is an ordered basis of (k n )— . Using the fact that [ ——

= ’AT , compute the matrix

A ]E

— — n’1

A (v)| · · · |(

P = [id]Bv ,E — = [v| A) (v)].

—

Note that if Bv is the dual ordered basis of Bv in k n , then [id]E,Bv = P T by (2.3). Compute

—

—n —n

( A ) (v), then compute the column vector v = [(

¯ A ) (v)]Bv by solving the matrix equation

—n —

Pv = (

¯ A ) (v). Let Q = [ A ]Bv .

We see that

®

0 0 ···

0 a0

’1 0 0 · · ·

a1

Q = 0 ’1 0 · · · ,

a2

.

. . .. .

.

. . .

.

. . . .

° »

0 0 · · · an’1

0

˜

where v = ’(a0 , . . . , an’1 )T . De¬ne A = ’QT = [

¯ A ]Bv . We claim that Y = AY + B and

—

˜ ˜ ˜

Y = AY + B are equivalent systems, where B = P T B. Indeed, one checks that the map

ψ : k n+1 ’ k n+1 given by ®

T

P 0

ψ(Y ) = ° »Y

ˆ ˆ

0 1

is an isomorphism that has the properties required for equivalence of inhomogeneous sys-

˜ ˜

tems. The system Y = AY + B has the form

®

···

0 1 0 0

···

0

0 1 0

Y = . Y + B,

˜ (3.18)

.

. . .

..

.

. . .

.

. . .

° »

’a1 ’a2 ··· ’an’1

’a0

where ai ∈ k. Write Y = (y1 , . . . , yn )T , B = (˜1 , . . . , ˜n )T . This yields

˜ b b

yi = yi+1 + ˜i for 1 ¤ i ¤ n ’ 1

b (3.19)

and

n

aj yj + ˜n .

yn = ’ b (3.20)

j=1

By solving for yi+1 in (3.19) and applying to (3.20), we may eliminate the variable yi+1

from (3.20) for i = n ’ 1, n ’ 2, . . . , 1. If we write y = y1 , we obtain an equation L(y) = b,

n’1

where L = Dn + ai and

i=0

®

j

n n’1

˜(n’i) + ° ˜(j+1’i) » ∈ k.

b= bi aj bi (3.21)

i=1 j=1 i=1

29

˜ ˜ ˜

We claim that Y = AY + B and Y = AY + (0, . . . , 0, b)T are equivalent systems. Indeed,

ˆ

de¬ne a map ψ : k n+1 ’ k n+1 given by

®

w

In

»Y ,

ψ(Y ) = °

ˆˆ ˆ

0 1

where

n

˜(n’i) )T ;

w = (0, ˜1 , ˜1 + ˜2 , ˜1 + ˜2 + ˜3 , . . . ,

bb bb b b bi

i=1

one checks that this mapping is an isomorphism that has the properties required for equiv-

alence of inhomogeneous systems. This shows that Y = AY + B is equivalent to L(y) = b,

as desired.

Given a homogeneous equation L(y) = 0, L = L1 —¦ L2 , L2 = Dm + am’1 Dm’1 + · · · +

˜

a1 D + a0 , L1 = Dn +an’1 Dn’1 + · · · + a1 D +a0 . Consider the following system of equations

˜ ˜

in y1 , . . . , yn+m :

m

’

ym = ai’1 yi + ym+1

˜

i=1

n

’

yn+m = ai’1 ym+i

i=1

yi+1 for i ∈ {m, n + m} .

yi = /

Written in matrix form, this system is

®

AL2 C0

Y =° » Y,

0 AL1

where C0 ∈ k m—n is the matrix having 1 in the m, 1 position and zero everywhere else.

In what follows, we will show how to compute the group of a system of the form

®

A2 C

Y =° » Y, A2 ∈ k m—m , A1 ∈ k n—n , C ∈ k m—n , (3.22)

0 A1

where Y = Ai Y is completely reducible for i = 1, 2 and C is an arbitrary given matrix.

