a system with a superior leading term. Much more can be said, however:

Theorem 9 Let (3.3) be a formal system with a leading term of the form

(3.9), normalized up to z ’n0 . Then we can ¬nd an unrami¬ed shearing

transformation producing a system with a superior leading term, except

when the coe¬cients An , 1 ¤ n ¤ n0 , are all diagonally blocked in the

block structure of A0 , i.e., when (3.3) is reduced up to z ’n0 .

Proof: For µ = 1 nothing remains to prove, so suppose otherwise. Ap-

plying Proposition 6 shows the existence of a shearing transformation as

required, except when all An , 1 ¤ n ¤ n0 , are lower triangularly blocked,

not necessarily in the block structure of A0 , but the coarser one with two

diagonal blocks, the second one of the same size as Nµ . It is not di¬cult

to see that an analogue to Proposition 6, with a shearing transformation

inverse to the one used there, produces a system with a superior leading

term, except when the above-mentioned coe¬cients are diagonally blocked

in the same coarser block structure. Repeating the same arguments for the

2

¬rst diagonal block then completes the proof.

According to the above results, we are now left to deal with a formal

system (3.3) with a leading term as in (3.9), and for some n0 to be chosen

later we may assume that (3.3) is reduced up to z ’n0 , i.e., for all n =

1, . . . , n0 and 1 ¤ j, k ¤ µ:

® (j,1) (j,s )

(j,2)

. . . an j

an an

0

0 ... 0

(jj) (jk)

An = . , An = 0 (j = k). (3.11)

. .

°. »

. .

. . .

0 0 ... 0

(jk)

Note that in case µ = 1, the condition on An for j = k becomes void!

Remark 5: On one hand, for theoretical arguments we may assume with-

out loss in generality that the number n0 introduced above is as large as we

need it to be; however, for more computational purposes it is convenient

to keep it as small as possible. So in the discussion to follow, we shall con-

sider an arbitrarily ¬xed value n0 , but keep in mind that we might have

arranged n0 ≥ ν, because if not, then we may apply Theorem 9 to obtain

a system with larger n0 , or one with superior leading term, and the latter

3.3 Nilpotent Leading Term 49

case can occur only ¬nitely many times. Hence after a ¬nite number of

steps we arrive at a system where, when normalizing it up to some power

z ’n0 , it automatically satis¬es (3.11). Also recall from Remark 4 (p. 47)

that increasing n0 can be done without changing the leading terms of the

3

system.

To proceed, let (3.3) be reduced up to z ’n0 , and consider the ¬nite set of

rational numbers of the form p/q, 1 ¤ p ¤ q ¤ s1 (recall that by assumption

s1 , the size of N1 , is at least as large as the sizes sj of the other blocks Nj

of A0 ). For each such p/q, assume from now on that we have chosen p and

q to be co-prime, i.e., having no nontrivial common divisor. The shearing

transformation

T (z) = diag [T1 (z), . . . , Tµ (z)],

diag [1, z p/q , z 2p/q , . . . , z (sj ’1)p/q ]

Tj (z) = (3.12)

then produces a rami¬ed system (except for p = q = 1), whose coe¬cient

ˆ

matrix may be denoted by Bp/q (z). The elements of this matrix in those

positions corresponding to 1™s in the matrix A0 then are formal Laurent

series in z ’1/q beginning with the term z r’p/q . Elements of Bp/q (z) in

ˆ

other positions may or may not involve higher rational powers of z “ if

they do, then we discard the corresponding rational p/q as inadmissible.

This may eliminate many values of p/q, but we observe that at least the

smallest possible value p/q = 1/s1 remains admissible. If several admissible

values remain, we take the largest one. For this admissible p/q, the matrix

ˆ

Bp/q (z) may be written in the form

∞

Bn z ’n/q ,

ˆ ˆ

Bp/q (z) = B(z) = z r (3.13)

n=p

If p/q = 1 (hence p = q = 1), then the transformed equation is unrami¬ed

and of Poincar´ rank r ’ 1, so we consider this an improvement, and then

e

apply to the resulting system the same arguments as above (depending

upon its new leading term having several distinct eigenvalues or not). If

this is not so, we show that the leading term Bp either is nilpotent or has

several distinct eigenvalues:

Lemma 5 Let a system (3.3) with coe¬cients as in (3.11) be given, let p/q

be determined as above, and let the transformed equation be written as in

(3.13). Then we have

e’2nπi/q Bn = D’1 Bn D, n ≥ p, (3.14)

with D = T (e2πi ) “ note that this is not the identity matrix except for

q = 1. In particular, the spectrum of the leading term Bp is closed with

respect to multiplication with e2πi/q , i.e. » is an eigenvalue of Bp if and

only if »e2πi/q is one, too. Hence, if p/q < 1, then Bp either is nilpotent or

has more than one eigenvalue.

50 3. Highest-Level Formal Solutions

Proof: It can be concluded from B(z) = T ’1 (z) [A(z) T (z) ’ zT (z)]

ˆ ˆ

and T (ze2πi ) = T (z) D that B(ze2πi ) = D’1 B(z) D, from which follows

ˆ ˆ

(3.14). This then implies det(Bp ’ xI) = e2πiνp/q det(Bp ’ e’2πip/q xI),

showing that the spectrum of Bp is closed with respect to multiplication

with e’2πip/q , and since p and q are assumed to be co-prime, the same

follows for e2πi/q . 2

Hence, according to the above lemma, the only case where we did not

make any progress is when Bp , for the maximal admissible value p/q, is

nilpotent. This case, however, cannot occur when the value n0 has been

large enough: Imagine that n0 ≥ ν, then for n ¤ ν the coe¬cients An are

diagonally blocked. In this case either 1 is admissible, or for the maximal

admissible value p/q < 1 we have a leading term Bp which is a direct sum

of matrices of the form Nj + Cj , with Nj as in (3.9), and Cj having zero

rows except for possibly the ¬rst one. According to maximality of p/q, not

all Cj can be the zero matrix, hence Bp cannot be nilpotent, as follows

from Exercise 1.

We summarize the result of the preceding discussion as follows:

Theorem 10 Given a formal system (3.3) with a nilpotent leading term,

then one of the following two cases occurs:

(a) There exists a terminating meromorphic transformation T (z) so that

the transformed system has Poincar´ rank smaller than r.

e

(b) There exist a q ∈ N, not larger than the order of nilpotency of the

leading term A0 , and a terminating q-meromorphic transformation

T (z), so that the transformed system has a non-nilpotent leading term

whose spectrum is closed under multiplication by e2πi/q .

The minimal value for q and a transformation T (z) can in both cases be

found in an algorithmic manner following the steps described below.

What we have shown so far can be summarized in algorithmic form: Let a

system (3.1), or more generally an arbitrary formal system (3.3), be given:

1. If the Poincar´ rank of (3.3) is zero, or if the dimension ν equals one,

e

stop. If not, ¬nd the number of distinct eigenvalues of the leading

term A0 ; if it turns out to be larger than one, stop. Otherwise, use

a scalar exponential shift to go to a system with a nilpotent leading

term and continue with step 2, taking the parameter n0 = 1.

2. Normalize the system up to z ’n0 .

(a) If (3.11) does not hold, use an unrami¬ed shearing transforma-

tion as in Proposition 6 (p. 47), resp. Theorem 9 to go to a

system with superior leading term, and continue with step 2

and value n0 = 1.

3.3 Nilpotent Leading Term 51

(b) If (3.11) holds, ¬nd the maximal admissible value p/q (¤ 1).

For p/q = 1 use the shearing transformation (3.12) to go to

a system of smaller Poincar´ rank, and continue with step 1;

e

otherwise, use the rami¬ed shearing transformation (3.12) and

a change of variable z = wq to go to a system whose leading

term either has several eigenvalues (in which case we accept the

transformed system and stop) or is nilpotent; in this case we

discard the transformed system and return to the previous one,

increase n0 by one and continue with step 2.

The above algorithm terminates after ¬nitely many steps. It produces

¬nitely many transformations that we may combine into a single one de-

˜

noted by T (z), and a resulting system

w = z 1/q , q ∈ N.

