M_{3}] possesses exactly one non-trivial invariant subspace and n_{1} < n_{2} holds.

(a) If κ is odd, then κ_{1} = (κ + 1)/2 and κ_{2} = (κ − 1)/2.

(b) If κ is even, then κ_{1} = κ_{2} = κ/2.

The results for m = 3 presented here are in accordance with the result obtained in [18] for the factorization problem. Actually, they are a generalization of the former result because the matrices E_{1}, E_{2}, E_{3} can be chosen less restrictive. (We require only non-resonance instead that the real parts of the eigenvalues are contained in an interval of length one.)

Corollary 6.5 Let [M_{1}, …, M_{4}] and [E_{1}, …, E_{4}] be admissible data of 2×2 matrices, and let a_{1}, …, a_{4} ∈ ℂ be distinct points. Assume that [M_{1}, …, M_{4}] is irreducible, or that [M_{1}, …, M_{4}] possesses exactly one non-trivial invariant subspace and n_{1} < n_{2} holds. Let ε_{k}^{(1)}, ε_{k}^{(2)} be the eigenvalues of E_{k}, k = 1, …, 4.

(a) If κ is odd, then κ_{1} = (κ + 1)/2 and κ_{2} = (κ − 1)/2.

(b) If κ is even, then either κ_{1} = κ_{2} = κ/2, or κ_{1} = κ/2 + 1 and κ_{2} = κ/2 − 1.

(b*) For κ even, the indices are κ_{1} = κ/2 + 1 and κ_{2} = κ/2 − 1 if and only if [M_{1}, …, M_{4}] is the monodromy of some second order linear differential equation with Fuchsian singularities a_{1}, …, a_{4} and local exponents given by {ε_{k}^{(1)}, ε_{k}^{(2)}} for z = a_{k}, k = 1, 2, 3, and {ε_{4}^{(1)} + 1 − κ/2, ε_{4}^{(2)} + 1 − κ/2} for z = a_{4}.

Proof. Again, assertions (a) and (b) are an immediate consequence of (6.9). In order to prove (b*) we argue as follows. The indices for the given data are κ_{1} = κ/2 + 1 and κ_{2} = κ/2 − 1 if and only if this data is associated to some system of standard form for which A(z) can be written as

A(z) = \frac{1}{p(z)} ( \begin{matrix}(κ/2 + 1)z^{3} + p_{1}(z) & p_{12}(z) \\ α & (κ/2 − 1)z^{3} + p_{2}(z) \end{matrix} )

with p(z) = (z − a_{1}) ⋯ (z − a_{4}), α ∈ ℂ, p_{1}, p_{2} ∈ P_{2} and p_{12} ∈ P_{4} (see Theorem 4.3). If α = 0, then we can apply Theorem 6.2 and conclude that the monodromy is reducible and n_{1} = κ/2 + 1 > κ/2 − 1 = n_{2}, which contradicts our assumption on the data. Hence α ≠ 0. Now we can use Theorem 5.4, which shows the desired equivalence. □

For the case where assumption (b) of the previous corollary applies, we are led to the monodromy problem for second order Fuchsian differential equations with four singular points. Note that the monodromy depends on the location of the singularities. Hence so will the corresponding indices. Unfortunately, no explicit answer is known to this monodromy problem. Only one particular case can be answered immediately by means of the remark made in the last paragraph of Section 5. Namely, if one of the matrices M_{1}, …, M_{4} is a scalar matrix (i.e., a multiple of I), then the indices are necessarily κ_{1} = κ_{2} = κ/2.

7 The 3 × 3 matrix case with 3 singularities

In this section we consider the 3 × 3 matrix case with 3 singularities. In principle, the results are similar to the 2 × 2 matrix case with 4 singularities, although it requires some more effort to describe them. In some cases of admissible data, the indices can be determined explicitly, in other cases they depend on the description of the monodromy of third order Fuchsian linear differential equations with three singularities.

7.1 Classification of reducibility type

We start again with some preparations related to the “reducibility type” of the data. Let [M_{1}, M_{2}, M_{3}] and [E_{1}, E_{2}, E_{3}] be admissible data of 3 × 3 matrices. Then the following (mutually exclusive) cases can occur:

(A) [M_{1}, M_{2}, M_{3}] is irreducible;

(B-1) [M_{1}, M_{2}, M_{3}] possesses exactly one non-trivial invariant subspace V_{1} such that dim V_{1} = 1;

(B-2) [M_{1}, M_{2}, M_{3}] possesses exactly one non-trivial invariant subspace V_{2} such that dim V_{2} = 2;

(B-3) [M_{1}, M_{2}, M_{3}] possesses exactly two non-trivial invariant subspaces V_{1} and V_{2} such that dim V_{1} = 1, dim V_{2} = 2 and V_{1} ∩ V_{2} = {0};

(C) [M_{1}, M_{2}, M_{3}] possesses exactly two non-trivial invariant subspaces V_{1} and V_{2} such that dim V_{1} = 1, dim V_{2} = 2 and V_{1} ⊂ V_{2};

(C-1) [M_{1}, M_{2}, M_{3}] possesses (at least) two one-dimensional invariant subspaces V_{1}^{(a)} and V_{1}^{(b)} and exactly one two-dimensional invariant subspace V_{2} = V_{1}^{(a)} ⊕ V_{1}^{(b)}; (C-2) [M_{1}, M_{2}, M_{3}] possesses (at least) two two-dimensional invariant subspaces V_{2}^{(a)}

and V_{2}^{(b)} and exactly one one-dimensional invariant subspace V_{1} = V_{2}^{(a)} ∩ V_{2}^{(b)};

(C-3) [M_{1}, M_{2}, M_{3}] possesses (at least) two one-dimensional invariant subspaces and two two-dimensional invariant subspaces, where all one-dimensional invariant subspaces are contained in a two-dimensional subspace and the intersection of all two-dimensional invariant subspaces is a one-dimensional subspace;

(D) [M_{1}, M_{2}, M_{3}] possesses (at least) three one-dimensional invariant subspaces V_{1}^{(a_{)}}, V_{1}^{(b)} and V_{1}^{(c)} such that V_{1}^{(a_{)}} ⊕ V_{1}^{(b)} ⊕ V_{1}^{(c)} = ℂ^{3}.

The readers may convince themselves that this classification is complete. In all these cases we define

κ = trace E_{1} + trace E_{2} + trace E_{3}.  (7.1)

In case (B-1), the triple [M_{1}, M_{2}, M_{3}] is equivalent to

[( \begin{matrix}μ_{1} & α_{1} \\ 0 & M_{1}^{#} \end{matrix} ) , ( \begin{matrix}μ_{2} & α_{2} \\ 0 & M_{2}^{#} \end{matrix} ) , ( \begin{matrix}μ_{3} & α_{3} \\ 0 & M_{3}^{#} \end{matrix} )]  (7.2)

with numbers μ_{k}, 1 × 2 vectors α_{k} and 2 × 2 matrices M_{k}^{#}, which satisfy μ_{1}μ_{2}μ_{3} = 1, M_{1}^{#}M_{2}^{#}M_{3}^{#} = I and

α_{1}M_{2}^{#}M_{3}^{#} + μ_{1}α_{2}M_{3}^{#} + μ_{1}μ_{2}α_{3} = 0.  (7.3)

The triple [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] is irreducible. For each k = 1, 2, 3, the value μ_{k} is uniquely determined and so is the eigenvalue ε_{k} of E_{k} that satisfies μ_{k} = exp(−2πiε_{k}). Hence the following integers are uniquely defined,

ν = ε_{1} + ε_{2} + ε_{3},  N = κ − ν.  (7.4)

In case (B-2), the triple [M_{1}, M_{2}, M_{3}] is equivalent to

[( \begin{matrix}M_{1}^{#} & α_{1} \\ 0 & μ_{1} \end{matrix} ) , ( \begin{matrix}M_{2}^{#} & α_{2} \\ 0 & μ_{2} \end{matrix} ) , ( \begin{matrix}M_{3}^{#} & α_{3} \\ 0 & μ_{3} \end{matrix} )]  (7.5)

with numbers μ_{k}, 2×1 vectors α_{k} and 2×2 matrices M_{k}^{#} having the same properties as above, but with (7.3) replaced by

α_{1}μ_{2}μ_{3} + M_{1}^{#}α_{2}μ_{3} + M_{1}^{#}M_{2}^{#}α_{3} = 0.  (7.6)

We also define the integers ν and N by (7.4).

In case (B-3), the triple [M_{1}, M_{2}, M_{3}] is equivalent both to (7.2) and (7.5) with α_{1} = α_{2} = α_{3} = 0. Again, the integers ν and N are uniquely defined by (7.4).

In case (C), the triple [M_{1}, M_{2}, M_{3}] is equivalent to

[( \begin{matrix}μ_{1}^{(1)} & α_{1} & γ_{1} \\ 0 & μ_{1}^{(2)} & β_{1} \\ 0 & 0 & μ_{1}^{(3)} \end{matrix} ) , ( \begin{matrix}μ_{2}^{(1)} & α_{2} & γ_{2} \\ 0 & μ_{2}^{(2)} & β_{2} \\ 0 & 0 & μ_{2}^{(3)} \end{matrix} ) , ( \begin{matrix}μ_{3}^{(1)} & α_{3} & γ_{3} \\ 0 & μ_{3}^{(2)} & β_{3} \\ 0 & 0 & μ_{3}^{(3)} \end{matrix} ) , ]  (7.7)

where the numbers μ_{k}^{(j)} are uniquely determined and satisfy μ_{1}^{(j)}μ_{2}^{(j)}μ_{3}^{(j)} = 1 for each j = 1, 2, 3 and the numbers α_{k}, β_{k}, γ_{k} satisfy the following conditions:

α_{1}μ_{2}^{(2)}μ_{3}^{(2)} + μ_{1}^{(1)}α_{2}μ_{3}^{(2)} + μ_{1}^{(1)}μ_{2}^{(1)}α_{3} = 0,  (7.8)
β_{1}μ_{2}^{(3)}μ_{3}^{(3)} + μ_{1}^{(2)}β_{2}μ_{3}^{(3)} + μ_{1}^{(2)}μ_{2}^{(2)}β_{3} = 0,  (7.9)
μ_{1}^{(1)}μ_{2}^{(1)}γ_{3} + μ_{1}^{(1)}γ_{2}μ_{3}^{(3)} + γ_{1}μ_{2}^{(3)}μ_{3}^{(3)} = −μ_{1}^{(1)}α_{2}β_{3} − α_{1}μ_{2}^{(2)}β_{3} − α_{1}β_{2}μ_{3}^{(3)}. (7.10)

Note that [M_{1}, M_{2}, M_{3}] is not equivalent to a triple of the form (7.7) where all α_{k} are zero or all β_{k} are zero. We introduce the integers n_{1}, n_{2}, n_{3} by

n_{j} = ε_{1}^{(j)} + ε_{2}^{(j)} + ε_{3}^{(j)},  j = 1, 2, 3,  (7.11)

where ε_{k}^{(j)} are the eigenvalues of E_{k} numbered in such a way that μ_{k}^{(j)} = exp(−2πiε_{k}^{(j)}). The integers n_{1}, n_{2}, n_{3} are uniquely defined in this case.

In case (C-1), the triple [M_{1}, M_{2}, M_{3}] is equivalent to (7.7) with α_{k} = 0 (but not all β_{k} or all γ_{k} can be made zero). The values μ_{3}^{(j)} are unique, but the values μ_{1}^{(j)} and μ_{2}^{(j)} can be interchanged. Hence the integer n_{3} is uniquely determined, and we will assume without loss of generality that n_{1} ≥ n_{2}.

