x

.

B=

’2x2 + 1

’2x(2x2 ’ 1) 4x2

2x

° »

2x(2x2 ’ 3) ’6x2 + 3 4x2 (2x2 ’ 3) ’6x(2x2 ’ 1)

˜ ˜ ˜

Therefore, the equation V + H V = ’C is equivalent to

˜

Z = KZ + B C.

(The reader can once again verify that K = B B ’1 + BHB ’1 .) In this example, the equiv-

alent inhomogeneous scalar equation is L(y) = ˆ where

ˆ b,

ˆ D 4 + h2 D 3 + h1 D 2 ,

L=

4x4 ’ 8x2 ’ 5

ˆ= 3 ’ 6x2 + 6 .

b

4x4 + 3

The eigenring command shows that the corresponding endomorphism ring has dimension

10 and yields a basis of this ring. Applying the command endomorphism charpoly to each

of these will yield a list of right factors and a simple calculation yields their corresponding

left factors. Despite the fact that in this case there is an in¬nite set of left factors, there is

a third order operator

8x6 ’ 12x4 + 6x2 + 3 2 8x6 + 12x4 ’ 6x2 + 3

8x6 + 3

D ’2 2

L0 = D 3 + D+2

x(4x4 + 3) x (4x4 + 3) x3 (4x4 + 3)

1

on this list of left factors such that L0 (y) = ˆ admits the rational solution y = ’ x(6x2 +5).

b

4

ˆ admits no rational solutions. We are

ˆ

Meanwhile, another computation shows L(y) = b

therefore able to avoid a calculation involving parameterized operators. Thus, we have

C—.

GL = C

40

Computing the group of Y = AY + B, Y = AY

3.4

completely reducible

The author would like to thank Daniel Bertrand for suggesting the approach given in this

section.

The goal of this section is to present an improvement on Algorithm I. We work in terms

of systems in this section, but these are interchangeable with equations provided C k.

We begin with two lemmas on modules over reductive groups. The ¬rst result is standard;

see, e.g., Proposition XVII.1.1 and Lemma XVIII.5.9 of [Lan84].

Lemma 3.4.1 Let G be a group de¬ned over C, and let V, V1 and V2 be irreducible ¬nite-

dimensional G-modules. Then:

1. If φ : V1 ’ V2 is a morphism of G-modules, then φ is either an isomorphism of V1

onto V2 or the zero map.

2. If ψ : V ’ V is a G-module automorphism, then ψ = c idV for some c ∈ C.

Proof. The ¬rst statement of the conclusion is known as Schur™s lemma. It follows from

the irreducibility hypothesis and the fact that the kernel (resp., the image) of a G-module

morphism φ is a G-submodule of the domain (resp., the range) of φ. The second statement

follows from the fact that ψ must have an eigenvalue in the algebraically closed ¬eld C; this

implies that the corresponding eigenspace is a G-submodule, which must be all of V.

Lemma 3.4.2 Given:

• G is a reductive linear algebraic group.

ˆˆ ˆ

• V1 , V2 , . . . , Vs are ¬nite-dimensional irreducible G-modules that are pairwise noniso-

morphic.

• V is a ¬nite-dimensional G-module such that

ˆ ˆ

Viνi = (v1,1 , . . . , vi,j , . . . , vs,νs )T : vi,j ∈ Vi .

T T T

V

1¤i¤s

• W is a submodule of V.

41

Then, the following statements hold:

1. There exist matrices Rl = (rl,ij ) ∈ C νl —νl for 1 ¤ l ¤ s such that

v = (v1,1 , . . . , vl,m , . . . , vs,νs )T ∈ V :

T T T

W =

νl

rl,mj vl,j = 0 for all l, m, 1 ¤ l ¤ s, 1 ¤ m ¤ νl .

j=1

ˆ ˆ

2. Suppose s = 1, and write ν = ν1 , V = V1 , vj = v1,j . Then there exists a vector subspace

S ⊆ C ν such that

±

ν

cj vj = 0 for all (c1 , . . . , cν ) ∈ S

(v1 , . . . , vν )T :

T T

W= .

j=1

ˆ

V ν’dim(S) .