˜

w˜ = A(w) x,

x ˜ (3.15)

As can be read o¬ from the results in this sections, the following two es-

sentially di¬erent cases occur:

1. The system (3.15) has Poincar´ rank r = 0; in this case all transfor-

e ˜

˜

mations used have been unrami¬ed (hence q = 1), and T (z) has the

form

˜

T (z) = eq(z) T1 (z),

with a polynomial q(z) of degree at most r and a terminating mero-

morphic transformation T1 (z).

2. The system (3.15) has Poincar´ rank r ≥ 1; in this case r and q are

e ˜ ˜

∞˜

˜ ’n

˜ r

co-prime, the leading term of A(w) = w 0 An w has several

˜

eigenvalues, and T (z) has the form

˜

T (z) = eq(z) T1 (z) T (z),

with a polynomial q(z) of degree at most r, a terminating meromor-

phic transformation T1 (z), and a rami¬ed shearing transformation

˜

T (z) as in (3.12). Moreover, the coe¬cients An satisfy

e’2(n’˜)πi/q An = T ’1 (e2πi ) An T (e2πi ),

˜ ˜ n ≥ 0.

r

Now, consider a convergent system (3.1) instead of a formal one. Then

the system (3.15) will also converge. If the second one of the above two

cases occurs, we may then apply the Splitting Lemma to (3.15). Doing so,

we obtain a diagonally blocked system, whose blocks are formal of Gevrey

order 1/˜. We shall show in the next section that this formal system can

r

then be transformed into a convergent one, using a diagonally blocked

formal analytic transformation of the same Gevrey order. Combining all

the transformations used into a single one, we then have obtained what we

shall de¬ne in the following section as a highest-level formal fundamental

solution.

52 3. Highest-Level Formal Solutions

Exercises:

1. For a nilpotent Jordan block N and a constant matrix C with nonzero

entries in the ¬rst row only, compute the characteristic polynomial of

A = N +C and show that A is nilpotent if and only if C = 0. Matrices

of this form will also be called companion matrices, although they are

slightly di¬erent from the ones named so before.

2. For the simplest nontrivial situation of ν = 2, verify Theorem 10

(p. 50) and give explicit conditions under which each case occurs.

3. For a system (3.3) with a nilpotent leading term, give examples show-

ing that the value for q in Theorem 10 (p. 50) can be any natural

number strictly smaller than the order of nilpotency of A0 .

4. For a system (3.1) with nilpotent leading term, let X(z) be an arbi-

trary fundamental solution of (3.1). For S as in Exercise 4 on p. 15,

show existence of c, a, δ > 0 so that

r’δ

X(z) ¤ c ea|z| z ∈ S.

,

3.4 Transformation to Rational Form

Given a system (3.1), then in case (b) of Theorem 10 (p. 50) we can ¬nd a

transformation which, after a change of variable, leads to a system of the

same kind, but with a leading term having several eigenvalues. In this case,

according to the Splitting Lemma, we can ¬nd a formal analytic transfor-

mation of Gevrey order s, with 1/s being the Poincar´ rank of the new

e

system, which splits this system into several smaller blocks. To continue,

we wish to show that these formal systems may always be transformed into

convergent ones, using formal analytic transformations of Gevrey order s

“ in fact, we shall show that the transformed system can be such that only

¬nitely many of its coe¬cients are nonzero, hence is a rational function

with poles (at most) at in¬nity and the origin.

Theorem 11 Let (3.3) be a formal system of Gevrey order s = 1/r. Then

for every su¬ciently large N, M ∈ N, a formal analytic transformation of

∞

Gevrey order s exists, which is of the form T (z) = I + n=N Tn z ’n , so

ˆ

that the transformed system has a coe¬cient matrix of the form

N +M

Bn z ’n .

r

B(z) = z

n=0

3.4 Transformation to Rational Form 53

ˆ

Proof: For the moment, assume that we had found T (z) and B(z) as

desired. Inserting formal expansions into (3.5) and equating coe¬cients

then shows Bn = An , for 0 ¤ n ¤ N ’ 1, and

n

’(n ’ r)Tn’r = An ’ Bn + (An’m Tm ’ Tm Bn’m ), (3.16)

m=N