In case (C-2), the triple [M_{1}, M_{2}, M_{3}] is equivalent to (7.7) with β_{k} = 0 (but not all α_{k} or all γ_{k} can be made zero). The values μ_{1}^{(j)} are unique, but the values μ_{2}^{(j)} and μ_{3}^{(j)} can be interchanged. Hence the integer n_{1} is uniquely determined, and we will assume without loss of generality that n_{2} ≥ n_{3}.

In case (C-3), the triple [M_{1}, M_{2}, M_{3}] is equivalent to (7.7) with β_{k} = γ_{k} = 0 (but not all α_{k} can be made zero). Here the values of μ_{k}^{(j)} are again unique for all j, k = 1, 2, 3. We define the integers n_{1}, n_{2}, n_{3} by (7.11), but to remark the different roles that these numbers play, we use the notation

ν_{1} = n_{1}, ν_{2} = n_{2}, ν_{#} = n_{3}.  (7.12)

Notice that (by permuting rows and columns) [M_{1}, M_{2}, M_{3}] is also equivalent to a triple of the form (7.7) with α_{k} = γ_{k} = 0 or with α_{k} = β_{k} = 0. This observation will play some role later on in treating this case. The numbers ν_{1}, ν_{2}, ν_{#} would then have to be defined appropriately.

Finally, in case (D), the triple [M_{1}, M_{2}, M_{3}] is diagonalizable, i.e., equivalent to (7.7) with α_{k} = β_{k} = γ_{k} = 0. The values of μ_{k}^{(j)} may be permuted and we will therefore assume without loss of generality that the numbers n_{1}, n_{2}, n_{3} defined as above are such that n_{1} ≥ n_{2} ≥ n_{3}.

7.2 Systems of triangular form

We will first consider monodromy data which is triangularizable, i.e., which corresponds to the cases (C)–(D). The generic case is (C), in which the following proposition provides some information.

Proposition 7.1 Let [M_{1}, M_{2}, M_{3}] be a triple of the form (7.7) with M_{1}M_{2}M_{3} = I, and assume that condition (C) is fulfilled. Let [M̃_{1}, M̃_{2}, M̃_{3}] be another triple of the same form and with the same entries except that γ_{1}, γ_{2}, γ_{3} are replaced by γ̃_{1}, γ̃_{2}, γ̃_{3}. Then these triples are equivalent if and only if there exist α, β, γ ∈ ℂ such that

1.) γ̃_{k} = γ_{k} + γ(μ_{k}^{(1)} − μ_{k}^{(3)})  in case (μ_{k}^{(1)})_{k=1}^{3} ≠ (μ_{k}^{(2)})_{k=1}^{3} ≠ (μ_{k}^{(3)})_{k=1}^{3},
2.) γ̃_{k} = γ_{k} + γ(μ_{k}^{(2)} − μ_{k}^{(3)}) + αβ_{k} in case (μ_{k}^{(1)})_{k=1}^{3} = (μ_{k}^{(2)})_{k=1}^{3} ≠ (μ_{k}^{(3)})_{k=1}^{3},
3.) γ̃_{k} = γ_{k} + γ(μ_{k}^{(1)} − μ_{k}^{(2)}) + βα_{k} in case (μ_{k}^{(1)})_{k=1}^{3} ≠ (μ_{k}^{(2)})_{k=1}^{3} = (μ_{k}^{(3)})_{k=1}^{3},
4.) γ̃_{k} = γ_{k} + αβ_{k} + βα_{k}  in case (μ_{k}^{(1)})_{k=1}^{3} = (μ_{k}^{(2)})_{k=1}^{3} = (μ_{k}^{(3)})_{k=1}^{3}.

Proof. These triples are equivalent if and only if there exists an invertible matrix S such that M̃_{k} = S^{−1}M_{k}S. It follows that the matrix S maps the invariant subspaces of [M̃_{1}, M̃_{2}, M̃_{3}] into the invariant subspaces of [M_{1}, M_{2}, M_{3}]. Because of the condition (C), V_{1} = ℂ ⊕ {0} ⊕ {0} and V_{2} = ℂ ⊕ ℂ ⊕ {0} are the only non-trivial invariant subspaces of both triples. Hence V_{1} and V_{2} are also invariant subspaces of S, and this implies that S is of triangular form,

S = ( \begin{matrix}s_{1} & α & γ \\ 0 & s_{2} & β \\ 0 & 0 & s_{3} \end{matrix} ) .  (7.13)

Now the equality M̃_{k} = S^{−1}M_{k}S gives the following two conditions,

α_{k} = \frac{s_{2}}{s_{1}} α_{k} + \frac{α}{s_{1}} (μ_{k}^{(1)} − μ_{k}^{(2)}),  (7.14)
β_{k} = \frac{s_{3}}{s_{2}} β_{k} + \frac{β}{s_{2}} (μ_{k}^{(2)} − μ_{k}^{(3)}),  (7.15)

and a third condition, which will be stated later. We first claim that s_{1} = s_{2}. Indeed, (7.14) can be rewritten as 0 = (s_{2} −s_{1})α_{k} +α(μ_{k}^{(1)} −μ_{k}^{(2)}). If s_{1} ≠ s_{2}, then Proposition 6.1 applied to the triple of 2 × 2 matrices obtained from [M_{1}, M_{2}, M_{3}] by removing the third rows and columns implies that [M_{1}, M_{2}, M_{3}] is equivalent to a triple of the form (7.7) with α_{k} = 0. This contradicts condition (C). Similarly, we obtain s_{2} = s_{3}. A simple thought shows that we may assume s_{1} = s_{2} = s_{3} = 1. Hence (7.14), (7.15) simplify to

α(μ_{k}^{(1)} − μ_{k}^{(2)}) = β(μ_{k}^{(2)} − μ_{k}^{(3)}) = 0,  (7.16)

and the third condition can be expressed as

γ̃_{k} = γ_{k} + γ(μ_{k}^{(1)} − μ_{k}^{(3)}) − αβ_{k} + βα_{k}.  (7.17)

The conditions (7.16) lead to four distinct cases. If μ_{k}^{(1)} ≠ μ_{k}^{(2)} for some k, then α = 0, and if μ_{k}^{(2)} ≠ μ_{k}^{(3)} for some k, then β = 0. Otherwise, α and β, resp., can be arbitrary. Combining these statements with formula (7.17) proves the assertion. □

We are now able to describe the sets of equivalence classes of triples [M_{1}, M_{2}, M_{3}], which satisfy condition (C) and for which all values appearing in (7.7) are kept fixed except γ_{1}, γ_{2}, γ_{3}. We first note that γ_{1}, γ_{2}, γ_{3} have to satisfy the linear condition (7.10).

In case 1, the set of equivalence classes can be identified with ℂ if μ_{k}^{(1)} ≠ μ_{k}^{(3)} for some k and with ℂ^{2} if μ_{k}^{(1)} = μ_{k}^{(3)} for all k, respectively.

In cases 2 and 3, the sets of equivalence classes are singletons. In fact, as to case 2, the vectors (β_{k}) and (μ_{k}^{(2)} − μ_{k}^{(3)}) are linearly independent. If they were not, i.e., if (β_{k}) is a multiple of (μ_{k}^{(2)} − μ_{k}^{(3)}), then Proposition 6.1 applied to the triple of 2 × 2 matrices obtained from [M_{1}, M_{2}, M_{3}] by removing the first rows and columns implies that [M_{1}, M_{2}, M_{3}] is equivalent to a triple (7.7) with β_{k} = 0. This conflicts with condition (C).

In case 4, the set of equivalence classes is a singleton if the vectors (α_{k}) and (β_{k}) are linearly independent. Otherwise, this set can be identified with ℂ.

In the next theorem, we consider all system of standard form Y′(z̃) = A(z)Y (z̃) with indices κ_{1} ≥ κ_{2} ≥ κ_{3}, for which A(z) can be written as

A(z) = \frac{1}{p(z)} ( \begin{matrix}κ_{1}z^{2} + p_{1}(z) & p_{12}(z) & p_{13}(z) \\ 0 & κ_{2}z^{2} + p_{2}(z) & p_{23}(z) \\ 0 & 0 & κ_{3}z^{2} + p_{3}(z) \end{matrix} ) , (7.18)

where p(z) = (z − a_{1})(z − a_{2})(z − a_{3}), p_{j} ∈ P_{1}, p_{jk} ∈ P_{1+κ_{j}−κ_{k}}. We will completely characterize the data associated to such systems.

Theorem 7.2 Let κ_{1}, κ_{2}, κ_{3} ∈ ℤ, κ_{1} ≥ κ_{2} ≥ κ_{3}, and [M_{1}, M_{2}, M_{3}] and [E_{1}, E_{2}, E_{3}] be admissible data of 3×3 matrices. Then the following two statements are equivalent:

(i) [M_{1}, M_{2}, M_{3}] and [E_{1}, E_{2}, E_{3}] is the data associated to some system of the form (7.18).

(ii) [M_{1}, M_{2}, M_{3}] is equivalent to some triple (7.7) with n_{j} = κ_{j} for j = 1, 2, 3, where the integers n_{j} are defined by (7.11).

Proof. (i)⇒(ii): Because A(z) is of triangular form, there exists a solution which is also of triangular form (see Lemma 5.1). Hence we can write Y′(z̃) = A(z)Y (z̃) as

( \begin{matrix}y′_{1} & y′_{12} & y′_{13} \\ 0 & y′_{2} & y′_{23} \\ 0 & 0 & Y′_{3} \end{matrix} ) = ( \begin{matrix}a_{1} & a_{12} & a_{13} \\ 0 & a_{2} & a_{23} \\ 0 & 0 & a_{3} \end{matrix} ) ( \begin{matrix}y_{1} & y_{12} & y_{13} \\ 0 & y_{2} & y_{23} \\ 0 & 0 & y_{3} \end{matrix} ) .  (7.19)

For j = 1, 2, 3, we have

a_{j}(z) = \frac{κ_{j}z^{2} + p_{j}(z)}{p(z)} = ∑_{k=1}^{3} \frac{ε_{k}^{(j)}}{z − a_{k}}  (7.20)

where the numbers ε_{1}^{(j)}, ε_{2}^{(j)}, ε_{3}^{(j)} are defined by the last equality and satisfy κ_{j} = ε_{1}^{(j)} + ε_{2}^{(j)} + ε_{3}^{(j)}. Because E_{k} is similar to the residue of A(z) at z = a_{k}, the numbers ε_{k}^{(1)}, ε_{k}^{(2)}, ε_{k}^{(3)} are the eigenvalues of E_{k}. Once it is shown that they are properly ordered, it follows that κ_{j} = n_{j} for j = 1, 2, 3.

The diagonal entries of the solution Y (z̃) satisfy y′_{j}(z̃) = a_{j}(z)y_{j}(z̃). Therefore they are given by y_{j}(z̃) = ∏_{k=1}^{3}(z̃− a_{k})^{ε_{k}^{(j)}} up to a redundant constant. Regardless of the precise expressions for the off-diagonal entries of Y (z̃) it follows that the corresponding monodromy representation is given by (7.7) with μ_{k}^{(j)} = exp(−2πiε_{k}^{(j)}) and with certain α_{k}, β_{k}, γ_{k}. Hence [M_{1}, M_{2}, M_{3}] is equivalent to (7.7) and the above relation between μ_{k}^{(j)} and ε_{k}^{(j)} implies that the eigenvalues of E_{k} are properly numbered.

