L(y) = an y (n) + an’1 y (n’1) + · · · + a1 y + a0 y.

We see that there is a one-to-one correspondence between homogeneous linear ordinary

di¬erential equations L(y) = 0 over k and elements L of D. The order ord(L) of a di¬er-

ential operator L is de¬ned to be the degree of D in L or, equivalently, the order of the

corresponding equation L(y) = 0.

The ring D is a left (resp., right) Euclidean domain. In particular, there is an extended

Euclidean algorithm that takes as input U, V ∈ D and computes operators A, B ∈ D with

ord A < ord V and ord B < ord U such that AU + BV = GCRD(U, V ).

Let K/k be an extension of di¬erential ¬elds with CK = Ck . Then we may view a

homogeneous linear ordinary di¬erential equation over k (resp., an operator L ∈ D = k[D])

as an equation over K (resp., an operator in K[D]). Let SolnK (L) denote the set of solutions

of L(y) = 0 in K. Then SolnK (L) is a C-vector space.

Lemma 2.3.1 Given L1 , L2 ∈ D, ord(Li ) = mi for i = 1, 2. Let K/k be a di¬erential ¬eld

extension with CK = Ck . Then, the following statements hold:

9

1. dimC SolnK (L1 ) ¤ m1 .

2. Suppose dimC SolnK (L1 ) = m1 and SolnK (L1 ) ⊆ SolnK (L2 ). Then L2 = L3 L1 for

some L3 ∈ D.

3. Suppose dimC SolnK (L2 ) = m2 and L2 = L3 L1 for some L3 ∈ D. Then:

(a) dimC SolnK (L1 ) = m1 .

(b) SolnK (L1 ) ⊆ SolnK (L2 ).

Proof. This is Lemma 2.1 of [Sin96].

Given L ∈ D, a Picard-Vessiot extension for L is a minimal extension KL /k of di¬erential

¬elds such that CKL = Ck and dimC SolnKL (L) = ord(L). Such an extension exists and is

unique up to isomorphism. If KL /k is a Picard-Vessiot extension, then the (full) solution

set of L in KL is VL = SolnKL (L). The Galois group G = GL = Gal(KL /k) of L over k is

the group of di¬erential automorphisms of KL which ¬x k elementwise. An automorphism

σ ∈ G maps VL to itself, and one can show that the action of σ on VL determines σ uniquely.

Thus, there is a faithful representation G ’ GL(VL ) that gives G the structure of linear

algebraic group. When discussing this representation, we often say that V is a G-invariant

vector space, or G-module.

Lemma 2.3.2 Let K/k be a Picard-Vessiot extension, G = Gal(K/k).

1. Let V ⊆ K be a ¬nite-dimensional C-vector space. Then, V = VL for some L ∈ D if

and only if V is G-invariant.

2. Suppose L ∈ D is a linear operator with VL ⊆ K. Let W ⊆ VL be a C-vector subspace.

˜

Then W = VL for some right factor L of L if and only if W is G-invariant.

˜

Proof. This is Lemma 2.2 of [Sin96] and an easy corollary.

10

Chapter 3

Computing the group of

L1 —¦ L2, L1, L2 completely

reducible

Connections and D-Modules

3.1

The following is taken from Sections 2.1-2.3 of [CS99]; see also [Hae87].

where M is a ¬nite-dimensional k-vector space

A connection over k is a pair (M, M ),

: M ’ M satis¬es the following properties for all u, v ∈ M, f ∈ k :

and

(u + v) = (u) + (v)

(f u) = f u + f (u).

We will omit the phrase “over k” when k is clear from context. We will use M (resp., )

to refer to (M, ) (resp., M) when the context is clear. The dimension of (M, ) is the

dimension of M as a k-vector space.

If (M1 , 1) and (M2 , 2) are connections, then a morphism from (M1 , 1) to (M2 , 2)

is de¬ned to be a map φ ∈ Homk (M1 , M2 ) such that —¦φ=φ—¦ 1.

2

Let D = k[D]. A connection (M, ) can be given a D-module structure via D.u = (u)

for u ∈ M. Conversely, a D-module M can be given a connection structure via (u) = D.u

for u ∈ M.

11

where n ∈ Z>0 , A ∈ k n—n and

One example of a connection is (k n , A ), A (Y )=

Y ’ AY for Y ∈ k n . Here, Y is viewed as a column vector and AY is the product of

multiplication of an n — n matrix by an n — 1 matrix. We have ) = 0 ” Y = AY,

A (Y

so this connection corresponds to a system of ¬rst-order linear di¬erential equations over k.