We have W

Proof. We prove the ¬rst statement as follows: Complete reducibility implies that W is

˜ ˜

the kernel of a projection mapping π : V ’ W for some subspace W ⊆ V. This projection

mapping is a morphism of G-modules. Applying a well-known result that characterizes the

endomorphism ring of a completely reducible module (see Proposition XVII.1.2 of [Lan84])

and the second statement of Lemma 3.4.1, we see that there exist matrices R1 , . . . , Rs such

that

π((v1,1 , . . . , vl,m , . . . , vs,νs )T ) = (

T T T

r1,1j v1,j , . . . , rl,mj vl,j , . . . , rs,νs j vs,j ).

j j j

The ¬rst statement of the conclusion of the lemma follows immediately. The second state-

ment then follows after de¬ning S ⊆ C ν = C ν1 to be the row space of the matrix R = R1 .

˜ ˜

We are interested in inhomogeneous systems of the form Y = AY + B such that the

˜

associated homogeneous system Y = AY is completely reducible. A system having this

property is equivalent to a system of the form

= AY + B,

Y

A = diag(A1 , A2 , . . . , As ),

(B1 , B2 , . . . , Bs )T ,

T T T

B =

(3.33)

Ai = diag(Mi , . . . , Mi ) (νi copies),

k mi —mi ,

∈

Mi

Z = Mi Z irreducible, 1 ¤ i ¤ s,

and pairwise inequivalent

42

For the remainder of this section, we make the following assumptions:

1. Y = AY + B is an inhomogeneous system of order n de¬ned over k such that the

associated homogeneous system Y = AY is completely reducible.

2. KI /k (resp., GI = Gal(KI /k)) is the Picard-Vessiot extension (resp., the group) of

Y = AY + B.

3. KH /k (resp., VH ⊆ KH ; GH = Gal(KH /k)) is the Picard-Vessiot extension (resp., the

n

full solution set; the group) of the associated homogeneous system Y = AY, with

K H ⊆ KI .

4. U is the subgroup of GI ¬xing KH elementwise, so that

Gal(KI /KH ) ⊆ GI

U

and GH GI /U.

Lemma 3.4.3 The following are equivalent for Y = AY + B :

1. The system admits a k-rational solution.

2. The system admits a KH -rational solution.

3. Every solution of the system is KH -rational.

4. The subgroup U is trivial, i.e., KH = KI .

Proof. It is clear that the ¬rst and third statements each imply the second statement. Since

the solution set of a system generates that system™s Picard-Vessiot extension, it is also clear

that the third and fourth statements are equivalent to each other.

Notice that any two solutions of Y = AY + B di¬er by an element of VH ⊆ KH . Using

n

this fact, one checks that the second statement implies the third statement.

We now show that the second statement implies the ¬rst statement. This implication is

proved in Proposition 3.2.1; for convenience we reproduce the proof here using the termi-

nology and notation of systems.

Suppose · is a KH -rational solution of Y = AY + B. Let W = VH + spanC {·} ⊆ KH .

n

Since σ ∈ GH maps · to · + v for some element v ∈ VH , we see that W is GH -invariant and

includes VH as a GH -invariant subspace. Moreover, W/VH is a trivial GH -module. Since GH

˜

is reductive by assumption, we see that VH has a one-dimensional complement V in W that

43

is trivial as a GH -module. Moreover, since W is spanned by · and VH , we may assume that

˜

V is generated by · = · + v0 for some v0 ∈ VH . It follows that · is a solution of Y = AY + B

˜ ˜

that is ¬xed by every element of GH and therefore is rational over k. This completes the

proof.

Lemma 3.4.4 Assume Y = AY + B is irreducible. Then, as a GH -module, U is either

trivial or isomorphic to VH .

Proof. If U is nontrivial, then Proposition 3.3.1 implies that U is isomorphic to a nonzero

G-submodule of VH . By hypothesis, VH is irreducible; the desired result follows.

Lemma 3.4.5 Suppose Y = AY + B is of the form (3.33). Then, for each i, the following