a monodromy factor corresponding to a path encircling in¬nity, since such

a matrix can be expressed as a product, in a suitable order, of the other

ones. These matrices generate a group, called the monodromy group of the

system, but this is of no importance here.

Obviously, a Fuchsian system is given in terms of n ν 2 free parameters,

namely, the matrices Aj . The monodromy matrices carry the same number

of parameters, and therefore it makes sense to ask the following question:

12.4 Birkho¬™s Reduction Problem 189

• Given the points aj , the paths γj , and invertible matrices Cγj , are

there matrices Aj , so that the corresponding Fuchsian system has the

Cγj as its monodromy factors?

The above problem is usually referred to as the Riemann-Hilbert problem,

or Hilbert™s 21st problem. It was believed to have been positively solved by

Plemelj [221], but Bolibruch [58, 59, 63] in 1989 showed that the answer is

in fact negative by giving an explicit counterexample. For a discussion of

the history of the problem, and a presentation of related results, see the

book of Anosov and Bolibruch [2]. Very recently, new results have been

obtained for the same problem, but lower triangular monodromy matrices,

by Vandamme [273], based on earlier work of Bolibruch [60].

12.4 Birkho¬™s Reduction Problem

Let a system of ordinary di¬erential equations of the form (3.1) (p. 37) be

given. One classical question concerning such systems is that of the behavior

of its solutions as z ’ ∞. Since analytic transformations essentially leave

this behavior unchanged, Birkho¬ [53, 56] in 1913 suggested the following

approach:

• Within an equivalence class of such systems, with respect to analytic

transformations, determine the system(s) that in some sense are the

simplest, and then study their solutions near the point in¬nity. This

approach is very much analogous to the question of normal forms

of constant matrices with respect to similarity, leading to Jordan

canonical form.

Birkho¬ conjectured that for every system (3.1) one can always ¬nd an

analytic transformation x = T (z) y, such that the transformed system z y =

B(z) y, B(z) = T ’1 (z) [A(z) T (z) ’ z T (z)], has a polynomial B(z) as its

coe¬cient matrix. In his honor, we shall call every system with polynomial

coe¬cient matrix a system in Birkho¬ standard form.

Birkho¬ himself showed in [54] that the answer to his question is positive

under the additional assumption that some monodromy matrix, around

the point in¬nity, of (3.1) is diagonalizable, but seemed to believe that

the same would hold in general. However, in 1959 Gantmacher [105] and

Masani [188] independently presented examples of systems (3.1), in the

smallest nontrivial dimension of ν = 2, for which no such transformation

exists. These counterexamples had triangular coe¬cient matrices, hence

the following harder problem arose:

• Calling (3.1) reducible if an analytic transformation exists for which

the transformed system is lower triangularly blocked, with square

diagonal blocks of arbitrary dimensions, is it so that every irreducible

system can be analytically transformed to Birkho¬ standard form?

190 12. Other Related Questions

This question was answered positively, ¬rst for dimension ν = 2 by Jurkat,

Lutz, and Peyerimho¬ [147], then for ν = 3 in [15], and ¬nally for any

dimension by Bolibruch [61, 62, 64].

While the problem stated above concerns linear systems of ODE, all the

attempts on proving it for various special cases are based on a general

result on factorization of holomorphic matrices which was independently

obtained by Hilbert [119] and Plemelj [220], as well as Birkho¬ [56]:

Suppose that we are given a ν — ν matrix function S(z), holo-

morphic for |z| > ρ, for some ρ ≥ 0, whose determinant does

not vanish there. Then

S(z) = T (z) E(z) z K , (12.1)

with an analytic transformation T (z), an entire matrix function

E(z) whose determinant does not vanish for any z ∈ C , and a

diagonal matrix of integers K = diag [k1 , . . . , kν ].

For ν = 1 one can easily show this through an additive decomposition

of log S(z); however, the proof for ν ≥ 2 is much more involved and shall

not be given here. For a very readable presentation of the basic ideas of

Birkho¬™s proof, see Sibuya [251].

The above factorization result applies to systems of ODE as follows: Let

X(z) be a fundamental solution of (3.1) with monodromy matrix M , so

that S(z) = X(z) z ’M is single-valued for |z| > ρ. Then det S(z) cannot

vanish for these z, because of Proposition 1 (p. 6). With T (z) as in (12.1),

the transformation x = T (z) x takes (3.1) into a system z x = B(z) x, of

˜ ˜ ˜

˜

the same Poincar´ rank r, and having the fundamental solution X(z) =

e

KM

E(z) z z . So

B(z) = z X (z) X ’1 (z) = [zE (z) + E(z) {K + z K M z ’K }] E ’1 (z),

˜ ˜

showing that B(z) is holomorphic and single-valued in C , except for the

˜

origin. Moreover, we see from the form of X that the origin is a regular-

singular point of the new system, and will be of ¬rst kind if z K M z ’K is

a polynomial. This certainly holds whenever M is diagonal. Hence we have

proven the following result:

Theorem 60 Every system (3.1) is analytically equivalent to one being

singular only at in¬nity and at the origin, with the origin being regular-

singular. In case (3.1) has a fundamental solution with diagonal monodromy

matrix, then there exists an analytically equivalent system in Birkho¬ stan-

dard form.

