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• GH ⊆ GLn (C) is de¬ned by H

48
ˆ ˜
• The action of Q ∈ GH on v ∈ C r i mi
˜
is given by Q.v = Qv (matrix-by-vector
i


multiplication), where

˜ ˜ ˜
Q= diag(Q1 , . . . , Qs ),
˜ ¯ ¯
Qi = diag(Q1 , . . . , Q1 ) (˜i copies),
r

¯
where Qi is as described above.

We prove the correctness of this algorithm as follows:
Proof. First, we recall that equivalent systems have identical Picard-Vessiot extensions
(resp., Galois groups). This algorithm computes the group of (3.33) and thus group of the
original system.
Computing GH is accomplished by [CS99], so by Proposition 3.3.1 we need to compute
the unipotent radical U and the action of GH on U. Write U = Ui as in Lemma 3.4.7.
i
˜ ˜
r
˜
VMii as modules over the group of Y = AY and thus
Then Lemma 3.4.8 implies that Ui
over GH . Correctness of Algorithm III is now immediate.




We now present examples of this algorithm.

Example 3.4.10 Consider the equation L(y) = b, where

L = D2 ’ 4xD + (4x2 ’ 2) = (D ’ 2x) —¦ (D ’ 2x)

and b ∈ C(x) as® Example 3.2.2. This equation is equivalent to the system Y = AY +
in 
0 1
B, where A = ° » and B = (0, b)T . A computation shows that another
’4x2 + 2 4x
˜ ˜˜ ˜ ˜ ˜
equivalent system is Y = AY + B, where A = diag(2x, 2x) and B = (’xb, b)T . The
˜
transformation from one system to the other is obtained by writing Y = P Y, where P =
® 
1 + 2x2 ’x
° » . Applying Lemma 3.4.8, we see that U C 2’r , where
’2x 1

dimC (c1 , c2 ) : y = 2xy ’ c1 xb + c2 b
r =

admits a C(x)-rational solution .

This computation is essentially identical to the one obtained in the ¬rst remark at the end
of Section 3.2. Thus, for this particular example, Algorithm III essentially coincides with
Algorithm I.

49
Example 3.4.11 Consider the ¬rst-order system

Y = diag(M, M, M, M, M )Y + (0, x2 , 0, x, 0, 1, 0, 1/x, 0, 1/x2 )T ,
® 
01
where M = ° » . In this case, we see that GH is the group of the equation y ’xy = 0.
x0
SL2 . To compute U, we consider the
From Example 3.3.4 above, we conclude that GH
equation
y ’ xy = c1 x2 + c2 x + c3 + c4 /x + c5 /x2 .

ˆ
Applying the algorithm given in the proof of Lemma 3.4.9, we consider the equation L(y) =
0, where
ˆ
L = LCLM(D ’ 2/x, D ’ 1/x, D, D + 1/x, D + 2/x) —¦ (D2 ’ x).

ˆ
A ratsols computation in Maple shows that the space of rational solutions of L(y) = 0 is
spanned by the elements 1 and x, which furthermore fail to satisfy y ’ xy = 0. Applying
C6 SL2 , where
Lemma 3.4.8, we now see that GI

T
Q.(v1 , v2 , v3 )T = (Qv1 )T , (Qv2 )T , (Qv3 )T
T T T



for Q ∈ SL2 , v1 , v2 , v3 ∈ C 2 , and Qvi is the standard matrix-by-vector product.

Example 3.4.12 Consider the matrix equation

diag(A1 , A2 )Y + (B1 , B2 )T ,
T T
Y =

®
0 1
»,
diag(M, M, M, M, M ), M = °
A1 =
x0
A2 = diag(2x, 2x),

(0, x2 , 0, x, 0, 1, 0, 1/x, 0, 1/x2 )T ,
B1 =

(’x, 1)T .
B2 =

Evidently GH is the group of the equation L(y) = 0, where L = LCLM(D2 ’ x, D ’ 2x).
We see that GH is a subgroup of SL2 — C — that projects surjectively onto each factor. As in
Example 3.3.5, the Theorem of [Kol68] yields GH = SL2 — C — . Next, Lemma 3.4.7 implies
U1 • U2 (direct sum of GH -modules), where U1 (resp., U2 ) is the unipotent radical
that U
of the system given in Example 3.4.11 (resp., Example 3.4.10 with b = 1). After applying
(SL2 — C — ), where
C7
the results of Example 3.4.11 and 3.2.2, we conclude that GI

T
(Q, t).(v1 , v2 , v3 , w)T = (Qv1 )T , (Qv2 )T , (Qv3 )T , tw

50
for (Q, t) ∈ SL2 — C — , v1 , v2 , v3 ∈ C 2 , w ∈ C.
Note that applying the method of Algorithm I to this system (or rather, more precisely,
to an equivalent inhomogeneous scalar equation) would involve computing all factorizations
of a twelfth-order completely reducible operator; this step alone would require solving a very
large system of equations, in contrast with the simple steps performed above.




