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z y = B(z) y, B(z) = diag [b1 (w)Is1 + N1 , . . . , bµ (w)Isµ + Nµ ], (8.5)

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

• 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
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 , . . . , eqµ (z) z Jµ ],

with z qm (z) = bm (w) ’ 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 ¤ j ¤ µ, 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 re¬‚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
(qκ (z), Jκ ) with qκ (z) = qm (ze2πi ) and Jm = Jκ . This κ 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.
ˆ ˆ
(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
ˆ ’1 ˆ
of any one of the FFS. Then F1 (z) F2 (z) is diagonally blocked of
type (s1 , . . . , sµ ) and constant.

Proof: To show existence of FFS, ¬rst assume that in¬nity is an al-
most regular-singular point. Then by de¬nition, 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
with a regular-singular point at in¬nity. According to the results of Sec-
tion 2, in particular Theorem 6 (p. 32),2 there exists another meromorphic
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

coe¬cient matrix B(z) ≡ 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 , . . . , Jµ ], 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 de¬nition that F (z) is a FFS, with data pairs (q(z), Jm ).
Observe that, according to Exercise 1 on p. 58, for dimension ν = 1 the
system (3.1) always is almost regular-singular. Hence we may now proceed
by induction with respect to ν, and we may assume that in¬nity is an
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 coe¬cient matrix B(z) having diagonal blocks Bm (z) of strictly
smaller dimension. By induction hypothesis, the systems with coe¬cient
matrices q ’1 Bm (wq ) have FFS; hence we obtain a FFS F2 (w) for the
transformed system as their direct sum. Setting F (z) = F1 (z) F2 (z 1/q ), it
ˆ ˆ ˆ
is not di¬cult to verify from the de¬nition 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 .
To show (a)“(c), conclude from the de¬nition that for any FFS F (z),
the matrix F (ze2πi ) is again a FFS whose data pairs are obtained from
those of the ¬rst one by continuation. Hence (b) follows from (a). Now, let
ˆ ’1
ˆ ˆ ˆ ˆ
F1 (z), F2 (z) be two FFS, and let T (z) = F1 (z) F2 (z). Observe that we
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
Fj (z), let
(j) (j) (j) (j)
Bj (z) = diag [b1 (w)Is(j) + N1 , . . . , bµ (w)Is(j) + Nµ ]

be as in (8.5). We then have that T (z) formally satis¬es (8.4). Blocking
rows, resp. colums, according to the block structure of B1 (z), resp. B2 (z),
ˆ ˆ
we obtain T (z) = [Tκm (z)], with not necessarily quadratic diagonal blocks.
From (8.4) we conclude, using Exercise 1 on p. 130, that Tκm (z) = 0 except
(1) (2) ˆ
when bκ (w) ≡ bm (w). Since T (z) is invertible, we conclude that to every
m there exists a κ = σ(m) so that this holds, and this κ even is unique, since
all bκ (w) are distinct. Hence, after a suitable permutation of the columns
of either one of the Fj (z), we may assume without loss of generality that
ˆ ˆ
σ(m) = m, i.e., T (z) is diagonally blocked. Using invertibility of T (z)
(1) (2)
again, we then ¬nd sm = sm , for every m. Moreover, the square diagonal
(1) ˆ (2)
ˆ ˆ
blocks satisfy q w(d/dw) Tmm (wq ) = Nm Tmm (wq ) ’ Tmm (wq ) Nm . From
(1) (2)
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
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
djm = deg(pj ’ pm ), if pj (z) ≡ pm (z), 1 ¤ j, m ¤ µ.
They form a ¬nite, possibly empty, set of rational numbers, which are all
positive, since by de¬nition of the pj their constant terms all vanish. We
write this set in the form {k1 > k2 > . . . > k > 0}, referring to it as
the level set of the FFS F (z). The integer ≥ 0 will be referred to as the
number of levels of F (z). In case ≥ 1, we shall call k1 the highest level
of F (z). From now on, consider data pairs with ≥ 1. For 1 ¤ µ ¤ , a
number d ∈ R will be called singular of level µ, provided that polynomials
pj , pm exist for which
pj (z) ’ pm (z) = pjm z kµ + lower powers, arg pjm = ’d kµ . (8.6)
In other words: exp[pj (z) ’ pm (z)] has degree kµ and maximal growth
rate along the ray arg z = d. Furthermore, we write j ∼ m of level µ,
if deg(pj ’ pm ) < kµ . This way, we obtain a number of equivalence rela-
tions that become ¬ner with increasing µ, 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 µ come
consecutively, for every µ = 1, . . . , . Note that we can always rearrange
the columns of the FFS F (z) and simultanuously permute its data pairs,
so that ¬rst polynomials whose indices are equivalent of level µ = 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 µ, there corresponds a block structure of arbitrary ν — ν
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
µth block structure, or the block structure of level µ. In particular, for µ = 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:
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 in¬nity is an almost
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 Poincar´ rank r of (3.1). Moreover, k1 = r holds if
and only if the leading term of (3.1) has several eigenvalues.

(c) Let in¬nity be an essentially irregular singularity of (3.1), so that
≥ 1. Then F (z) can be factored as
ˆ ˆ ˆ
F (z) = F1 (z) · . . . · F (z),
where each Fµ (z) is a kµ -summable q-meromorphic formal transfor-
mation, whose singular directions are among those given by (8.6). In
particular, F (z) is of Gevrey order not larger than s = 1/k , and
F1 (z) is an HLFFS of (3.1).
ˆ ˆ
(d) In case of ≥ 2, let G1 (z) · . . . · G (z) be another factorization of
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
ˆ ˆ
Fm (z) Tm+1 (z) = Tm (z) Gm (z), 1 ¤ m ¤ . (8.7)

(e) If the data pairs of F (z) are ordered, then one can ¬nd a factorization
as above, for which each factor Fj (z) is diagonally blocked in the block
structure of level j ’ 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 su¬ces 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
in¬nity is an almost regular-singular point, one can read o¬ 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),
ˆ ˆ ˆ ˆ
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 de¬nition of, resp. the algorithm to construct, an
HLFFS, one can read o¬ that the leading terms of these data pairs coincide
with what was introduced on p. 56 as the data pairs of the HLFFS F1 (z).
This implies that the data pairs of F (z) are ordered, and their highest-
level k1 coincides with the number 1/s de¬ned there. From Theorem 42 we
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)“(c), and (e). To
show (d), de¬ne Tm (z) by (8.7), and note that then
ˆ ’1 ˆ ’1
Fm (z) Tm+1 (z) G’1 (z) = Fm’1 (z) · . . . · F1 (z) G1 (z) · . . . · Gm’1 (z),
ˆ ˆm ˆ ˆ

for 1 ¤ m ¤ . 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/km’1 , while the left-hand side is km -summable,
so both sides converge according to Theorem 37 (p. 106). This, however,
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
di¬erently, before Ramis created the theory of k-summability.

Exercises: In the following exercises, consider a ¬xed formal system (3.3)
(p. 40) that is elementary.
1. Setting T0 = I, de¬ne Tn recursively by
’n Tn = n ≥ 1.
An+r’m Tm ,

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


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