Conversely, given a ¬rst-order system of equations

Y = AY, Y = (y1 , y2 , . . . , yn )T , A ∈ k n—n ,

we may consider the connection (k n , A ).

Given a di¬erential ¬eld extension K/k with CK = Ck , we may view Y = AY as a

system over K. That is, we may consider the connection (K n , A ). The solution space of

the system in K n is the set of all Y ∈ K n such that ) = 0; it is a C-vector space of

A (Y

dimension at most n. A Picard-Vessiot extension K/k for the system is a minimal extension

containing the full n-dimensional set of solutions of the system. G = Gal(K/k) acts on K n

by

σ.ζ = (σ(ζ1 ), σ(ζ2 ), . . . , σ(ζm )) (3.1)

for all σ ∈ G, ζ = (ζ1 , ζ2 , . . . , ζm ) ∈ K m .

Given a connection (M, ), let E = {e1 , . . . , en } be an ordered k-basis of M. For 1 ¤

i, j ¤ n, de¬ne aij by

n

(ei ) = ’ aji ej .

j=1

ui ei ∈ M, ui ∈ k, then ’

It follows that if u = (u) = i (ui aij uj )ei . We call

i j

the matrix [ ]E = (aij ) ∈ k n—n the matrix of (M, ) with respect to E. Observe that if

A = [ ]E , then u ’ [u1 u2 · · · un ]T de¬nes an isomorphism from (M, ) onto (kn , A ). If

N ⊆ M is a k-subspace such that (N ) ⊆ N , then (N , ) is a connection; moreover,

N

+ N) = (m) + N .

one checks that (M/N , M/N ) is a connection, where M/N (m

In the following observations and de¬nitions, (M1 , 1) and (M2 , 2) are two connec-

tions; E = {e1 , . . . , em } (resp., F = {f1 , . . . , fn }) is a basis of M1 (resp., M2 ); A = (aij ) =

[ 1 ]E and B = [ 2 ]F .

(M1 • M2 , • 2) is a connection.

1

(M1 — M2 , 1 — idM2 + idM1 — 2) is a connection. It can be shown that has

M1 —M2

matrix

A — In + Im — B

[ M1 —M2 ]E—F =

12

®

···

a11 In a12 In a1m In

···

a21 In a22 In

a2m In

=

. . .

..

. . .

.

. . .

° »

· · · amm In

am1 In am2 In

®

B 0 ··· 0

0 B ··· 0

+ .

. . (3.2)

. . ..

. .

. .

. .»

°

0 0 ··· B

(Homk (M1 , M2 ), Hom ), where

—¦φ’φ—¦

Hom (φ) = 1,

2

is a connection. Suppose φ ∈ Homk (M1 , M2 ) is such that [φ]E,F = U. Then a calculation

shows that

= U ’ BU + U A.

[ Hom (φ)]E,F (3.3)

(M— , —

) is a connection, where M— = Homk (M1 , k) is the vector space dual of M1

1 1

—

—¦φ’φ—¦

and (φ) = 1. Observe that this de¬nition coincides with the de¬nition of

—

= (k, f ’ f ). One checks that [ ]E — = ’AT .

in the case (M2 , 2)

Hom

There is a natural isomorphism Ψ : Homk (M1 , M2 ) ’ M— — M2 . Given a homomor-

1

uji e— — fj . In the

phism φ ∈ Homk (M1 , M2 ) with [φ]E,F = U = (uij ), then Ψ(φ) = i

i,j

ordered basis given by (2.2), we have

[Ψ(φ)]E — —F = (u11 , u21 , . . . , un1 , u12 , u22 , . . . , un2 ,

. . . , u1m , u2m , . . . , unm )T

(U1 , U2 , . . . , Un )T ,

T T T

= (3.4)

where Ui is the ith column vector of U.

We apply the preceding observations to obtain the following matrix for M— — M2 :

1

’AT — In + Im — B

[ M— —M2 ]E — —F =

®

1

a11 In a21 In · · · am1 In

a12 In a22 In · · ·

am2 In

= ’

. . .

..

. . .

.

. . .

° »

···

a1m In a2m In amm In

13

®

···

B 0 0

···

0 B 0

.

+ (3.5)

. . .