51
52
Chapter 4


Computing the group of
D3 + aD + b, a, b ∈ C[x]

4.1 De¬nitions and main results

In this chapter, except where otherwise speci¬ed, C is an algebraically closed constant ¬eld
of characteristic zero.
We de¬ne the defect of a connected group, using the de¬nition given in [Sin99], as
follows: Assume G is a connected linear algebraic group with Levi decomposition G = Ru P
(semidirect product of subgroups), where Ru is the unipotent radical and P a Levi subgroup
of G. Note that the group Ru /(Ru , Ru ) is commutative and unipotent, hence isomorphic to a
vector group C n for some n. We see that the conjugation action of P on Ru leaves (Ru , Ru )
invariant and therefore induces a representation of P on the vector group Ru /(Ru , Ru ).
U1 1 • · · · • Us s , where each Ui is an irreducible P -module. Suppose
n n
Write Ru /(Ru , Ru )
U1 is the trivial one-dimensional P -module. Then the defect of G is the number n1 .


Proposition 4.1.1 Given L ∈ C(x)[D] of order three such that all singularities of L in the
¬nite plane are apparent singularities and the group GL of L(y) = 0 over C(x) is included
in SL3 . Then GL is connected and has defect zero.


Proof. This result follows from Proposition 11.20 and Theorem 11.21 of [dPS].



53
The following well-known result allows us to apply Proposition 4.1.1 to operators of the
form L = D3 + aD + b, a, b ∈ C[x].


Lemma 4.1.2 Given L = Dn + an’1 Dn’1 + · · · + a1 D + a0 ∈ D. Then the group GL is
isomorphic to a subgroup of SLn if and only if an’1 = f /f for some f ∈ C(x).


Proof. Let B = {y1 , . . . , yn } be a basis of VL . Then the fundamental solution matrix
(i’1)
associated to B is ZB = (yj ) ∈ GLn . It is well-known (see, e.g., [Mag94]) that if Z = ZB
is a fundamental solution matrix of L, then an’1 is the logarithmic derivative of det Z. Also,
given σ ∈ GL , one checks that σ(det Z) = det([σ]B ) det Z. It then follows from basic Galois
theory that det Z is contained in C(x) if and only if det([σ]B ) = 1 for all σ ∈ GL , i.e.,
[GL ]B ⊆ SLn . The desired result then follows easily.




Corollary 4.1.3 Given L = D3 + aD + b ∈ C(x)[D], a, b ∈ C[x]. Then GL is included in
SL3 , is connected, and has defect zero.


Proof. This result follows easily from Proposition 4.1.1 and Lemma 4.1.2.




In light of the above results, we de¬ne a subgroup G ⊆ SL3 to be admissible if it is
connected and has defect zero. Theorem 4.1.5 below enumerates the admissible subgroups
of SL3 up to conjugation.
Before proceeding, we de¬ne some speci¬c algebraic subgroups of SL3 using certain
parameterizations. Remark that these parameterizations are noncanonical; we use them
in this section for clarity of description only. See Lemmas 4.2.2 and 4.3.1 for equivalent
de¬nitions of these subgroups that do not use parameterizations.


diag(y d1 , y d2 , y ’d1 ’d2 ) : y ∈ C — ,
T(d1 ,d2 ) =

d1 , d2 ∈ Z, d1 ≥ d2 ≥ 0, d1 > 0, GCD(d1 , d2 ) = 1
±® 

10x 
 
 
  
 
:x∈C
1
0 1 0
U(0,0) =
° 
»
 
 
001 

54
±® 

 
 
12
 
2x
1 x

 


x :x∈C
1

U(1,1) = 0 1
° 
»
 
 
 
0 0 1
±® 

 
 
 
1 x y
  
 
0  : x, y ∈ C
2

=

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. 12
( 33 .)



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