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O(z rв€’n0 в€’1 ), 1 в‰¤ k в‰¤ Вµв€’1, or we have a shearing transformation producing
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 в€’n0 . Then we can п¬Ѓnd an unramiп¬Ѓed shearing
transformation producing a system with a superior leading term, except
when the coeп¬ѓcients An , 1 в‰¤ n в‰¤ n0 , are all diagonally blocked in the
block structure of A0 , i.e., when (3.3) is reduced up to z в€’n0 .

Proof: For Вµ = 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 в‰¤ n в‰¤ 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 NВµ . It is not diп¬ѓcult
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 coeп¬ѓcients are diagonally blocked
in the same coarser block structure. Repeating the same arguments for the
2
п¬Ѓrst 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 в€’n0 , i.e., for all n =
1, . . . , n0 and 1 в‰¤ j, k в‰¤ Вµ:
пЈ® (j,1) (j,s ) пЈ№
(j,2)
. . . an j
an an
пЈЇ0 пЈє
0 ... 0
пЈЇ пЈє
(jj) (jk)
An = пЈЇ . пЈє , An = 0 (j = k). (3.11)
. .
пЈ°. пЈ»
. .
. . .
0 0 ... 0
(jk)
Note that in case Вµ = 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 п¬Ѓxed value n0 , but keep in mind that we might have
arranged n0 в‰Ґ ОЅ, 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 п¬Ѓnitely many times. Hence after a п¬Ѓnite number of
steps we arrive at a system where, when normalizing it up to some power
z в€’n0 , it automatically satisп¬Ѓes (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 в€’n0 , and consider the п¬Ѓnite set of
rational numbers of the form p/q, 1 в‰¤ p в‰¤ q в‰¤ 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), . . . , TВµ (z)],
diag [1, z p/q , z 2p/q , . . . , z (sj в€’1)p/q ]
Tj (z) = (3.12)
then produces a ramiп¬Ѓed system (except for p = q = 1), whose coeп¬ѓcient
Л†
matrix may be denoted by Bp/q (z). The elements of this matrix in those
positions corresponding to 1вЂ™s in the matrix A0 then are formal Laurent
series in z в€’1/q beginning with the term z rв€’p/q . Elements of Bp/q (z) in
Л†
other positions may or may not involve higher rational powers of z вЂ“ 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
Л†
Bp/q (z) may be written in the form
в€ћ
Bn z в€’n/q ,
Л† Л†
Bp/q (z) = B(z) = z r (3.13)
n=p

If p/q = 1 (hence p = q = 1), then the transformed equation is unramiп¬Ѓed
and of PoincarВґ rank r в€’ 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 coeп¬ѓcients 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
eв€’2nПЂi/q Bn = Dв€’1 Bn D, n в‰Ґ p, (3.14)
with D = T (e2ПЂi ) вЂ“ 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 e2ПЂi/q , i.e. О» is an eigenvalue of Bp if and
only if О»e2ПЂi/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 в€’1 (z) [A(z) T (z) в€’ zT (z)]
Л† Л†
and T (ze2ПЂi ) = T (z) D that B(ze2ПЂi ) = Dв€’1 B(z) D, from which follows
Л† Л†
(3.14). This then implies det(Bp в€’ xI) = e2ПЂiОЅp/q det(Bp в€’ eв€’2ПЂip/q xI),
showing that the spectrum of Bp is closed with respect to multiplication
with eв€’2ПЂip/q , and since p and q are assumed to be co-prime, the same
follows for e2ПЂi/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 в‰Ґ ОЅ, then for n в‰¤ ОЅ the coeп¬ѓcients 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 п¬Ѓrst 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 PoincarВґ rank smaller than r.
e
(b) There exist a q в€€ 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 e2ПЂi/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 PoincarВґ rank of (3.3) is zero, or if the dimension ОЅ equals one,
e
stop. If not, п¬Ѓnd 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 в€’n0 .
(a) If (3.11) does not hold, use an unramiп¬Ѓed 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, п¬Ѓnd the maximal admissible value p/q (в‰¤ 1).
For p/q = 1 use the shearing transformation (3.12) to go to
a system of smaller PoincarВґ rank, and continue with step 1;
e
otherwise, use the ramiп¬Ѓed 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 п¬Ѓnitely many steps. It produces
п¬Ѓnitely many transformations that we may combine into a single one de-
Лњ
noted by T (z), and a resulting system
w = z 1/q , q в€€ N.
Лњ
wЛњ = A(w) x,
x Лњ (3.15)
As can be read oп¬Ђ from the results in this sections, the following two es-
sentially diп¬Ђerent cases occur:
1. The system (3.15) has PoincarВґ rank r = 0; in this case all transfor-
e Лњ
Лњ
mations used have been unramiп¬Ѓed (hence q = 1), and T (z) has the
form
Лњ
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 PoincarВґ rank r в‰Ґ 1; in this case r and q are
e Лњ Лњ
в€ћЛњ
Лњ в€’n
Лњ r
co-prime, the leading term of A(w) = w 0 An w has several
Лњ
eigenvalues, and T (z) has the form
Лњ
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 ramiп¬Ѓed shearing transformation
Лњ
T (z) as in (3.12). Moreover, the coeп¬ѓcients An satisfy
eв€’2(nв€’Лњ)ПЂi/q An = T в€’1 (e2ПЂi ) An T (e2ПЂi ),
Лњ Лњ n в‰Ґ 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/Лњ. 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 deп¬Ѓne 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 п¬Ѓrst 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 diп¬Ђerent from the ones named so before.

2. For the simplest nontrivial situation of ОЅ = 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, Оґ > 0 so that
rв€’Оґ
X(z) в‰¤ c ea|z| z в€€ S.
,

3.4 Transformation to Rational Form
Given a system (3.1), then in case (b) of Theorem 10 (p. 50) we can п¬Ѓnd 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 п¬Ѓnd a formal analytic transfor-
mation of Gevrey order s, with 1/s being the PoincarВґ 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
вЂ“ in fact, we shall show that the transformed system can be such that only
п¬Ѓnitely many of its coeп¬ѓcients are nonzero, hence is a rational function
with poles (at most) at inп¬Ѓnity and the origin.

Theorem 11 Let (3.3) be a formal system of Gevrey order s = 1/r. Then
for every suп¬ѓciently large N, M в€€ N, a formal analytic transformation of
в€ћ
Gevrey order s exists, which is of the form T (z) = I + n=N Tn z в€’n , so
Л†
that the transformed system has a coeп¬ѓcient matrix of the form
N +M
Bn z в€’n .
r
B(z) = z
n=0
3.4 Transformation to Rational Form 53

Л†
Proof: For the moment, assume that we had found T (z) and B(z) as
desired. Inserting formal expansions into (3.5) and equating coeп¬ѓcients
then shows Bn = An , for 0 в‰¤ n в‰¤ N в€’ 1, and
n
в€’(n в€’ r)Tnв€’r = An в€’ Bn + (Anв€’m Tm в€’ Tm Bnв€’m ), (3.16)
m=N

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