(ii)⇒(i): Suppose we are given [M_{1}, M_{2}, M_{3}], which is equivalent to a triple of the form (7.7), and assume that the numbers ε_{k}^{(j)} and n_{j} = κ_{j} are defined appropriately. We introduce a_{j}(z) by (7.20) and consider systems (7.19) with those a_{j} and yet unspecified a_{12}, a_{13}, a_{23}. We have to show that there exist a_{12}, a_{13}, a_{23} (being the ratio p_{jk}/p of polynomials with properties as described above) such that the monodromy of this system is given by [M_{1}, M_{2}, M_{3}].

We first consider some trivial cases. Assume that [M_{1}, M_{2}, M_{3}] is equivalent to (7.7) with α_{k} = 0. The proof of Theorem 6.2 tells us that there exists an a_{13} such that the system

( \begin{matrix}y′_{1} & y′_{13} \\ 0 & y′_{3} \end{matrix} ) = ( \begin{matrix}a_{1} & a_{13} \\ 0 & a_{3} \end{matrix} ) ( \begin{matrix}y_{1} & y_{13} \\ 0 & y_{3} \end{matrix} )  (7.21)

has monodromy given by

[( \begin{matrix}μ_{1}^{(1)} & γ_{1} \\ 0 & μ_{1}^{(3)} \end{matrix} ) , ( \begin{matrix}μ_{2}^{(1)} & γ_{2} \\ 0 & μ_{2}^{(3)} \end{matrix} ) , ( \begin{matrix}μ_{3}^{(1)} & γ_{3} \\ 0 & μ_{3}^{(3)} \end{matrix} )] .  (7.22)

Moreover, there exists an a_{23} such that the system

( \begin{matrix}y′_{2} & y′_{23} \\ 0 & y′_{3} \end{matrix} ) = ( \begin{matrix}a_{2} & a_{23} \\ 0 & a_{3} \end{matrix} ) ( \begin{matrix}y_{2} & y_{23} \\ 0 & y_{3} \end{matrix} )  (7.23)

has monodromy given by

[( \begin{matrix}μ_{1}^{(2)} & β_{1} \\ 0 & μ_{1}^{(3)} \end{matrix} ) , ( \begin{matrix}μ_{2}^{(2)} & β_{2} \\ 0 & μ_{2}^{(3)} \end{matrix} ) , ( \begin{matrix}μ_{3}^{(2)} & β_{3} \\ 0 & μ_{3}^{(3)} \end{matrix} )] .  (7.24)

In this connection we remark that (7.22) and (7.24) are indeed “admissible” triples in the sense that the product of the occurring matrices is equal to I. Moreover, a_{13} = p_{13}/p and a_{23} = p_{23}/p where the polynomials p_{13} and p_{23} have the proper degree. Now we consider the system (7.19) with those a_{13} and a_{23} and with a_{12} = 0. This system has a solution of triangular form as given in (7.19) with y_{12} = 0. The monodromy representation with respect to this solution is of the form (7.7) with desired values μ_{k}^{(j)} and α_{k} = 0. However, we may have different values β̃_{k} and γ̃_{k} (instead of the desired values β_{k} and γ_{k}). Analyzing the relation between the monodromy of (7.19) and the monodromy of the “subsystems” (7.21) and (7.23), it follows that (7.22) is equivalent to (7.22) with γ_{k} replaced by γ̃_{k} and that (7.24) is equivalent to (7.24) with β_{k} replaced by β̃_{k}. We apply Proposition 6.1 and obtain that

γ̃_{k} = λ^{(1)}γ_{k} + ρ^{(1)}(μ_{k}^{(1)} − μ_{k}^{(3)}),  β̃_{k} = λ^{(2)}β_{k} + ρ^{(2)}(μ_{k}^{(2)} − μ_{k}^{(3)})  (7.25)

with certain λ^{(1)}, λ^{(2)} ∈ ℂ \ {0} and ρ^{(1)}, ρ^{(2)} ∈ ℂ. A simple thought shows that a representation (7.7) with α_{k} = 0 and having above β_{k} and γ_{k} as entries is equivalent to the same representation but with above β̃_{k} and γ̃_{k}. In fact, the equivalence is established by a certain upper triangular matrix having (1, 2)-entry equal to zero. Hence, the monodromy of the constructed system (7.19) is given by [M_{1}, M_{2}, M_{3}], and this concludes this case.

The other trivial case, where [M_{1}, M_{2}, M_{3}] is equivalent to a triple of the form (7.7) with β_{k} = 0 can be treated in the same way. Here we consider the system (7.19) with a_{23} = 0 and Y_{23} = 0 and the appropriate “subsystems”.

Now assume that [M_{1}, M_{2}, M_{3}] does not fall into these trivial classes. In other words, [M_{1}, M_{2}, M_{3}] satisfies the condition (C) concerning the reducibility type. Arguing similar as above, we find an appropriate function a_{12} such that the system

( \begin{matrix}y′_{1} & y′_{12} \\ 0 & y′_{2} \end{matrix} ) = ( \begin{matrix}a_{1} & a_{12} \\ 0 & a_{2} \end{matrix} ) ( \begin{matrix}y_{1} & y_{12} \\ 0 & y_{2} \end{matrix} )  (7.26)

has monodromy given by

[( \begin{matrix}μ_{1}^{(1)} & α_{1} \\ 0 & μ_{1}^{(2)} \end{matrix} ) , ( \begin{matrix}μ_{2}^{(1)} & α_{2} \\ 0 & μ_{2}^{(2)} \end{matrix} ) , ( \begin{matrix}μ_{3}^{(1)} & α_{3} \\ 0 & μ_{3}^{(2)} \end{matrix} )] .  (7.27)

and we also find a function a_{23} such that the system (7.23) has the monodromy (7.24). With those functions a_{12} and a_{23} and an arbitrary function a_{13}^{#} we consider the system (7.19). This system has a solution with certain functions y_{1}, y_{2}, y_{3} and y_{12}, y_{23}, y_{13}^{#}. The monodromy representation of this solution is given by (7.7) with the desired values μ_{k}^{(j)}, but possibly different values α̃_{k}, β̃_{k} and γ̃_{k}^{#}. Analyzing the relation to the “subsystems”, for which we know the monodromy, it follows again from Proposition 6.1 that

α̃_{k} = λ^{(1)}α_{k} + ρ^{(1)}(μ_{k}^{(1)} − μ_{k}^{(2)}),  β̃_{k} = λ^{(2)}β_{k} + ρ^{(2)}(μ_{k}^{(2)} − μ_{k}^{(3)})  (7.28)

with certain λ^{(1)}, λ^{(2)} ∈ ℂ \ {0} and ρ^{(1)}, ρ^{(2)} ∈ ℂ. By means of the construction of a suitable triangular matrix (as above), we obtain that [M_{1}, M_{2}, M_{3}] is equivalent to a triple (7.7) having off-diagonal entries α̃_{k}, β̃_{k} and certain γ̃_{k}.

In other word, the monodromy of the system we have constructed so far is “almost” as desired, only the values γ̃_{k}^{#} might not yet be the desired ones γ̃_{k}. In fact, in cases 2 and 3 and also in the “generic” subcase of case 4, which has been singled out in Proposition 7.1, we are already done because (as has been remarked in the paragraphs following this proposition) there exists only a single equivalence class of such triples. Hence the monodromy of the constructed system is equivalent to [M_{1}, M_{2}, M_{3}].

In the remaining cases we must still modify the system, and we proceed as follows. Again the proof of Theorem 6.2 tells us that there exists a suitable function a_{13}^{$} such that the system

( \begin{matrix}y′_{1} & y_{13}^{$}′ \\ 0 & y′_{3} \end{matrix} ) = ( \begin{matrix}a_{1} & a_{13}^{$} \\ 0 & a_{3} \end{matrix} ) ( \begin{matrix}y_{1} & y_{13}^{$} \\ 0 & y_{3} \end{matrix} )  (7.29)

has monodromy given by

[( \begin{matrix}μ_{1}^{(1)} & γ̃_{1} − γ̃_{1}^{#} \\ 0 & μ_{1}^{(3)} \end{matrix} ) , ( \begin{matrix}μ_{2}^{(1)} & γ̃_{2} − γ̃_{2}^{#} \\ 0 & μ_{2}^{(3)} \end{matrix} ) , ( \begin{matrix}μ_{3}^{(1)} & γ̃_{3} − γ̃_{3}^{#} \\ 0 & μ_{3}^{(3)} \end{matrix} )] .  (7.30)

In this connection we remark that both γ̃_{k} and γ̃_{k}^{#} satisfy a linear relation (7.10) with the same right hand side. The differences satisfy the homogeneous linear relation, which corresponds to (6.3). Hence the (7.30) is an admissible triple, i.e., the product of the matrices is equal to one.

The monodromy representation of the solution in (7.29) is also given by (7.30) except that the upper right entries are replaced by certain values γ̃_{k}^{$}. The relation between these and the former values is (see Proposition 6.1)

γ̃_{k}^{$} = λ(γ̃_{k} − γ̃_{k}^{#}) + ρ(μ_{k}^{(1)} − μ_{k}^{(3)})  (7.31)

with certain λ ∈ ℂ \ {0} and ρ ∈ ℂ.

Now we consider a new system (7.19) where A(z) has the same entries as the old one except that a_{13}^{#} is replaced by a_{13} where we define a_{13} := a_{13}^{#} + λ^{−1}a_{13}^{$}. It is easy to see that the solution of this new system has the same entries as the old solution except that y_{13}^{#} is replaced by a certain y_{13}. Moreover, the monodromy representation with respect to this solution is of the form (7.7) with off-diagonal entries α̃_{k}, β̃_{k} (as in the old one) and certain γ̃_{k}^{&}.

What remains to show is that this monodromy representation is equivalent to [M_{1}, M_{2}, M_{3}]. Because we already pointed out that [M_{1}, M_{2}, M_{3}] is equivalent to (7.7) with off-diagonal entries α̃_{k}, β̃_{k} and γ̃_{k}, we are done as soon as we have shown that γ̃_{k} and γ̃_{k}^{&} are related in the way described in Proposition 7.1.

