0387986995Basic TheoryC

so that A' is a diagonal matrix with n distinct diagonal entries. Then, a transformation similar to (T') can be found so that (E") is changed to (E') with suitable constants ao, al, ... , an_ I .

XIII-6. The Hukuhara-Turrittin theorem

In this section, we explain a theorem due to M. Hukuhara and H. L. TLrrittin that clearly shows the structure of solutions of a system of linear differential equations of the form

(XIII.6.1)

where the entries of the n x n matrix A are in K (cf. §XIII-5). In order to state this theorem, we must introduce a field extension f- of K. To define L, we first set

+a

+a

where 7L is the set of all integers. For any element a = F a,nxn'/° of K,,, we

M=M

+00

define xji by x da = F \ v) amx'nl '. Then, K, is a differential field. The field

M=M

+oo

L is given by L = U K which is also a differential field containing K as a subfield.

L=1

Furthermore, L is algebraically closed. The Hukuhara-T rrittin theorem is given as follows.

Theorem XIII-6-1 ({Huk4j and [Tu1j). There exists a transformation

such that

(i)the entries of the matrix U are in L and det U ;j-1 0,

(ii)transformation (XIII.6.2) changes system (X111.6.1) to

where B is an n x n matrix in the Jordan canonical form

B=diag[Bl,B2,...,Bpj, Bj=diag[BJi,Bj2i...,B,,n,],

(XIII.6.4)

Bjk =AI In,,. + Jn,,,.

430 XIII. SINGULARITIES OF THE SECOND KIND

Case 2. Assume that there is no n x n matrix S with entries in X such that det.S 0 and the transformation w = Su' changes (XM.6.9) to xdu = A(x)u, where the entries of A are in C([x]]. Since

a cyclic vector can be found for system (XIII.6.9) by using the matrix W defined

by

triangular and the diagonal entries are {1, ... ,1}, i.e.,

where 1k E )C (k = 0, 1,... , n - 1). In particular, from (XIII.6.13), it follows that On-1 = 0. Under our assumption, not all Qt are in C[[x]]. Set 9 = (t: at V C[[x]]}

positive integer, 1: /3tmxm E C[[xJ] and #to 54 0. Set m=0

k=max(n µt t :IE91.

J

Then, ut < k(n - t) for every t E J and µt = k(n - t) for some t E J. This implies

for some t such that k(n - t) is a positive integer, ct is a nonzero number in C, qt E C[[x]], and Qt,m E C. We may assume without any loss of generality that

k = h for some positive integers h and q.

9

Let us change system (XIII.6.12) by the transformation

v" = diag [1, x-k, ... , x-(n-1)kI u".

Then,

(XIII.6.14)

where

m>-k(n-t)

In particular, 7n-1 = 0.

+oo

Setting F = E xm/9Fm, where the entries of F,,, are in C, we obtain m=o

0 1 0 0 0 01

Fo=

C1 C2 C3 ... Cn_2 0

CO

where the constants co, cl ... , cn_2 are not all zero. This implies that the matrix

F0 must have at least two distinct eigenvalues. Hence, there exists an n x n matrix

T such that

(1) T = XM19Tm, where the entries of the matrices Tm are in C and To is

changes system (XIII.6.14) to a system xd = x-kGxi with a matrix G in

a block-diagonal form G = [ 0 0 , where Gl and G2 are respectively

G2 J

n1 x nl and n2 x n2 matrices with entries in C[[x1/91and that nl + n2 = n and n., > 0 (j = 1, 2) (cf. §XIII-5). Therefore, the proof of Theorem XIII-6-1 can be completed recursively on n. 0

Observation XIII-6-2. In order to find a fundamental matrix solution of (XIII.6.1), let us construct a fundamental matrix solution of (XIII.6.3) in the following way:

Step 1. For each (j, k), set

4'1k = xA,0 eXp[AJ (x)] exp[(log x)Jf, ],

where

if dj = 0,

-t/s if d1>0.

