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z y = B(z) y, B(z) = diag [b1 (w)Is1 + N1 , . . . , bВµ (w)IsВµ + NВµ ], (8.5)

with w = z 1/q and B(z) in normalized Jordan canonical form, meaning
that
вЂў the bm (w) are distinct polynomials of degree at most r, with constant
Лњ
terms having a real part in the half-open interval [0, 1/q), and
132 8. Solutions of Highest Level

вЂў the Nm are nilpotent Jordan matrices whose blocks are ordered so
that they increase in size.
Therefore, we shall always assume that for a formal fundamental solution
Л†
F (z) the transformed system already is of the form (8.5). The system (8.5)
then has a unique fundamental solution of the form

Y (z) = diag [eq1 (z) z J1 , . . . , eqВµ (z) z JВµ ],

with z qm (z) = bm (w) в€’ bm (0), and Jm = bm (0)Ism + Nm . So each qm
is a polynomial in the qth root of z without constant term, and Jm a
constant matrix in Jordan form, with a single eigenvalue whose real part
is in [0, 1/q). In analogy to the phrase used for HLFFS, we call the pairs
(qj (z), Jj ), 1 в‰¤ j в‰¤ Вµ, the data pairs of the FFS. Recall that the bm (w)
were all distinct to see the same for the data pairs. Moreover, note that
the enumeration of the data pairs is always chosen to correspond to the
enumeration of the bm (w) in (8.5). In particular, a renumeration of the
data pairs reп¬‚ects in a suitable permutation of the columns of the FFS. As
for HLFFS, the data pairs of a FFS will be shown to be closed with respect
to continuation, meaning that to every pair (qm (z), Jm ) there exists a pair
(qОє (z), JОє ) with qОє (z) = qm (ze2ПЂi ) and Jm = JОє . This Оє even is uniquely
determined by m, because the data pairs are all distinct!
We now show existence of FFS. For simplicity, we restrict ourselves to
convergent systems (3.1); however, one can check that the same statements
remain correct for formal systems (3.3).
Proposition 20 Every system (3.1) (p. 37) has a FFS of the form de-
scribed above. Moreover, the following always holds:
(a) For any two FFS of (3.1), the data pairs coincide up to renumeration.
(b) The data pairs of any FFS are closed with respect to continuation.
Л† Л†
(c) For any two FFS F1 (z), F2 (z), assume that their data pairs coincide,
which can always be brought about by suitably permuting the columns
Л† в€’1 Л†
of any one of the FFS. Then F1 (z) F2 (z) is diagonally blocked of
type (s1 , . . . , sВµ ) and constant.

Proof: To show existence of FFS, п¬Ѓrst assume that inп¬Ѓnity is an al-
most regular-singular point. Then by deп¬Ѓnition, a terminating meromor-
phic transformation T (z) and a polynomial q(z) of degree at most r exist,
so that the combined transformation x = eq(z) T (z) x leads to a system
Лњ
with a regular-singular point at inп¬Ѓnity. According to the results of Sec-
tion 2, in particular Theorem 6 (p. 32),2 there exists another meromorphic
Лњ
transformation T (z), transforming this new system into one with constant

2 Use a change of variable z = 1/w to transfer the singularity to the origin.
8.4 Factorization of Formal Fundamental Solutions 133

coeп¬ѓcient matrix B(z) в‰Ў B0 , having eigenvalues with real parts in [0, 1).
Finally, a constant transformation may be used to put B0 into Jordan
canonical form J = diag [J1 , . . . , JВµ ], where each Jm contains all blocks
corresponding to one eigenvalue, those blocks being ordered according to
size. Since the scalar exponential shift eq(z) commutes with every other
transformation, this altogether shows existence of a meromorphic transfor-
mation F (z), transforming (3.1) into z y = B(z) y, with B(z) = z q (z)+J.
It follows from the deп¬Ѓnition that F (z) is a FFS, with data pairs (q(z), Jm ).
Observe that, according to Exercise 1 on p. 58, for dimension ОЅ = 1 the
system (3.1) always is almost regular-singular. Hence we may now proceed
by induction with respect to ОЅ, and we may assume that inп¬Ѓnity is an
Л†
essentially irregular singularity. Let then F1 (z) be a q-meromorphic HLFFS
of (3.1). This transformation takes (3.1) into a system with diagonally
blocked coeп¬ѓcient matrix B(z) having diagonal blocks Bm (z) of strictly
smaller dimension. By induction hypothesis, the systems with coeп¬ѓcient
matrices q в€’1 Bm (wq ) have FFS; hence we obtain a FFS F2 (w) for the
Л†
transformed system as their direct sum. Setting F (z) = F1 (z) F2 (z 1/q ), it
Л† Л† Л†
is not diп¬ѓcult to verify from the deп¬Ѓnition that this is a FFS of (3.1), with
data pairs obtained from those of F2 (w) by change of variable w = z 1/q .
Л†
Л†
To show (a)вЂ“(c), conclude from the deп¬Ѓnition that for any FFS F (z),
the matrix F (ze2ПЂi ) is again a FFS whose data pairs are obtained from
Л†
those of the п¬Ѓrst one by continuation. Hence (b) follows from (a). Now, let
Л† в€’1
Л† Л† Л† Л†
F1 (z), F2 (z) be two FFS, and let T (z) = F1 (z) F2 (z). Observe that we
Л†
may assume without loss in generality that both Fj (z) are q-meromorphic,
with q independent of j, and then set z = wq . Corresponding to each FFS
Л†
Fj (z), let
(j) (j) (j) (j)
Bj (z) = diag [b1 (w)Is(j) + N1 , . . . , bВµ (w)Is(j) + NВµ ]
Вµ
1