In order to show this, we have to consider the defining relations for the (1, 3)- entries of the solutions and of the monodromy representations explicitly. As to the old system, we have

y_{13}^{#}′ = a_{1}y_{13}^{#} + a_{12}y_{23} + a_{13}^{#}y_{3},  (7.32)
y_{13}^{#}(σ_{k}^{−1}(z̃)) = y_{1}(z̃)γ̃_{k}^{#} + y_{12}(z̃)β̃_{k} + y_{13}^{#}(z̃)μ_{k}^{(3)},  (7.33)

where σ_{k}, k = 1, 2, 3, are the deck transformations. The corresponding relations for the new system are

y′_{13} = a_{1}y_{13} + a_{12}y_{23} + a_{13}y_{3},  (7.34)
y_{13}(σ_{k}^{−1}(z̃)) = y_{1}(z̃)γ̃_{k}^{&} + y_{12}(z̃)β̃_{k} + y_{13}(z̃)μ_{k}^{(3)}.  (7.35)

We also have to state the conditions on the (1, 2)-entries in (7.29) and (7.30),

y_{13}^{$}′ = a_{1}y_{13}^{$} + a_{13}^{$}y_{3},  (7.36)
y_{13}^{$}(σ_{k}^{−1}(z̃)) = y_{1}(z̃)γ̃_{k}^{$} + y_{13}^{$}(z̃)μ_{k}^{(3)}.  (7.37)

Combining (7.32) and (7.36) and using a_{13} = a_{13}^{#} + λ^{−1}a_{13}^{$}, we obtain that

(y_{13}^{#} + λ^{−1}y_{13}^{$})′ = a_{1}(y_{13}^{#} + λ^{−1}y_{13}^{$}) + a_{12}y_{23} + a_{13}y_{3}.  (7.38)

Comparing this with (7.34) we may conclude without loss of generality that y_{13} = y_{13}^{#} + λ^{−1}y_{13}^{$}. (The solution of the differential equation is not unique but we may take any of them.) Using this identity and combining (7.33) with (7.37), we arrive at

y_{13}(σ_{k}^{−1}(z̃)) = y_{1}(z̃)(γ̃_{k}^{#} + λ^{−1}γ̃_{k}^{$}) + y_{12}(z̃)β̃_{k} + y_{13}(z̃)μ_{k}^{(3)}.  (7.39)

Comparing this with (7.35) we conclude that

γ̃_{k}^{&} = γ̃_{k}^{#} + λ^{−1}γ̃_{k}^{$}
= γ̃_{k} + ρλ^{−1}(μ_{k}^{(1)} − μ_{k}^{(3)}),  (7.40)

where the last equality follows from (7.31). Now we conclude from Proposition 7.1 that the triple (7.7) with α̃_{k}, β̃_{k} and γ̃_{k}^{&}, which is the monodromy representation for the new system, is equivalent to the triple (7.7) with α̃_{k}, β̃_{k} and γ̃_{k}, which in turn is equivalent to [M_{1}, M_{2}, M_{3}]. Hence the monodromy of the new system is given by [M_{1}, M_{2}, M_{3}]. □

7.3 Systems of block-triangular form

In what follows, we are going to consider 3 × 3 systems of certain block-triangular forms, which are not of triangular form. There exists two types of them relating to two kinds of block-triangular partitions of the 3 × 3 matrix function A(z).

In order to describe these systems and the data associated to them, we have first to introduce some definitions and to state some results related to 2 × 2 systems and the corresponding data.

Let [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] and [E_{1}^{#}, E_{2}^{#}, E_{3}^{#}] be admissible data of 2 × 2 matrices. We call this data quasi-block data if either

(I) [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] is irreducible, or,

(II) [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] possesses exactly one non-trivial invariant subspace and we have n_{1}^{#} < n_{2}^{#}, where n_{1}^{#} and n_{2}^{#} are defined by (6.4).

The next result is an immediate consequence of Theorem 6.2. It describes the data of 2 × 2 systems with three singularities which are not equivalent to a system of triangular form. We remark that such systems have necessarily indices κ_{1}^{#}, κ_{2}^{#} ∈ ℤ for which κ_{1}^{#} − κ_{2}^{#} ∈ {0, 1}.

Lemma 7.3 Let [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] and [E_{1}^{#}, E_{2}^{#}, E_{3}^{#}] be admissible data of 2 × 2 matrices. Then the following two statements are equivalent:

(i) [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] and [E_{1}^{#}, E_{2}^{#}, E_{3}^{#}] is the data of a 2×2 system of standard form with three singularities which is not equivalent to a system of the form (6.7).

(ii) [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] and [E_{1}^{#}, E_{2}^{#}, E_{3}^{#}] is quasi-block data.

Proof. Quasi-block data is exactly the data which is not covered by (ii) of Theorem 6.2. As was already remarked in the paragraph after Theorem 6.2, the data which is covered by this theorem is exactly the data for which [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] is reducible with the condition n_{1}^{#} ≥ n_{2}^{#} or is diagonalizable without another condition. □

Let [M_{1}, M_{2}, M_{3}] and [E_{1}, E_{2}, E_{3}] be admissible data of 3 × 3 matrices. We call this data (1, 2)-quasi-block data if [M_{1}, M_{2}, M_{3}] is equivalent to a triple

[( \begin{matrix}μ_{1} & α_{1} \\ 0 & M_{1}^{#} \end{matrix} ) , ( \begin{matrix}μ_{2} & α_{2} \\ 0 & M_{2}^{#} \end{matrix} ) , ( \begin{matrix}μ_{3} & α_{3} \\ 0 & M_{3}^{#} \end{matrix} )]  (7.41)

and [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] and [E_{1}^{#}, E_{2}^{#}, E_{3}^{#}] is quasi-block data. Here μ_{k} are numbers, α_{k} are 1 × 2 vectors, and M_{k}^{#} are 2 × 2 matrices. The matrices E_{k}^{#} and numbers ε_{k} are supposed to satisfy M_{k}^{#} ∼ exp(−2πiE_{k}^{#}) and μ_{k} = exp(−2πiε_{k}) and are chosen in such a way that the eigenvalues of E_{k}^{#} and the number ε_{k} are exactly the eigenvalues of E_{k}. We note that for given M_{k}^{#} and μ_{k}, the number ε_{k} is uniquely determined and so is the matrix E_{k}^{#} up to similarity (because of the non-resonance of E_{k}). We also define the integers

ν = ε_{1} + ε_{2} + ε_{3},  N = trace (E_{1}^{#} + E_{2}^{#} + E_{3}^{#}).  (7.42)

We remark that if [M_{1}, M_{2}, M_{3}] possesses two different one-dimensional invariant subspaces, then [M_{1}, M_{2}, M_{3}] is equivalent to different triples of the form (7.41) (with different μ_{k} and M_{k}^{#} in particular). In this case we obtain in general also different values for ε_{k}, E_{k}^{#}, ν and N.

Proposition 7.4 Let [M_{1}, M_{2}, M_{3}] be a triple of the form (7.41). Assume that

(a) [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] is not diagonalizable;

(b) If [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] possess exactly one non-trivial invariant subspace, then

(b1) for at least one k, M_{k}^{#} possesses two different eigenvalues;

(b2) for at least one k, μ_{k} is not equal to the eigenvalue of M_{k}^{#} which corresponds to this invariant subspace.

Let [M̃_{1}, M̃_{2}, M̃_{3}] be another triple of the same form (7.41) but with α_{1}, α_{2}, α_{3} replaced by α̃_{1}, α̃_{2}, α̃_{3}. Then these triples are equivalent if and only if there exist λ ∈ ℂ \ {0} and c ∈ ℂ^{1×2} such that

α_{k} = λα̃_{k} + cM_{k}^{#} − μ_{k}c  for each k = 1, 2, 3.  (7.43)

Proof. The triples [M_{1}, M_{2}, M_{3}] and [M̃_{1}, M̃_{2}, M̃_{3}] are equivalent if and only if there exists a matrix C ∈ Gℂ^{3×3} such that M̃_{k} = CM_{k}C^{−1}. We write C in the same block form as the matrices in the triple (7.41),

C = ( \begin{matrix}c_{11} & c_{12} \\ c_{21} & c_{22} \end{matrix} ) ,  (7.44)

and inspect the (2, 1)-block entry in the resulting equality M̃_{k}C = CM_{k}. We obtain M_{k}^{#}c_{21} = c_{21}μ_{k}. Because of (a) and (b2), it follows that c_{21} = 0. Otherwise, Span {c_{21}} is an invariant subspace for all M_{k}^{#} with the eigenvalue μ_{k}. Hence C is also of block triangular form.

Now we look at the (2, 2)-block entry of M̃_{k}C = CM_{k}, and it follows that M_{k}^{#}c_{22} = c_{22}M_{k}^{#}. We claim that c_{22} is a scalar matrix. Assume the contrary. Then c_{22} is either similar to a diagonal matrix with two different diagonal entries or it is similar to a Jordan block. We may assume without loss of generality that c_{22} is actually equal to such a diagonal matrix or to a Jordan block. (Otherwise one applies an appropriate similarity transformation simultaneously to c_{22} and M_{1}^{#}, M_{2}^{#}, M_{3}^{#}, which will equally result in a contradiction.) By simple computations it follows in the first case that all M_{1}^{#}, M_{2}^{#}, M_{3}^{#} are of diagonal form. In the second case, it follows that all M_{1}^{#}, M_{2}^{#}, M_{3}^{#} are either Jordan blocks or scalar matrices. Both contradict the assumption on the triple [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}].

With this information about the entries of C, it is now easy to conclude that M̃_{k} = CM_{k}C^{−1} is equivalent to (7.43). □

Now we are able to identify the set of equivalence classes for triples of the form (7.41) with μ_{k} and M_{k}^{#} being fixed and satisfying the assumptions stated in the previous proposition. All such triples are parameterized by 1 × 2 vectors α_{1}, α_{2}, α_{3}.

It is convenient to introduce the 1 × 6 vector α = (α_{1}, α_{2}, α_{3}). First we have to observe the linear condition (7.3) which can be restated as

α ( \begin{matrix}M_{2}^{#}M_{3}^{#} \\ μ_{1}M_{3}^{#} \\ μ_{1}μ_{2}I_{2} \end{matrix} ) = 0,  (7.45)

where I_{2} is the 2 × 2 identity matrix. Because the 6 × 2 matrix appearing in this linear condition has rank 2, the vector α is actually subject to two linear conditions. Hence all “admissible” vectors α are taken from a 4-dimensional linear subspace. The equivalence relation (7.43) can shortly be written as

α = λα̃ + c ( M_{1}^{#} − μ_{1}I_{2}, M_{2}^{#} − μ_{2}I_{2}, M_{3}^{#} − μ_{3}I_{2} )  (7.46)

with λ ∈ ℂ \ {0} and c ∈ ℂ^{1×2}. Because of the assumptions in Proposition 7.4, the 2 × 6 matrix appearing there has either rank 2 or rank 1. It has rank 1 if and only if [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] is equivalent to a triple

[( \begin{matrix}μ̂_{1} & α̂_{1} \\ 0 & μ_{1} \end{matrix} ) , ( \begin{matrix}μ̂_{2} & α̂_{1} \\ 0 & μ_{2} \end{matrix} ) , ( \begin{matrix}μ̂_{3} & α̂_{1} \\ 0 & μ_{3} \end{matrix} )]  (7.47)

with certain μ̂_{k} and α̂_{k}. In other words, the above matrix has rank 1 if and only if [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] possesses exactly one non-trivial invariant subspace and, for each k = 1, 2, 3, the number μ_{k} is equal to the eigenvalue of M_{k}^{#} which does not correspond to this invariant subspace. The second part in (7.46) gives a “linear” factorization of vector spaces and thus yields either a two-dimensional or a three-dimensional quotient space. Finally, there is another “multiplicative” equivalence. So we arrive at the statement that the set of equivalence classes of such triples [M_{1}, M_{2}, M_{3}] can be identified either with ℙ^{1}∪{0} or with ℙ^{2}∪{0} (corresponding to the above distinction), where ℙ^{n} is the n-dimensional complex projective space and {0} denotes the single equivalence class containing the vector α = 0. This single equivalence class consists precisely of the triples which are “block-diagonalizable”.

Next we consider systems Y′(z̃) = A(z)Y (z̃) of standard form with indices κ_{1} ≥ κ_{2} ≥ κ_{3} for which A(z) can be written as

A(z) = \frac{1}{p(z)} ( \begin{matrix}κ_{1}z^{2} + p_{1}(z) & p_{12}(z) & p_{13}(z) \\ 0 & κ_{2}z^{2} + p_{2}(z) & p_{23}(z) \\ 0 & p_{32}(z) & κ_{3}z^{2} + p_{3}(z) \end{matrix} ) , (7.48)

where p(z) = (z −a_{1})(z −a_{2})(z −a_{3}), p_{k} ∈ P_{1}, p_{jk} ∈ P_{1+κ_{j}−κ_{k}}. If such a system is not equivalent to a system of the form (7.18), then it will be called a system of (1, 2)-block form. We remark that necessarily κ_{2} − κ_{3} ≤ 1 because otherwise p_{32}(z) = 0 and the system is of the form (7.18).

