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

that

• 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

ˆ

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

ˆ

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

ˆ

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 .

ˆ

ˆ

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

ˆ

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

ˆ

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

ˆ

Fj (z), let

(j) (j) (j) (j)

Bj (z) = diag [b1 (w)Is(j) + N1 , . . . , bµ (w)Is(j) + Nµ ]

µ

1

ˆ

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

(1)

all bκ (w) are distinct. Hence, after a suitable permutation of the columns

ˆ

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)

ˆ

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 ’ 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

ˆ

the level set of the FFS F (z). The integer ≥ 0 will be referred to as the

ˆ

number of levels of F (z). In case ≥ 1, we shall call k1 the highest level

ˆ

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

ˆ

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:

ˆ

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

ˆ

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

e

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

ˆ

≥ 1. Then F (z) can be factored as

ˆ ˆ ˆ

F (z) = F1 (z) · . . . · F (z),

ˆ

where each Fµ (z) is a kµ -summable q-meromorphic formal transfor-

mation, whose singular directions are among those given by (8.6). In

ˆ

particular, F (z) is of Gevrey order not larger than s = 1/k , and

ˆ

F1 (z) is an HLFFS of (3.1).

ˆ ˆ

(d) In case of ≥ 2, let G1 (z) · . . . · G (z) be another factorization of

ˆ

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

ˆ ˆ

Fm (z) Tm+1 (z) = Tm (z) Gm (z), 1 ¤ m ¤ . (8.7)

ˆ

(e) If the data pairs of F (z) are ordered, then one can ¬nd a factorization

ˆ

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

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

ˆ

with what was introduced on p. 56 as the data pairs of the HLFFS F1 (z).

ˆ

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

ˆ

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,

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

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’1

’n Tn = n ≥ 1.

An+r’m Tm ,

m=0

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

∞