Л†
be as in (8.5). We then have that T (z) formally satisп¬Ѓes (8.4). Blocking
rows, resp. colums, according to the block structure of B1 (z), resp. B2 (z),
Л† Л†
we obtain T (z) = [TОєm (z)], with not necessarily quadratic diagonal blocks.
From (8.4) we conclude, using Exercise 1 on p. 130, that TОєm (z) = 0 except
(1) (2) Л†
when bОє (w) в‰Ў bm (w). Since T (z) is invertible, we conclude that to every
m there exists a Оє = Пѓ(m) so that this holds, and this Оє even is unique, since
(1)
all bОє (w) are distinct. Hence, after a suitable permutation of the columns
Л†
of either one of the Fj (z), we may assume without loss of generality that
Л† Л†
Пѓ(m) = m, i.e., T (z) is diagonally blocked. Using invertibility of T (z)
(1) (2)
again, we then п¬Ѓnd sm = sm , for every m. Moreover, the square diagonal
(1) Л† (2)
Л† Л†
blocks satisfy q w(d/dw) Tmm (wq ) = Nm Tmm (wq ) в€’ Tmm (wq ) Nm . From
(1) (2)
Л†
Exercise 6 we conclude that Tmm (z) is constant, so Nm and Nm are
similar. Owing to our assumption that their blocks are ordered according
to size, they must in fact be equal. This observation completes the proof
2
of statements (a) and (c).
134 8. Solutions of Highest Level

The above result shows the existence, and describes the degree of free-
dom, of FFS. To describe their summability properties, it is necessary to
introduce some notation: For an arbitrary nonzero polynomial p(z) in some
root of z, let deg p stand for the rational exponent of the highest power of z
occurring in p(z). Given the data pairs of a FFS, we introduce the rational
numbers
djm = deg(pj в€’ pm ), if pj (z) в‰Ў pm (z), 1 в‰¤ j, m в‰¤ Вµ.
They form a п¬Ѓnite, possibly empty, set of rational numbers, which are all
positive, since by deп¬Ѓnition of the pj their constant terms all vanish. We
write this set in the form {k1 > k2 > . . . > k > 0}, referring to it as
Л†
the level set of the FFS F (z). The integer в‰Ґ 0 will be referred to as the
Л†
number of levels of F (z). In case в‰Ґ 1, we shall call k1 the highest level
Л†
of F (z). From now on, consider data pairs with в‰Ґ 1. For 1 в‰¤ Вµ в‰¤ , a
number d в€€ R will be called singular of level Вµ, provided that polynomials
pj , pm exist for which
pj (z) в€’ pm (z) = pjm z kВµ + lower powers, arg pjm = в€’d kВµ . (8.6)
In other words: exp[pj (z) в€’ pm (z)] has degree kВµ and maximal growth
rate along the ray arg z = d. Furthermore, we write j в€ј m of level Вµ,
if deg(pj в€’ pm ) < kВµ . This way, we obtain a number of equivalence rela-
tions that become п¬Ѓner with increasing Вµ, in the sense that fewer numbers
m are equivalent. We now say that the data pairs (qm (z), Jm ) are ordered
provided that polynomials with indices that are equivalent of level Вµ come
consecutively, for every Вµ = 1, . . . , . Note that we can always rearrange
Л†
the columns of the FFS F (z) and simultanuously permute its data pairs,
so that п¬Ѓrst polynomials whose indices are equivalent of level Вµ = 1 come
consecutively, then further permute pairs within each equivalence class of
level 1 to make polynomials come together whenever their indices are equiv-
alent of level 2, and so on. In other words, every system (3.1) has a FFS
with ordered data pairs.
Consider now an ordered set of data pairs. Then to each equivalence
relation of level Вµ, there corresponds a block structure of arbitrary ОЅ Г— ОЅ
matrices, by grouping their rows and columns together once their indices
are equivalent of that level. This block structure will be referred to as the
Вµth block structure, or the block structure of level Вµ. In particular, for Вµ = 1
we speak of the block structure of highest level. For convenience, we also
introduce the trivial block structure of level zero, where we do not subdivide
matrices at all. These block structures were studied in [34, 35] under the
name iterated block structure.
Using this terminology, we can now formulate the following basic result
on FFS:
Л†
Theorem 43 For an arbitrary FFS F (z) of a system (3.1) (p. 37), the
following holds:
8.4 Factorization of Formal Fundamental Solutions 135

(a) The FFS has empty level set if and only if inп¬Ѓnity is an almost
Л†
regular-singular point of (3.1), and in this case F (z) converges.