The next theorem describes the data which is associated to such systems.

Theorem 7.5 Let κ_{1}, κ_{2}, κ_{3} ∈ ℤ, κ_{1} ≥ κ_{2} ≥ κ_{3}, and assume κ_{2} − κ_{3} ≤ 1. Let [M_{1}, M_{2}, M_{3}] and [E_{1}, E_{2}, E_{3}] be admissible data of 3×3 matrices. Then the following two statements are equivalent:

(i) [M_{1}, M_{2}, M_{3}] and [E_{1}, E_{2}, E_{3}] is the data associated to a system of (1, 2)-block form with indices κ_{1}, κ_{2}, κ_{3}.

(ii) [M_{1}, M_{2}, M_{3}] and [E_{1}, E_{2}, E_{3}] is (1, 2)-quasi-block data, and ν = κ_{1}, N = κ_{2} + κ_{3}, where ν and N are defined by (7.42).

Proof. (i)⇒(ii): Because A(z) is of block-triangular form, there exists a solution which is also of block triangular form. Hence let us write Y′(z̃) = A(z) = Y (z̃) as

( \begin{matrix}y′_{1}(z̃) & y′_{12}(z̃) \\ 0 & Y′_{2}(z̃) \end{matrix} ) = ( \begin{matrix}a_{1}(z) & a_{12}(z) \\ 0 & A_{2}(z) \end{matrix} ) ( \begin{matrix}y_{1}(z̃) & y_{12}(z̃) \\ 0 & Y_{2}(z̃) \end{matrix} ) .  (7.49)

We first consider the scalar subsystem y′_{1}(z̃) = a_{1}(z)y_{1}(z̃). We know that a_{1}(z) = (κ_{1}z^{2} +p_{1}(z))/p(z). If ε_{1}, ε_{2}, ε_{3} denote the residues of a_{1}(z) at z = a_{1}, a_{2}, a_{3}, then the monodromy is given by [μ_{1}, μ_{2}, μ_{3}], where μ_{k} = exp(−2πiε_{k}). Moreover, it follows that κ_{1} = ε_{1} + ε_{2} + ε_{3}.

Now we consider the 2 × 2 subsystem Y′_{2}(z̃) = A_{2}(z)Y_{2}(z̃), where A_{2}(z) is given by the lower right block in (7.48). We claim that this system is not equivalent to a system of the form (6.7). Indeed, if it were equivalent, then a simple thought shows that the system Y′(z̃) = A(z)Y (z̃) is equivalent to a system of the form (7.18), which contradicts the assumption that it is a system of (1, 2)-block form. Hence we are in a position to apply Lemma 7.3. We obtain that the data of the system Y′_{2}(z̃) = A_{2}(z)Y_{2}(z̃), [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] and [E_{1}^{#}, E_{2}^{#}, E_{3}^{#}], is quasi-block data. Note that M_{k}^{#} ∼ exp(−2πiE_{k}^{#}) and that E_{k}^{#} is similar to the residue of A_{2}(z) at z = a_{k}. Hence, in particular, κ_{2} + κ_{3} = trace (E_{1}^{#} + E_{2}^{#} + E_{3}^{#}).

Combining the information about the two subsystems, we obtain immediately that the monodromy data [M_{1}, M_{2}, M_{3}] of the original system is equivalent to a triple (7.41) with certain α_{1}, α_{2}, α_{3} ∈ ℂ^{1×2}. The eigenvalues of the matrix E_{k}, which is similar to the residue of A(z) at z = a_{k}, are given by the number ε_{k} and the eigenvalues of E_{k}^{#} because they are the residues of a_{1}(z) and A_{2}(z), respectively. Hence, since the data of the 2 × 2 subsystem is quasi-block data, the data of the original system is (1, 2)-quasi-block data. Moreover, from the values for κ_{1} and κ_{2} + κ_{3} given above, it follows that κ_{1} = ν and κ_{2} + κ_{3} = N.

(ii)⇒(i): Assume that [M_{1}, M_{2}, M_{3}] is a triple of the form (7.41) such that (on defining ε_{k} and E_{k}^{#} as well as ν and N by (7.42)) the appropriate conditions for (1, 2)-quasi-block data are satisfied. In particular, [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] and [E_{1}^{#}, E_{2}^{#}, E_{3}^{#}] is quasi-block data, and ν = κ_{1}, N = κ_{2} + κ_{3}. We have to show that there exists a system of (1, 2)-block form which has monodromy [M_{1}, M_{2}, M_{3}], the indices κ_{1}, κ_{2}, κ_{3}, and the residues of A(z) at z = a_{k} are similar to E_{k}.

We first define

a_{1}(z) = ∑_{k=1}^{3} \frac{ε_{k}}{z − a_{k}} = \frac{κ_{1}z^{2} + p_{1}(z)}{p(z)},  (7.50)

noting that κ_{1} = ν = ε_{1} + ε_{2} + ε_{3}. Furthermore, because [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] and [E_{1}^{#}, E_{2}^{#}, E_{3}^{#}] is quasi-block data, Lemma 7.3 shows that there exists a 2 × 2 system having this data with singularities a_{1}, a_{2}, a_{3} such that this system is not equivalent to a system of the form (6.7). Denote this system by Y′_{2}(z̃) = A_{2}(z)Y_{2}(z̃), and thus define the matrix function A_{2}(z). Let κ_{2}^{#} and κ_{3}^{#} be the indices of this 2 × 2 system. We claim that κ_{2} = κ_{2}^{#} and κ_{3} = κ_{3}^{#}. Indeed, because the 2 × 2 system is not of triangular form, we have 0 ≤ κ_{2}^{#} − κ_{3}^{#} ≤ 1, and by assumption we also have 0 ≤ κ_{2} −κ_{3} ≤ 1. Moreover, κ_{2} +κ_{3} = N = trace (E_{1}^{#} +E_{2}^{#} +E_{3}^{#}) = κ_{2}^{#} +κ_{3}^{#}, where the last relation follows from formula (4.2) applied in the case of the 2 × 2 system. These relations for the integers prove the stated equality.

Now we consider systems Y′(z̃) = A(z)Y (z̃) given by (7.49) with those a_{1}(z) and A_{2}(z) and yet unspecified a_{12}(z), the entries of which have, of course, to be the ratio of appropriate polynomials. We note that these systems are of the form (7.48). They are also systems of standard form with indices κ_{1}, κ_{2}, κ_{3}, because, in particular, κ_{1} ≥ κ_{2} ≥ κ_{3} is assumed. The monodromy representation of a suitable solution of these systems is given by (7.41) with certain α_{1}, α_{2}, α_{3}. The residue Ê_{k} of A(z) at z = a_{k} has the eigenvalues given by ε_{k} and the eigenvalues of E_{k}^{#}. Hence, as the eigenvalues of E_{k} are also given in this way, it follows that E_{k} is similar to Ê_{k}. (In case of coinciding eigenvalues, one has also to employ that M_{k} ∼ exp(−2πiE_{k}) and M_{k} ∼ exp(−2πiÊ_{k}).)

What remains to show is, firstly, that the systems constructed in this way are indeed of (1, 2)-block form, i.e., that they are not equivalent to systems (7.18). Secondly, one has to show that if a_{12}(z) runs through all admissible functions, then corresponding vectors α_{1}, α_{2}, α_{3} appearing in the monodromy representation (7.41) run through all possibilities modulo the equivalence described in Proposition 7.4.

Let us turn to the first problem. Assume the contrary, namely that our system with A(z) is equivalent to a system with Ã(z) being of the form (7.18). Theorem 3.5 says that both systems have the same indices and that the equivalence is established by a matrix function V (z), which has a particular structure. We also know that the 2 × 2 subsystem A_{2}(z) of A(z) is not equivalent to a triangular system, and we are going to show a contradiction to this assertion. If κ_{2} > κ_{3}, then the (3, 1)- and (3, 2)- entries of V (z) vanish. Using A(z) = V^{−1}(z)Ã(z)V (z) − V^{−1}(z)V′(z), it follows that A(z) has (3, 2)-entry equal to zero, which is a contradiction. If κ_{1} > κ_{2}, then the V (z) is of the same block structure as A(z). Now using Ã(z) = V (z)A(z)V^{−1}(z) + V′(z)V^{−1}(z), it is easy to see that the lower right 2 × 2 submatrix in V (z) establishes an equivalence between the subsystem A_{2}(z) and a triangular 2 × 2 system, namely the one which is given by the lower right 2 × 2 matrix in Ã(z). In the remaining case, κ_{1} = κ_{2} = κ_{3}, the function V (z) is a constant matrix V. As before, let Ê_{k} be the residues of A(z) at z = a_{k}, and let Ẽ_{k} be those of Ã(z). The equivalence between A(z) and Ã(z) can now be restated as Ẽ_{k} = V Ê_{k}V^{−1}. We claim that each one-dimensional invariant subspace of [Ẽ_{1}, Ẽ_{2}, Ẽ_{3}] is contained in some twodimensional invariant subspace of this triple. Indeed, this is obvious if this triple has at least two one-dimensional invariant subspaces. If it has exactly one, then it is necessarily the space ℂ ⊕ {0} ⊕ {0}, which is contained in ℂ ⊕ ℂ ⊕ {0} (because of the triangular form of Ẽ_{k}). The above relation between Ẽ_{k} and Ê_{k} establishes a one-to-one correspondence between invariant subspaces of the previous triple and [Ê_{1}, Ê_{2}, Ê_{3}]. Hence each one-dimensional invariant subspace of the latter triple is contained in some two-dimensional invariant subspace. A simple thought shows that the 2 × 2 matrices obtained from [Ê_{1}, Ê_{2}, Ê_{3}] by removing the first rows and columns are reducible. But those matrices are the residues of the system A_{2}(z), and thus we obtain a contradiction.

Finally, we turn to the problem of showing that if a_{12} runs through all admissible vector functions, then the vector α = (α_{1}, α_{2}, α_{3}) ∈ ℂ^{1×6} runs through all possibilities modulo the equivalence stated in Proposition 7.4 (see also (7.45) and (7.46)). Before we have to show that the assumptions of Proposition 7.4 are satisfied.

We know that [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] and [E_{1}^{#}, E_{2}^{#}, E_{3}^{#}] is quasi-block data. Hence [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] is not diagonalizable. If this triple possesses one non-trivial invariant subspace and all M_{k}^{#} have only a single eigenvalue, then so have E_{k}^{#}. It follows that n_{1}^{#} = n_{2}^{#}, which is a contradiction (see (II) in the definition of quasi-block data). As to condition (b2), assume that (b2) is not satisfied. Then it is easy to see (using the nonresonance of E_{k}) that ν = n_{1}^{#}. Because N = trace (E_{1}^{#}+E_{2}^{#}+E_{3}^{#}) = n_{1}^{#}+n_{2}^{#} and n_{1}^{#} < n_{2}^{#} by (II) of the definition of quasi-block data, it follows that N > 2n_{1}^{#}. On the other hand, κ_{1} = ν and κ_{2} + κ_{3} = N. We obtain that κ_{1} = n_{1}^{#} and κ_{2} + κ_{3} > 2n_{1}^{#}. Hence κ_{2} + κ_{3} > 2κ_{1}, which conflicts with κ_{1} ≥ κ_{2} ≥ κ_{3}. So we have proved that the assumptions of Proposition 7.4 are satisfied.