In order to obtain Bolibruch™s result on irreducible systems, we show a

somewhat di¬erent factorization result:

12.4 Birkho¬™s Reduction Problem 191

Lemma 23 Suppose that we are given a ν — ν matrix function S(z), holo-

morphic for |z| > ρ, for some ρ ≥ 0, whose determinant does not vanish

there. Then

S(z) = T (z) z K E(z),

with an analytic transformation T (z), an entire matrix function E(z) whose

determinant does not vanish for any z ∈ C , and a diagonal matrix of

integers K = diag [k1 , . . . , kν ] satisfying k1 ≥ k2 ≥ . . . ≥ kν .

Proof: For the proof, we factor S(z) as in (12.1), and let F (z) = E(z) z K .

Then det F (z) = e(z) z k , k = k1 + . . . + kν , e(z) = det E(z), hence e(z) = 0

˜

for every z ∈ C . The rows of F (z) can be written as fj = ej (z) z kj , with

˜

˜

kj ∈ Z, ej (z) holomorphic at the origin, and ej (0) = 0. Without loss of

˜ ˜

˜j are weakly decreasing with respect to

generality, we may assume that k

j, since otherwise, we may permute the rows of F (z) and columns of T (z)

˜˜ ˜

accordingly. Note F (z) = z K E(z), with E(z) having rows ej (z). Hence,

˜

˜ ˜

det F (z) = z k e(z), and k ¤ k, e(z) = 0 for every z except possibly z = 0.

˜ ˜

˜ = k if and only if e(0) = 0, in which case the proof

Moreover, we have k ˜

˜ ˜

is completed. Suppose k < k. This occurs if and only if the rows of E(0)

are linearly dependent; however, note that no row vanishes, owing to the

˜

choice of kj . In this situation, we choose j ≥ 2 minimally, so that the jth

˜

row of E(0) is linearly dependent on the earlier ones. We now add multiples

of the th row of F (z) to the jth one, for 1 ¤ ¤ j ’ 1, the factor used

˜˜

being a constant times z kj ’k . This operation is nothing but multiplication

from the left with a special analytic transformation. Choosing the constants

properly, we can achieve that the new matrix, which for simplicity is again

˜˜ ˜

denoted by F (z), has the form z K E(z), where now the jth row of E(0)

¯¯ ¯¯

vanishes. Consequently, we factor F (z) as z K E(z), with K, F (z) as above,

¯ ¯ ˜

but k1 + . . . + kν > k. Repeating this ¬nitely many times, the proof can be

2

completed.

We now show Bolibruch™s result:

Theorem 61 Every irreducible system (3.1) is analytically equaivalent to

one in Birkho¬ standard form.

Proof: Choose a fundamental solution of (3.1) of the form X(z) = S(z) z J ,

with a monodromy matrix J in lower triangular Jordan form, and S(z)

single-valued and holomorphic for |z| > ρ. Let D = diag [d1 , . . . , dν ] have

integer diagonal elements with dj ’ dj+1 > r (ν ’ 1), and apply the above

lemma to S(z) z ’D to obtain X(z) = T (z) z K E(z) z D z J . For B(z) =

T ’1 (z) [A(z) T (z) ’ z T (z)], the system z y = B(z) y then has the funda-

mental solution Y (z) = z K E(z) z D z J . This implies

z ’K B(z) z K = K + [z E (z) + E(z) (D + z D J z ’D )] E ’1 (z).

192 12. Other Related Questions

Because of J lower triangular and dj decreasing, we ¬nd the right-hand

side to be holomorphic at the origin. If there was a j with kj ’ kj+1 > r,

then the fact that z y = B(z) y has Poincar´ rank r would imply B(z)

e

triangularly blocked. This, however, would contradict the irreducibility of

(3.1). Hence, kj ’ kj+1 ¤ r for 1 ¤ j ¤ ν ’ 1 follows. Using Exercise 2, we

˜

conclude Y (z) = T (z) Y (z), with Y (z) = E(z) z K+D z J , where E ± (z) are

¯ ˜ ˜ ¯ ¯

˜

entire, and K is equal to K but for a permutation of its diagonal elements.

˜

Owing to our choice of D, we ¬nd that the diagonal elements of K + D

are decreasing, so that z Y (z) Y ’1 (z) is holomorphic at the origin, i.e., in

˜ ˜

2

fact, is a polynomial.