(b) If the number of its levels is positive, then its highest level is not
larger than the PoincarВґ rank r of (3.1). Moreover, k1 = r holds if
e
and only if the leading term of (3.1) has several eigenvalues.

(c) Let inп¬Ѓnity be an essentially irregular singularity of (3.1), so that
Л†
в‰Ґ 1. Then F (z) can be factored as
Л† Л† Л†
F (z) = F1 (z) В· . . . В· F (z),
Л†
where each FВµ (z) is a kВµ -summable q-meromorphic formal transfor-
mation, whose singular directions are among those given by (8.6). In
Л†
particular, F (z) is of Gevrey order not larger than s = 1/k , and
Л†
F1 (z) is an HLFFS of (3.1).
Л† Л†
(d) In case of в‰Ґ 2, let G1 (z) В· . . . В· G (z) be another factorization of
Л†
F (z) as described in (c). Then there exist convergent meromorphic
transformations T2 (z), . . . , T (z) so that with T1 (z) = T +1 (z) = I we
have
Л† Л†
Fm (z) Tm+1 (z) = Tm (z) Gm (z), 1 в‰¤ m в‰¤ . (8.7)

Л†
(e) If the data pairs of F (z) are ordered, then one can п¬Ѓnd a factorization
Л†
as above, for which each factor Fj (z) is diagonally blocked in the block
structure of level j в€’ 1.

Proof: First, observe that if the theorem holds for some FFS, then it is
correct for every other FFS, owing to parts (a) and (c) of Proposition 20.
So it suп¬ѓces to construct a particular FFS. To do so, follow the same
steps, and use the same notation, as in the proof of Proposition 20. If
inп¬Ѓnity is an almost regular-singular point, one can read oп¬Ђ from there
existence of a convergent FFS with empty level set. In the other case, an
FFS exists having the form F (z) = F1 (z) F2 (z 1/q ), with an HLFFS F1 (z),
Л† Л† Л† Л†
Л†
while F2 (w) is the direct sum of at least two blocks. Each of these blocks
is an FFS of strictly smaller systems, and by induction hypothesis we can
assume that the theorem holds for those. In particular, we may assume
that the data pairs of the blocks are ordered and a factorization as in (e)
exists. Making the change of variable w = z 1/q , and then combining the
data pairs of the diagonal blocks in their natural order, gives the data pairs
for the FFS. From the deп¬Ѓnition of, resp. the algorithm to construct, an
HLFFS, one can read oп¬Ђ that the leading terms of these data pairs coincide
Л†
with what was introduced on p. 56 as the data pairs of the HLFFS F1 (z).
Л†
This implies that the data pairs of F (z) are ordered, and their highest-
level k1 coincides with the number 1/s deп¬Ѓned there. From Theorem 42 we
Л†
then conclude that F1 (z) is k1 -summable, with singular directions that are
136 8. Solutions of Highest Level

among those introduced above as being singular of level 1. This, together
with the induction hypothesis, completes the proof of (a)вЂ“(c), and (e). To
show (d), deп¬Ѓne Tm (z) by (8.7), and note that then
Л† в€’1 Л† в€’1
Fm (z) Tm+1 (z) Gв€’1 (z) = Fmв€’1 (z) В· . . . В· F1 (z) G1 (z) В· . . . В· Gmв€’1 (z),
Л† Л†m Л† Л†

for 1 в‰¤ m в‰¤ . Assume that we have shown Tm+1 (z) to converge, which
holds trivially for m = . Then, the right-hand side of this identity is of
Gevrey order at most 1/kmв€’1 , while the left-hand side is km -summable,
so both sides converge according to Theorem 37 (p. 106). This, however,
2
implies convergence of Tm (z).
The above theorem shows that FFS with more than one level cannot
be k-summable for any k > 0, because of Theorem 37 (p. 106). Moreover,
if only one level occurs, the notions of FFS and HLFFS coincide. In this
situation, the summability of FFS was essentially known, but formulated
diп¬Ђerently, before Ramis created the theory of k-summability.

Exercises: In the following exercises, consider a п¬Ѓxed formal system (3.3)
(p. 40) that is elementary.
1. Setting T0 = I, deп¬Ѓne Tn recursively by
nв€’1
в€’n Tn = n в‰Ґ 1.
An+rв€’m Tm ,
m=0

Show that all Tn commute with one another as well as with all Am ,
в€ћ
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