Next we examine the question when two system with A(z) and Ã(z) given by (7.49) with a_{1}(z) and A_{2}(z) being introduced above, but with possibly different functions a_{12}(z) and ã_{12}(z), respectively, are equivalent. Assume that

a_{12}(z) = (\frac{p_{12}(z)}{p(z)}, \frac{p_{13}(z)}{p(z)}) ,  ã_{12}(z) = (\frac{p̃_{12}(z)}{p(z)}, \frac{p̃_{13}(z)}{p(z)}) ,  (7.51)

where p_{12}, p̃_{12} ∈ P_{1+κ_{1}−κ_{2}} and p_{13}, p̃_{13} ∈ P_{1+κ_{1}−κ_{3}}. This is the case (see Theorem 3.5) if there exists a function V (z) of a particular form such that Ã = V AV^{−1} + V′V^{−1}.

We first claim that the (2, 1)- and (3, 1)-entries of V (z), V_{21} and V_{31}, are equal to zero. This is obvious in the case κ_{1} > κ_{2}. In case κ_{1} = κ_{2} > κ_{3}, we have V_{31} = V_{32} = 0 and the upper left 2 × 2 matrix of V (z) is constant. In order to show that V_{21} = 0, we write ÃV = V A + V′, and inspect the (3, 1)-entries. It is equal to zero for V′ and also for V A. The (3, 1)-entry of ÃV equals V_{21} times the (3, 2)-entry of Ã(z). We conclude that V_{21} = 0 because otherwise the (3, 2)-entry of Ã(z) is zero, which contradicts the assumption that Ã(z) is not of triangular form. In the case, κ_{1} = κ_{2} = κ_{3}, we have Ã(z) = V A(z)V^{−1} with a constant matrix V. This can be rephrased in terms of the residues Ê_{k} and Ẽ_{k} of A(z) and Ã(z), respectively, namely Ẽ_{k} = V Ê_{k}V^{−1}. The triples [Ê_{1}, Ê_{2}, Ê_{3}] and [Ẽ_{1}, Ẽ_{2}, Ẽ_{3}] possess exactly one one-dimensional invariant subspace ℂ ⊕ {0} ⊕ {0}. (If there were another, then we could argue similar to above and conclude that the 2 × 2 matrices obtained from these triples by removing the first rows and columns are reducible.) The matrix V must map this invariant subspace into itself. Hence it is also an invariant subspace of V, which implies that V_{21} = V_{31} = 0.

So we have shown that

V (z) = ( \begin{matrix}v_{1} & v(z) \\ 0 & V_{2}(z) \end{matrix} ) ,  (7.52)

with the same block structure as A(z) and Ã(z) and with entries having certain properties. The next claim is that V_{2}(z) is a constant scalar matrix. From Ã = V AV^{−1} + V′V^{−1}, it follows that A_{2} = V_{2}A_{2}V_{2}^{−1} + V′_{2}V_{2}^{−1}, and we know that the 2 × 2 system with A_{2}(z) is not equivalent to a triangular system. If the indices κ_{2}, κ_{3} coincide, then V_{2}(z) is a constant matrix, and the above can be rephrased in terms of the residues Ê_{k}^{#} of A_{2}(z). Namely, Ê_{k}^{#} = V_{2}Ê_{k}^{#}V_{2}^{−1}. If V_{2} is not a scalar matrix, then we can argue similar as in the second paragraph of the proof of Proposition 7.4 and conclude that the triple [Ê_{1}^{#}, Ê_{2}^{#}, Ê_{3}^{#}] is reducible, which contradicts the assumption on A_{2}(z). In case κ_{2} − κ_{3} = 1, V_{2}(z) is of triangular form with constant values on the diagonal. Now a straightforward analysis of A_{2} = V_{2}A_{2}V_{2}^{−1} + V′_{2}V_{2}^{−1} (using that A_{2}(z) is not triangular) implies that V_{2} is a scalar constant matrix.

Having this information about V (z), we can now reformulate the condition Ã = V AV^{−1} + V′V^{−1} as follows. The systems A(z) and Ã(z) are equivalent if and only if

ã_{12}(z) = λa_{12}(z) + v(z)A_{2}(z) − a_{1}(z)v(z) + v′(z),  (7.53)

for some λ ∈ ℂ \ {0} and some v(z) = (v_{12}(z), v_{13}(z)) where v_{12} ∈ P_{κ_{1}−κ_{2}} and v_{13} ∈ P_{κ_{1}−κ_{3}}. Referring to (7.51) this can be expressed as

(p̃_{12}, p̃_{13}) = λ(p_{12}, p_{13}) + p(v_{12}, v_{13})A_{2} − a_{1}p(v_{12}, v_{13}) + p(v′_{12}, v′_{13}). (7.54)

Hence the two systems with (p_{12}, p_{13}) and (p̃_{12}, p̃_{13}) are equivalent if and only if there exists λ ∈ ℂ \ {0} such that

(p̃_{12}, p̃_{13}) − λ(p_{12}, p_{13}) ∈ im Ξ,  (7.55)

where im Ξ is the image of the linear mapping defined by

Ξ : P_{κ_{1}−κ_{2}} ⊕ P_{κ_{1}−κ_{3}} → P_{1+κ_{1}−κ_{2}} ⊕ P_{1+κ_{1}−κ_{3}}
(v_{12}, v_{13}) ↦ p(v_{12}, v_{13})A_{2} − a_{1}p(v_{12}, v_{13}) + p(v′_{12}, v′_{13}).  (7.56)

Now we decompose P_{1+κ_{1}−κ_{2}} ⊕ P_{1+κ_{1}−κ_{3}} = im Ξ ⊕ X as a direct sum. We arrive at the following statement: If (p_{12}, p_{13}) ∈ X and (p̃_{12}, p̃_{13}) ∈ X, then the corresponding systems are equivalent if and only if (p̃_{12}, p̃_{13}) = λ(p_{12}, p_{13}) for some λ ∈ ℂ \ {0}.

Obviously, we have dim X ≥ 2. However, we claim that dim X ≥ 3 in the case where the 2 × 6 matrix appearing in (7.46) has rank 1, i.e., if [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] is equivalent to a triple (7.47). Indeed, this is shown as soon as one has shown that the kernel of Ξ is non-trivial. For this we consider the 2 × 2 system Ŷ′_{2}(z̃) = Â_{2}(z)Ŷ_{2}(z̃) with Â_{2}(z) = A_{2}(z) − a_{1}(z)I_{2}. It is easy to see that this system is also of standard form with indices κ_{2} − κ_{1} and κ_{3} − κ_{1}, and with data [M_{1}^{#}μ_{1}^{−1}, M_{2}^{#}μ_{2}^{−1}, M_{3}^{#}μ_{3}^{−1}] and [E_{1}^{#} − ε_{1}I_{2}, E_{2}^{#} − ε_{2}I_{2}, E_{3}^{#} − ε_{3}I_{2}]. The solution of this system is given by Ŷ_{2}(z̃) = Y_{2}(z̃)y_{1}^{−1}(z̃). From Ŷ′_{2} = Â_{2}Ŷ_{2} it follows immediately that Ŷ_{2}^{−1}Â_{2} + (Ŷ_{2}^{−1})′ = 0 by multiplying on both sides with Ŷ_{2}^{−1}. We may assume that the monodromy representation of Ŷ_{2}(z̃) is given by

[( \begin{matrix}μ̂_{1}μ_{1}^{−1} & α̂_{1}μ_{1}^{−1} \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}μ̂_{2}μ_{2}^{−1} & α̂_{1}μ_{2}^{−1} \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}μ̂_{3}μ_{3}^{−1} & α̂_{1}μ_{3}^{−1} \\ 0 & 1 \end{matrix} )] .  (7.57)

Now we define (v_{12}, v_{13}) to be the last row in Ŷ_{2}^{−1}. Obviously, this vector function satisfies (v_{12}, v_{13})(A_{2}(z) − a_{1}(z)I_{2}) + (v′_{12}, v′_{13}) = 0. What remains to show is that v_{12} and v_{13} are polynomials of appropriate degree. From the above monodromy representation for Ŷ_{2}, it follows that (v_{12}, v_{13}) is a single valued function. This function is analytic on ℂ\{a_{1}, a_{2}, a_{3}}. Because of the non-resonance of [E_{1}, E_{2}, E_{3}] and because μ_{k} is an eigenvalue of M_{k}^{#}, it follows that ε_{k} is an eigenvalue of E_{k}^{#}. Hence the matrices E_{k}^{#} − ε_{k}I_{2} have an eigenvalue equal to zero and are also non-resonant. But these matrices describe the local behavior of Ŷ_{2} (and of its inverse) near the point a_{k}. A straightforward computation (using the fact that (v_{12}, v_{13}) is single valued) shows that (v_{12}, v_{13}) is actually analytic at z = a_{k}. Hence we have shown that (v_{12}, v_{13}) is entire analytic. That v_{12} and v_{13} are polynomials of degree not greater than κ_{1} − κ_{2} and κ_{1} −κ_{3}, respectively, can be obtained by employing the behavior of Ỹ_{2} at infinity, which is described by the integers κ_{2} −κ_{1} and κ_{3} −κ_{1}. In summary, we can conclude that (v_{12}, v_{13}) ≠ 0 lies in the kernel of Ξ.

Similar as in the proof of Theorem 6.2, we can define a linear mapping Λ : X → ℂ^{1×6} from the space X into the set of all “admissible” vectors α = (α_{1}, α_{2}, α_{3}) (see also (7.45)). This mapping is characterized by the following property. For a system of the form (7.49) with above defined a_{1}(z) and A_{2}(z) and with a_{12} given by (7.51), where (p_{12}, p_{13}) ∈ X, the corresponding monodromy is given by (7.41), where α_{1}, α_{2}, α_{3} are given by α = (α_{1}, α_{2}, α_{3}) = Λ((p_{12}, p_{13})).

We have a one-to-one correspondence between equivalence classes of systems of standard form and equivalence classes of data (see Corollary 6.3). In our situation we can apply Proposition 7.4 (see also (7.46)) and what has been stated above about the equivalence of the systems under consideration. We obtain the following statement. For (p_{12}, p_{13}), (p̃_{12}, p̃_{13}) ∈ X, the vectors Λ((p_{12}, p_{13})) and Λ((p̃_{12}, p̃_{13})) are equivalent in the sense of (7.46) (or, (7.43)) if and only if there exists a λ ∈ ℂ \ {0} such that (p̃_{12}, p̃_{13}) = λ(p_{12}, p_{13}).

The kernel of Λ is trivial. Indeed, Λ((p_{12}, p_{13})) = 0 = Λ(0) implies immediately (p_{12}, p_{13}) = 0. Hence dim im Λ ≥ 2 or even dim im Λ ≥ 3 depending on whether the 2 × 6 matrix appearing in (7.46) has rank 2 or 1. So we are done as soon as we have shown that the sum of im Λ and the following linear subspace of ℂ^{1×6},

{ c ( M_{1}^{#} − μ_{1}I_{2}, M_{2}^{#} − μ_{2}I_{2}, M_{3}^{#} − μ_{3}I_{2} ) : c ∈ ℂ^{1×2} } ,  (7.58)

exhausts the whole four-dimensional subspace of ℂ^{1×6} consisting of all admissible vectors α (see again (7.46)). As we have estimates for the dimensions, this follows from the statement that the intersection of (7.58) with im Λ contains only the zero vector. Indeed, if α = Λ((p_{12}, p_{13})) is contained in the above set, then α is equivalent to 0 = Λ(0), whence again follows that (p_{12}, p_{13}) = 0. Hence α = 0. □

Finally, we are going to consider data and systems which are also of block form, but where the block partition of the matrices are (2, 1) rather than (1, 2). The proofs of the corresponding statements are analogous, and will therefore be omitted. However, we will state definitions and the results completely.