As we pointed out above, the problem of Birkho¬ standard form arose

in the study of behavior of solutions of (3.1) for z ’ ∞. This behavior

is not too drastically altered even when using meromorphic transforma-

tions instead of analytic ones. So it is natural to ask whether transfor-

mation to Birkho¬ standard form is always possible using meromorphic

transformations. The answer to this question is positive, once we allow the

transformation to increase the Poincar´ rank of the system. However, it is

e

more natural to restrict to meromorphic transformations leaving the rank

the same. For dimensions ν = 2, resp. ν = 3, Jurkat, Lutz, and Peyer-

imho¬ [147], resp. Balser [14], have shown the answer to this question to

be positive. For general dimensions, but under the additional assumption

of the leading matrix A0 of (3.1) having distinct eigenvalues, Turrittin [271]

also obtained a positive answer, but in general this problem is still open.

For numerous su¬cient conditions under which a positive answer is known,

see Balser and Bolibruch [30].

Exercises: Let E(z) be an entire ν—ν matrix function with det E(0) = 0,

and let kj ∈ Z, k1 ≥ . . . ≥ kν be given.

˜

1. Show E(z) = P (z) E(z) R, with:

• P (z) = [pj ] is a lower triangular matrix with pjj (z) ≡ 1, and

pj (z) polynomials of degree at most k ’ kj , for 1 ¤ < j ¤ ν;

˜

• E(z) = [˜j (z)] is entire, with ej (z) vanishing at the origin at

e ˜

least of order k ’ kj + 1 for 1 ¤ < j ¤ ν;

• R is a permutation matrix.

˜

¯ ¯

2. With K = diag [k1 , . . . , kν ], show z K E(z) = T (z) E(z) z K , with an

¯ ¯ ¯

analytic transformation T (z), E(z) entire, det E(z) = 0 everywhere,

˜

and K a diagonal matrix of integers, di¬ering from K only by per-

mutation of its diagonal elements.

12.5 Central Connection Problems 193

12.5 Central Connection Problems

Generally speaking, a connection problem is concerned with two fundamen-

tal solutions X1 (z), X2 (z) of the same system of ODE, say, of the form (1.1)

(p. 2). The solutions may be given in terms of power series, or integrals, con-

verging in regions G1 , G2 ‚ G, and usually have certain natural properties

there. According to Theorem 1 (p. 4), both solutions Xj (z) can be holomor-

phically continued into all of G, and then are related as X1 (z) = X2 (z) „¦,

with a unique invertible constant matrix „¦. This matrix then is the cor-

responding connection matrix, and its computation is referred to as the

connection problem. For example, the system may be of the form (3.1)

(p. 37), and the fundamental solutions can be two consecutive highest-level

normal solutions Xj (z) and Xj’1 (z), which were introduced in Section 9.1

and are characterized through their Gevrey asymptotic in the correspond-

ing sectors Sj , Sj’1 . In this setting, the connection problem is the same

as the computation of the corresponding Stokes multiplier of highest level

and has been discussed in Chapter 9. While such problems sometimes are

called lateral connection problems, we shall here be concerned with another

type: Consider a system that is singular at two points z1 , z2 . Assume z1 to

be of ¬rst kind, or at least regular-singular; then a fundamental solution

∞

X1 (z) can be obtained as X1 (z) = 0 Sm (z ’z1 )m I+M , |z ’z1 | < ρ, with

a monodromy matrix M and matrix coe¬cients Sn that at least theoret-

ically can be computed from the system. Then the problem arises of how

this fundamental solution behaves as we approach the other singularity z2 .

To study this behavior is what we call the central connection problem.

Such problems, with z2 also being regular-singular, have been treated,

e.g., by Sch¨fke [238], resp. Sch¨fke and Schmidt [241, 253]. Here, we shall

a a

instead assume that the point z2 is irregular-singular. This situation, under

various additional assumptions, has been investigated by, among others,

Newell [202], Kazarino¬ and Kelvey [150], Knobloch [152], Wasow [280],

Kohno [156“159], Wyrwich [285, 286], Okubo [206], Naundorf [198“200],

Bakken [6, 7], Jurkat [143], Sch¨fke [237, 239, 240], Paris and Wood [216“

a

218], Paris [215], Lutz [175], Kovalevski [163, 164], Balser, Jurkat, and

Lutz [37, 41], Yokoyama [287“289], Balser [13], Lutz and Sch¨fke [178],

a

Okubo, Takano, and Yoshida [208], Sibuya [251], and Reuter [231, 232].

For numerical investigations and applications to problems in physics, see a

recent article by Lay and Slavyanov [166] and the literature quoted there.

Here, we shall study the central connection problem in the following