Step 2. For each j, set

where J. = diag [J,,,,, J,,72 , ... J,mJ ] .

p",,(0) are n eigenvectors of A(O). Denote by P(x) the n x n matrix whose column vectors are p1(x), p2(x), ... , p,+(x). Then, detP(O) 36 0 and P(x)-lA(x)P(x) = diag[A1(x),A2(x),... ,An(x)]. This implies that the transformation y = P(x)ii changes system (XIII.6.19) to

(XIII.6.20) xd+1 dx {diagiAi(x)A2(x).... , An(x)] - xd+1P(x)-1 d ( )

It is easy to construct another n x n matrix Q(x) so that (a) the entries of Q(x) are in C[(x]], (b) Q(0) = 4,, and (c) the transformation i = Q(x)v changes system

(XIII.6.20) to

where µ1(x), µ2(x), ... , i (x) are polynomials in x of degree at most d such that A, (x) = it) (x) + O(xd+1) ( j = 1, 2, ... , n). Therefore, in this case, the entries of the matrix U of transformation (XIII.6.2) are in 1C.

Observation XIII-6-6. Assume that the entries of A(x) of (XIII.6.19) are in

C([x]]. Assume also that A(O) is invertible. Then, upon applying Theorem XIII-6-1

Theorem XIII-6-7. Let Q;i1(x) and Qi2(x) be two solutions of a system

where d is a positive integer, the entries of the n x n matrix A(x) and the C"-valued function 1 *(x) are holomorphic in a neighborhood of x = 0, and A(O) is invertible.

Assume that for each j = 1, 2, the solution ¢,(x) admits an asymptotic expansion in powers of x as x - 0 in a sector S3 = {x E C : lxl < ro, aj < arg x < b3 }, where ro is a positive number, while aJ and bi are neat numbers. Suppose also that S1 n S2 0. Then, there exist positive numbers K and A and a closed subsector S =

in S.

Proof.

Since the matrix A(O) is invertible, the asymptotic expansions of1(x) and ¢2(x) are identical. Set 1 (x) = ¢1(x) -$2(x). Then, the C"-valued function >G(x) satisfies system (XIII.6.19) in S, n$2 and ii(x) ^_- 6 as x 0 in S1nS2. By virtue of Theorem

XIII-6-4, a constant vector 66 E C" can be found so that r%i(x) = U4D(x)6, where lb(x) is given by (XIII.6.17). Now, using Observation XIII-6-6, we can complete the proof of Theorem XIII-6-7.

Definition XIII-7-2. The polygon N is called the Newton polygon of system

(XIII.6.1) at x = 0.

Observation XIII-7-3. In §XIII-5, it was shown that system (XIII.6.1) is equiv- alent to an n-th-order linear differential equation

the Newton polygon N(C) is defined in the following way.

If a power series a = E c,,x'n E C((xj] is not 0, we set v(a) = min{m : cn

m=0

0}. If a = 0, set v(0) = +oo. For operator (XIII.7.4), consider n + 1 points

(Q, v(at)) (t = 0, 1, ... , n) on an (X, Y)-plane. Set

Pt = ((X, Y) : 0 < X < e, Y > v(at)}, and P = UP,.

t=o

Definition XIII-7-4. The boundary curve C of the smallest convex set containing P is called the Newton polygon of the operator C at x = 0.

Denote by N(C) the Newton polygon of C at x = 0.

Definition XIII-7-5. Two Newton polygons are said to be identical if the two polygons become the same by moving one or the other upward in the direction of the Y-axis.

Now, we prove the following theorem.

Theorem XIII-7-6. If system (XIII.6.1) and differential equation (X111.7.3) are equivalent in the sense of Theorem X111-5-4. then the two Newton polygons N and

N(C) are identical.

Proof.

The proof of this theorem will be completed if the following three statements are verified:

(a) If./V(C) has only one nonvertical side with slope k, then differential equation

whose entries are power series in x11", and A(0) is invertible if k > 0, where s is a positive integer such that sk is an integer.