Let [M_{1}, M_{2}, M_{3}] and [E_{1}, E_{2}, E_{3}] be admissible data of 3 × 3 matrices. We call this data (2, 1)-quasi-block data if [M_{1}, M_{2}, M_{3}] is equivalent to a triple

[( \begin{matrix}M_{1}^{#} & α_{1} \\ 0 & μ_{1} \end{matrix} ) , ( \begin{matrix}M_{2}^{#} & α_{2} \\ 0 & μ_{2} \end{matrix} ) , ( \begin{matrix}M_{3}^{#} & α_{3} \\ 0 & μ_{3} \end{matrix} )]  (7.59)

and [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] and [E_{1}^{#}, E_{2}^{#}, E_{3}^{#}] is quasi-block data. Here μ_{k} are numbers, α_{k} are 2 × 1 vectors, and M_{k}^{#} are 2 × 2 matrices. The matrices E_{k}^{#} and numbers ε_{k} are supposed to satisfy M_{k}^{#} ∼ exp(−2πiE_{k}^{#}) and μ_{k} = exp(−2πiε_{k}) and are chosen in such a way that the eigenvalues of E_{k}^{#} and the number ε_{k} are the exactly eigenvalues of E_{k}. For given M_{k}^{#} and μ_{k}, the number ε_{k} is uniquely determined and so is the matrix E_{k}^{#} up to similarity (because of the non-resonance of E_{k}). Again we define the integers

ν = ε_{1} + ε_{2} + ε_{3},  N = trace (E_{1}^{#} + E_{2}^{#} + E_{3}^{#}).  (7.60)

As above, we remark that if [M_{1}, M_{2}, M_{3}] possesses two different two-dimensional invariant subspaces, then [M_{1}, M_{2}, M_{3}] is equivalent to different triples of the form (7.59) (with different μ_{k} and M_{k}^{#} in particular). In general, we obtain also different values for ε_{k}, E_{k}^{#}, ν and N.

The counterpart to Proposition 7.4 is the following result.

Proposition 7.6 Let [M_{1}, M_{2}, M_{3}] be a triple of the form (7.59). Assume that

(a) [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] is not diagonalizable;

(b) If [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] possess exactly one non-trivial invariant subspace, then (b1) for at least one k, M_{k}^{#} possesses two different eigenvalues;

(b2) for at least one k, μ_{k} is not equal to the eigenvalue of M_{k}^{#} which does not correspond to this invariant subspace.

Let [M̃_{1}, M̃_{2}, M̃_{3}] be another triple of the same form (7.59) but with α_{1}, α_{2}, α_{3} replaced by α̃_{1}, α̃_{2}, α̃_{3}. Then these triples are equivalent if and only if there exist λ ∈ ℂ \ {0} and c ∈ ℂ^{2×1} such that

α_{k} = λα̃_{k} + M_{k}^{#}c − cμ_{k}  for each k = 1, 2, 3.  (7.61)

The set of equivalence classes for triples of the form (7.59) with μ_{k} and M_{k}^{#} being fixed can be identified similar as before. All such triples are parameterized by 2 × 1 vectors α_{1}, α_{2}, α_{3}, and we introduce the 6 × 1 vector α = (α_{1}^{T} , α_{2}^{T} , α_{3}^{T})^{T}. We have to take the linear condition (7.6) into account. It can be restated as

( μ_{2}μ_{3}I_{2}, M_{1}^{#}μ_{3}, M_{1}^{#}M_{2}^{#} )α = 0.  (7.62)

The 2 × 6 matrix appearing here has rank 2. Hence the vector α is again subject to two linear conditions, and all “admissible” vectors α are taken from a 4-dimensional linear subspace of ℂ^{6×1}. The equivalence relation (7.61) can be rewritten as

α = λα̃ + ( \begin{matrix}M_{1}^{#} − μ_{1}I_{2} \\ M_{2}^{#} − μ_{2}I_{2} \\ M_{3}^{#} − μ_{3}I_{2} \end{matrix} ) c  (7.63)

with λ ∈ ℂ \ {0} and c ∈ ℂ^{2×1}. The 6 × 2 matrix appearing here has again either rank 2 or rank 1. It has rank one if and only if [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] possesses exactly one non-trivial invariant subspace and, for each k = 1, 2, 3, the number μ_{k} is equal to the eigenvalue of M_{k}^{#} which corresponds to this invariant subspace. In conclusion, we obtain that the set of equivalence classes of the above triples can be identified with either ℙ^{1} ∪ {0} or with ℙ^{2} ∪ {0}. The single equivalence class containing the vector α = 0 consists of the triples which are “block-diagonalizable”.

Next we consider systems Y′(z̃) = A(z)Y (z̃) of standard form with indices κ_{1} ≥ κ_{2} ≥ κ_{3} for which A(z) can be written as

A(z) = \frac{1}{p(z)} ( \begin{matrix}κ_{1}z^{2} + p_{1}(z) & p_{12}(z) & p_{13}(z) \\ p_{21}(z) & κ_{2}z^{2} + p_{2}(z) & p_{23}(z) \\ 0 & 0 & κ_{3}z^{2} + p_{3}(z) \end{matrix} ) , (7.64)

where p(z) = (z − a_{1})(z − a_{2})(z − a_{3}), p_{k} ∈ P_{1}, p_{jk} ∈ P_{1+κ_{j}−κ_{k}}. If such a system is not equivalent to a system of the form (7.18), then it will be called a system of (2, 1)-block form. Note that necessarily κ_{2} − κ_{3} ≤ 1 because otherwise p_{32}(z) = 0 and the system is of the form (7.18).

The data which is associated to such systems is described in the following theorem.

Theorem 7.7 Let κ_{1}, κ_{2}, κ_{3} ∈ ℤ, κ_{1} ≥ κ_{2} ≥ κ_{3}, and assume κ_{1} − κ_{2} ≤ 1. Let [M_{1}, M_{2}, M_{3}] and [E_{1}, E_{2}, E_{3}] be admissible data of 3×3 matrices. Then the following two statements are equivalent:

(i) [M_{1}, M_{2}, M_{3}] and [E_{1}, E_{2}, E_{3}] is the data associated to a system of (2, 1)-block form with indices κ_{1}, κ_{2}, κ_{3}.

(ii) [M_{1}, M_{2}, M_{3}] and [E_{1}, E_{2}, E_{3}] is (2, 1)-quasi-block data, and N = κ_{1} + κ_{2}, ν = κ_{3}, where ν and N are defined by (7.60).

We remark that the cases of (1, 2)-block form systems and of (2, 1)-block form systems are not exclusive. They overlap exactly if κ_{1} = κ_{2} = κ_{3} and if the system is equivalent to a system of a “block-diagonal form”. This can be deduced from the argumentation in the proof of the main results given below.

7.4 The main results in the case n = m = 3

The main results for the case m = n = 3, i.e., the identification of the indices corresponding to given data is given in the following theorem. We have to resort to the classification of the reducibility type of [M_{1}, M_{2}, M_{3}] and the definition of the various integers κ, ν, N, n_{1}, n_{2}, n_{3}, ν_{1}, ν_{2}, ν_{#} stated at the beginning of this section. Recall that we have assumed n_{1} ≥ n_{2} in case (C-1), n_{2} ≥ n_{3} in case (C-2), and n_{1} ≥ n_{2} ≥ n_{3} in case (D) without lost of generality. In the other cases, the integers of interest were defined uniquely.

Theorem 7.8 Let [M_{1}, M_{2}, M_{3}] be admissible data of 3 × 3 matrices. Then in the following cases the indices κ_{1}, κ_{2}, κ_{3} ∈ ℤ, κ_{1} ≥ κ_{2} ≥ κ_{3}, are as follows.

(1) In cases (B-1) or (B-3) with 2ν ≥ N, the indices are κ_{1} = ν, κ_{2} = N/2, κ_{3} = N/2 if N is even, or,

κ_{1} = ν, κ_{2} = (N + 1)/2, κ_{3} = (N − 1)/2 if N is odd.

(2) In cases (B-2) or (B-3) with 2ν ≤ N, the indices are κ_{1} = N/2, κ_{2} = N/2, κ_{3} = ν if N is even, or,

κ_{1} = (N + 1)/2, κ_{2} = (N − 1)/2, κ_{3} = ν if N is odd.

(3) In case (C) with n_{1} ≥ n_{2} ≥ n_{3} or in case (C-1) with n_{2} ≥ n_{3} or in case (C-2)

with n_{1} ≥ n_{2} or in case (D), the indices are κ_{1} = n_{1}, κ_{2} = n_{2}, κ_{3} = n_{3}.

(4) In case (C-3) with ν_{1} ≥ ν_{2}, the indices are κ_{1} = ν_{1}, κ_{2} = ν_{2}, κ_{3} = ν_{#} if ν_{2} ≥ ν_{#}, or,

κ_{1} = ν_{1}, κ_{2} = ν_{#}, κ_{3} = ν_{2} if ν_{1} ≥ ν_{#} ≥ ν_{2}, or,
κ_{1} = ν_{#}, κ_{2} = ν_{1}, κ_{3} = ν_{2} if ν_{#} ≥ ν_{1}.

(5) In cases (C) or (C-1) with n_{2} < n_{3} and 2n_{1} ≥ n_{2} + n_{3}, the indices are κ_{1} = n_{1}, κ_{2} = (n_{2} + n_{3})/2, κ_{3} = (n_{2} + n_{3})/2 if n_{2} + n_{3} is even, or, κ_{1} = n_{1}, κ_{2} = (n_{2} + n_{3} + 1)/2, κ_{3} = (n_{2} + n_{3} − 1)/2 if n_{2} + n_{3} is odd.

(6) In cases (C) or (C-2) with n_{1} < n_{2} and n_{1} + n_{2} ≥ 2n_{3}, the indices are κ_{1} = (n_{1} + n_{2})/2, κ_{2} = (n_{1} + n_{2})/2, κ_{3} = n_{3} if n_{1} + n_{2} is even, or, κ_{1} = (n_{1} + n_{2} + 1)/2, κ_{2} = (n_{1} + n_{2} − 1)/2, κ_{3} = n_{3} if n_{1} + n_{2} is odd.

(7) In case (C-3) with ν_{1} < ν_{2} and 2ν_{#} ≥ ν_{1} + ν_{2}, the indices are κ_{1} = ν_{#}, κ_{2} = (ν_{1} + ν_{2})/2, κ_{3} = (ν_{1} + ν_{2})/2, if ν_{1} + ν_{2} is even, or, κ_{1} = ν_{#}, κ_{2} = (ν_{1} + ν_{2} + 1)/2, κ_{3} = (ν_{1} + ν_{2} − 1)/2, if ν_{1} + ν_{2} is odd.