Proposition 3.3.1 Given a system of the form (3.16). Suppose that the associated homo-

geneous system Y = AY is completely reducible. Let U = Gal(KI /KH ) ⊆ GI . Then the

following statements hold:

1. Fix an arbitary particular solution · ∈ KI of Y = AY + B. Then, the map ¦· : U ’

n

VH given by ¦· („ ) = „.· ’ · is an injective GH -module homomorphism, where the

30

action of GH on U (resp., on VH ) is by conjugation (resp., by the usual representation

of Galois group on solution space). In particular, U is a vector group over C.

2. GI is isomorphic to U GH , where GH acts on U by conjugation. This is a Levi

decomposition of GI , i.e., U is the unipotent radical of GI and GI has a maximal

reductive subgroup isomorphic to GH .

Proof. These statements follow from the proof of Proposition 3.2.1, suitably rewritten in

terms of ¬rst-order systems.

The following result is adapted from Lemma 1 and the discussion in Section 2 of [Ber90].

A fundamental solution matrix of a ¬rst-order homogeneous system is a matrix whose column

vectors form a basis of the solution space.

Lemma 3.3.2 Let Y1 (resp., Y2 ) be a fundamental solution matrix of Y = A1 Y (resp.,

Y = A2 Y ). Then, the matrix

«

U Y1

Y2

Y = (3.23)

0 Y1

is a fundamental solution matrix of equation (3.22) if and only if U satis¬es

U = A2 U ’ U A1 + C. (3.24)

Proof. This statement is veri¬ed by direct calculation.

The following de¬nitions provide an interpretation of (3.24) in terms of connections. Let

Mi = (D/DLi )— for i = 1, 2. Then [ = Ai , where Ei is a suitable basis of Mi for

Mi ]Ei

i = 1, 2. Let ψ0 : M1 ’ M2 be such that [ψ0 ]E1 ,E2 = C. Applying (3.3), we conclude that

(3.24) is a matrix expression of the equation Hom (φ) = ψ0 , where φ is an unknown member

of Homk (M1 , M2 ).

We compute another matrix expression of this equation as follows: Let K/k be a ¬eld

extension with CK = Ck . Given V ∈ K m—n , let vi be the ith column of V. Then de¬ne

˜ ˜

V ∈ C mn by V = (v1 , . . . , vn )T . Formulas (3.4) and (3.5) imply that V satis¬es V =

T T

˜ ˜ ˜

A2 V ’ V A1 if and only if V satis¬es V = (’AT — Im + In — A2 )V . Furthermore, if Y1

1

31

(resp., Y2 ) is a fundamental solution matrix of the system Y = A1 Y (resp., Y = A2 Y ),

then a calculation shows that (Y1’1 )T — Y2 is a fundamental solution matrix of the system

˜ ˜ ˜

V = (’AT — Im + In — A2 )V , where V ∈ C mn is a column vector of unknowns. De¬ne

1

ci ∈ C m to be the ith column vector of C. We see that the equation Hom (φ) = ψ0 also has

the matrix expression

˜ ˜ ˜

V = (’AT — Im + In — A2 )V + C, (3.25)

1

˜

where C = (cT , . . . , cT )T .

1 n

Lemma 3.3.3 Assume that C k. Let Y = A1 Y and Y = A2 Y be completely reducible

systems and let K/k be the Picard-Vessiot extension of k corresponding to equation (3.22).

Let F/k, F ‚ K, be the Picard-Vessiot extension corresponding to

®

A2 0

Y =° » Y. (3.26)

0 A1

Then, the following statements hold:

1. The system

˜ ˜

V = (’AT — Im + In — A2 )V (3.27)

1

admits a full set of F -rational solutions, so there exists a tower of ¬elds k ⊆ E ⊆ F

with E/k a Picard-Vessiot extension for (3.27). In particular, Gal(E/k) is a quotient

of Gal(F/k).

2. K = F (˜), where · is a particular solution of (3.25).

· ˜

3. E(˜)/k is a Picard-Vessiot extension with Gal(E(˜)/k)

· · W Gal(E/k), where W is a