(8) In case (C-3) with ν_{1} < ν_{2} and ν_{1} + ν_{2} ≥ 2ν_{#}, the indices are κ_{1} = (ν_{1} + ν_{2})/2, κ_{2} = (ν_{1} + ν_{2})/2, κ_{3} = ν_{#} if ν_{1} + ν_{2} is even, or, κ_{1} = (ν_{1} + ν_{2} + 1)/2, κ_{2} = (ν_{1} + ν_{2} − 1)/2, κ_{3} = ν_{#} if ν_{1} + ν_{2} is odd.

In the remaining cases, i.e.,

(9) in case (A),

(10) in case (B-1) with 2ν < N,

(11) in case (B-2) with 2ν > N,

(12) in case (C) with n_{1} + n_{2} < 2n_{3} and 2n_{1} < n_{2} + n_{3},

(13) in case (C-1) with 2n_{1} < n_{2} + n_{3},

(14) in case (C-2) with n_{1} + n_{2} < 2n_{3}, the indices are determined by the following statements:

(a) If κ ≡ 1 mod 3, then κ_{1} = (κ + 2)/3, κ_{2} = κ_{3} = (κ − 1)/3.

(b) If κ ≡ −1 mod 3, then κ_{1} = κ_{2} = (κ + 1)/3, κ_{3} = (κ − 2)/3.

(c) If κ ≡ 0 mod 3, then either κ_{1} = κ_{2} = κ_{3} = κ/3 or κ_{1} = κ/3 + 1, κ_{2} = κ/3, κ_{3} = κ/3 − 1.

(c*) If κ ≡ 0 mod 3, then the indices are κ_{1} = κ/3 + 1, κ_{2} = κ/3, κ_{3} = κ/3 − 1 if and only if [M_{1}, M_{2}, M_{3}] is the monodromy of some third order linear differential equation with Fuchsian singularities a_{1}, a_{2}, a_{3} and local exponents given by {ε_{k}^{(1)}, ε_{k}^{(2)}, ε_{k}^{(3)}} for z = a_{k}, k = 1, 2, and {ε_{3}^{(1)} + 1 − κ/3, ε_{k}^{(2)} + 1 − κ/3, ε_{k}^{(3)} + 1 − κ/3} for z = a_{3}, where ε_{k}^{(1)}, ε_{k}^{(2)}, ε_{k}^{(3)} are the eigenvalues of E_{k}, k = 1, 2, 3.

Proof. First of all, the reader can easily verify that the classification into the cases (1)–(14) is complete. Some of these cases do overlap (namely, case (B-3) with 2ν = N is stated in (1) and (2), case (C-3) with ν_{1} ≥ ν_{2} and ν_{1} = ν_{#} or ν_{2} = ν_{#} is stated multiply in (4), and case (C-3) with ν_{1} < ν_{2} and ν_{1} + ν_{2} = 2ν_{#} is given in both (7) and (8)), but in these cases, the description of the partial indices is the same. We remark in particular that the cases given in (9)–(14) are exactly the cases which are not contained in (1)– (8).

In Theorem 7.2 we have completely characterized the data which is associated to systems of the form (7.18). We obtain that given data is associated to such a system if and only if [M_{1}, M_{2}, M_{3}] is equivalent to a certain triple (7.7) such that for the corresponding integers n_{1}, n_{2}, n_{3} (which are defined by resorting to the properly numbered eigenvalues of E_{1}, E_{2}, E_{3}) the inequality n_{1} ≥ n_{2} ≥ n_{3} holds. This inequality follows from the statement that κ_{k} = n_{k} for k = 1, 2, 3, and from the ordering κ_{1} ≥ κ_{2} ≥ κ_{3}. We note that in some cases, there exist essentially different ways to establish an equivalence between [M_{1}, M_{2}, M_{3}] and a triangular triple (7.7), which in turn results into different definitions of n_{1}, n_{2}, n_{3}. The statement (i) of Theorem 7.2 is true if and only if such a relationship exists.

Examining the different reducibility types for a triple (7.7) under the condition n_{1} ≥ n_{2} ≥ n_{3} we arrive at the following cases. First of all, only the reducibility types (C)–(D) are covered. Case (C) is covered exactly for n_{1} ≥ n_{2} ≥ n_{3} as the enumeration of these integers is unique. Case (C-1) is covered exactly for n_{2} ≥ n_{3} as n_{3} is defined uniquely and n_{1} ≥ n_{2} without loss of generality. Similarly, case (C-2) is covered exactly for n_{1} ≥ n_{2} as n_{1} is unique and n_{2} ≥ n_{3} without loss of generality. The case (D) is obviously covered completely. So we arrive at the statement (3).

However, there are some other possibilities leading to case (C-3) and statement (4). In case (C-3), the integers ν_{1}, ν_{2}, ν_{3} are uniquely defined, but an equivalence with a triple (7.7) can be established in three different ways. The first way is if the triple has entries β_{k} = γ_{k} = 0. This gives n_{1} = ν_{1}, n_{2} = ν_{2}, n_{3} = ν_{#}, and hence the first subcase in (4). The second possibility is if the triple has entries α_{k} = β_{k} = 0. Here we obtain n_{1} = ν_{1}, n_{2} = ν_{#}, n_{3} = ν_{2}, which is the second subcase in (4). Finally, we may have α_{k} = γ_{k} = 0, which leads to n_{1} = ν_{#}, n_{2} = ν_{1}, n_{3} = ν_{2}, and the third subcase in (4).

We have now identified all the data associated to system (7.18) in terms of the classification established at the beginning of this section.

Next we do the same for the systems of (1, 2)-block form (see (7.48)). The data of such systems has been described in Theorem 7.5. With the numbers ν and N defined by (7.42) we obtain that the indices are κ_{1} = ν, κ_{2} = [(N + 1)/2], and κ_{3} = [N/2]. The latter follows from the fact that κ_{2} +κ_{3} = N and 0 ≤ κ_{2} −κ_{3} ≤ 1. The ordering κ_{1} ≥ κ_{2} ≥ κ_{3} can be rephrased by ν ≥ [(N + 1)/2], which is equivalent to 2ν ≥ N. We arrive at conclusion that the data of such systems is precisely the data which is of (1, 2)-quasi-block form and satisfies 2ν ≥ N.

Now we have to examine what does this mean in terms of the classification given at the beginning of this section. In the triple (7.41), there appears a triple [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] which along with [E_{1}^{#}, E_{2}^{#}, E_{3}^{#}] has to be quasi-block data. We have to distinguish the cases (I) and (II) stated in the definition of quasi-block data.

In case (I), the triple (7.41) possesses exactly one one-dimensional invariant subspace ℂ ⊕ {0} ⊕ {0} and no two-dimensional invariant subspace containing it. Hence the data [M_{1}, M_{2}, M_{3}] and [E_{1}, E_{2}, E_{3}] is precisely the data of reducibility type (B-1) or (B-3) with the condition 2ν ≥ N, where ν and N are now defined by (7.4). In fact, these numbers are the same as the former ones defined in (7.48). So we arrive at the statement (1).

In case (II), we may assume [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] to be of triangular form. We have N = n_{1}^{#} + n_{2}^{#} and n_{1}^{#} < n_{2}^{#} with the integers n_{1}^{#} and n_{2}^{#} defined in terms of the eigenvalues of [E_{1}^{#}, E_{2}^{#}, E_{3}^{#}]. Hence the data is given by triples (7.48) with 2ν ≥ n_{1}^{#} + n_{2}^{#} and n_{1}^{#} < n_{2}^{#}. These triples are of reducibility type (C), (C-1) or (C-3). We do not obtain (C-2) or (D) because of the non-diagonalizability of [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}]. In case of reducibility type (C), we have n_{1} = ν, n_{2} = n_{1}^{#} and n_{3} = n_{2}^{#}, which gives the condition 2n_{1} ≥ n_{2} + n_{3} and n_{2} < n_{3}. In case (C-1) we have the same identification, but in addition we may also have (interchanging n_{1} and n_{2}) n_{2} = ν, n_{1} = n_{1}^{#} and n_{3} = n_{2}^{#}. This would give the condition 2n_{2} ≥ n_{1} + n_{3} and n_{1} < n_{3}, which, however, conflicts with the assumption n_{1} ≥ n_{2}. So we arrive at the statement (5).

In case (C-3), the only possibility not contradicting the non-diagonalizability of [M_{1}^{#}, M_{2}^{#}, M_{3}^{#}] is if the triple [M_{1}, M_{2}, M_{3}] is equivalent to a triple (7.7) with α_{k} = γ_{k} = 0. This results in the identification ν_{#} = ν, ν_{1} = n_{1}^{#} and ν_{2} = n_{2}^{#}. Hence we obtain the conditions 2ν_{#} ≥ ν_{1} + ν_{2} and ν_{1} < ν_{2}, and this gives the statement (7).

So we have also settled the cases of data corresponding to systems of (1, 2)-block form.

The analysis of the data corresponding to systems of (2, 1)-block form (7.64) is completely analogous and will therefore be omitted. Here we arrive at the statements (2), (6) and (8) instead of (1), (5) and (7).

Finally, we have to examine the indices for the data in the remaining cases. We know that this data is associated to certain systems which are neither of (1, 2)-block form, (2, 1)-block form nor of triangular form.

It follows that κ_{1} − κ_{2} ≤ 1 because otherwise (by Theorem 4.3) the (2, 1)- and (3, 1)-entries of the matrix A(z) of the system vanish. Similarly, it follows that κ_{2} − κ_{3} ≤ 1. From the fact that κ_{1} ≥ κ_{2} ≥ κ_{3} and κ_{1} + κ_{2} + κ_{3} = κ, we immediately obtain the assertions (a), (b) and (c).

In order to prove (c*), we have to examine when the indices are such that κ_{1}−κ_{2} = κ_{2} − κ_{3} = 1. Again by Theorem 4.3, this is exactly the case if the system is of the form

A(z) = \frac{1}{p(z)} ( \begin{matrix}(κ/3 + 1)z^{2} + p_{1}(z) & p_{12}(z) & p_{13}(z) \\ α & κ/3z^{2} + p_{2}(z) & p_{23}(z) \\ 0 & β & (κ/3 − 1)z^{2} + p_{3}(z) \end{matrix} ) ,

where p(z) = (z − a_{1})(z − a_{2})(z − a_{3}), p_{k} ∈ P_{1}, p_{jk} ∈ P_{1+κ_{j}−κ_{k}} and α, β ∈ ℂ. If α = 0 or β = 0, then A(z) is of block-triangular or even triangular form, which has been excluded. Hence αβ ≠ 0. (In fact, if αβ ≠ 0, then A(z) is not equivalent to a system of triangular or block-triangular form.) Applying Theorem 5.4, we obtain the desired assertion (c*). □

In the cases where assumption (c) of the previous theorem applies, we are led to the monodromy problem for third order Fuchsian differential equations with three singular points. Here the monodromy does not depend on the location of the singularities, and neither do the corresponding indices. Again, as far as the authors know, no explicit answer exists to this monodromy problem. However, some particular cases can be tackled directly by help of the remark made in the last paragraph of Section 5. Namely, if one of the matrices M_{1}, M_{2} or M_{3} has the property that min(M_{k}) ≤ 2, then the indices are necessarily κ_{1} = κ_{2} = κ_{3} = κ/3. We note that in the cases (9)–(14), matrices [M_{1}, M_{2}, M_{3}] with this property really exist.

Acknowledgments. The authors want to thank A. A. Bolibruch and A. R. Its for fruitful discussions with one of them (T.E.) about various aspects of the monodromy and the Riemann-Hilbert problem during their stay at MSRI.

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