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¬ 
· 
(Int t) ¬ 0 1 d · =  td 
0 1
° »
» °
0 0 1
001

as in Lemma 4.5.6 with σ = id, T(d1 ,d2 ) = T(1,0) .

Q— — Q— .
¯ ¯
(c) Examples satisfying T

115
0. Let L = LCLM(D ’ x, D ’ 1, D + x + 1). Here,
i. Examples satisfying Ru
Q— — Q— .
¯ ¯
it is easy to see that GL
¯
Q. Let L = LCLM((D ’ x) —¦ (D ’ 1), D + x + 1).
ii. Examples satisfying Ru
A Maple computation shows that n1 = n2 = 1. It is easy to see that T
Q— — Q— . We conclude from Theorem 4.1.5 that GL (Q— — Q— ), with
¯ ¯ ¯ ¯ ¯
Q
(Int(t1 , t2 ))(u) = t2 t2 u as in Item 1 of Lemma 4.5.4.
1

¯
Q2 . Let L = (LCLM(Dx’x, Dx’1))—¦(D +x+1).
iii. Examples satisfying Ru
A Maple computation shows that n1 = 1, n2 = 2. It is easy to see that
Q— — Q— . We conclude from Theorem 4.1.5 that GL (Q— —
¯ ¯ ¯ ¯
Q2
T
Q— ), with (Int(t1 , t2 ))(u, v) = (t1 t’1 u, t2 t2 v) as in Item 2 of Lemma 4.5.4.
¯
1
2

Similar computations show that the group of adj L is as described in Item 3
of Lemma 4.5.4.
U3 . Let L = D3 + (’x2 ’ x ’ 2)D + (x2 + x + 1) =
iv. Examples satisfying Ru
(D + x + 1) —¦ (D ’ x) —¦ (D ’ 1). Maple computations show that n1 = n2 = 1
Q— — Q— . We conclude that GL
¯ ¯ T3 © SL3 .
and that T

SL2 .
2. Examples satisfying H

(a) Examples satisfying T 1.

0. Let L = LCLM(D2 ’ x, D). A Maple com-
i. Examples satisfying Ru
putation shows that n1 = n2 = 1; we see that GL is nonsolvable and that
SL2 . From Theorem 4.1.5 we conclude that GL SL2 .
GD2 ’x
¯
Q2 . Let L = D3 ’ xD ’ 1 = D —¦ (D2 ’ x). Maple
ii. Examples satisfying Ru
SL2 ; we conclude that
computations show that n1 = 1, n2 = 0, and GD2 ’x
¯
Q2 SL2 , with the unique conjugation action. Similar computations
GL
¯
show that adj L satis¬es n1 = 0, n2 = 1 and also has group Q2 SL2 with
the unique conjugation action. See Lemma 4.5.8.

Q— , so that P
¯ GL2 .
(b) Examples satisfying T

0. Let L = D3 + (’3x2 + 1)D + (2x3 ’ 4x) =
i. Examples satisfying Ru
LCLM(D2 + xD ’ 2x2 , D ’ x). Maple computations show that n1 = n2 = 1,
GL2 . We conclude that
that GL is nonsolvable and that GD2 +xD’2x2
GL2 .
GL
¯
Q2 . Let L = D3 + (’x4 + 2x + 1)D ’ x2 =
ii. Examples satisfying Ru
(D ’ x2 ) —¦ (D2 + x2 D + 1). Maple computations show that n1 = 1, n2 = 0.

116
GL2 with M.v = (det M )’1 (M ’1 )T v as
¯
Q2
It is easy to see that GL
in Item 2(b) of Lemma 4.5.8. Similar computations show that the group of
adj L is as described in Lemma 4.5.8.

PSL2 . Let L = D3 ’ 4xD ’ 2 = (D2 ’ x) 2
3. Examples satisfying H . A Maple
PSL2 .
2
computation shows that L has order 5. We conclude that GL

SL3 . Let L = D3 ’ x. A Maple computation shows that L 2
4. Examples satisfying H
SL3 .
is irreducible of order 6. We conclude that GL




117
118
Chapter 5


Bibliography




119
120
Bibliography

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F.B. et al., editor, P-adic analysis (Trento, 1989), volume 1454 of Lecture Notes in
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[Ber92] Daniel Bertrand. Un analogue di¬´rentiel de la th´orie de kummer. In P. Philippon,
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[Bro96] Manuel Bronstein. Symbolic Integration I: Transcendental Functions. Algorithms
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[Gri90] D. Y. Grigoriev. Complexity of factoring and calculating the gcd of linear ordinary
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[Hae87] A. Hae¬‚iger. Local theory of meromorphic connections in dimension one (Fuchs
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[HK71] K. Ho¬man and R. Kunze. Linear Algebra. Prentice Hall, Englewood Cli¬s, New
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1976.

[Kat87] Nicholas Katz. A simple algorithm for cyclic vectors. Amer. J. Math, 109:65“70,
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[Kol68] E. Kolchin. Algebraic groups and algebraic dependence. Amer. J. Math, 90:1151“
1164, 1968.

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123
Appendix




124
Appendix A


Maple code, documentation

A.1 README ¬le
File o3np.README: information about o3np.mpl, by Peter Berman.
Version 1.0. July 23, 2001. Compatible with Maple 6.


EXAMPLES:

> read("o3np.mpl");
Warning, the protected names norm and trace have been redefined and
unprotected
Warning, the name adjoint has been redefined

> dom := [Dx,x];
dom := [Dx, x]

> L := mult(Dx^2 - 2*x, Dx, dom);
3
L := Dx - 2 x Dx

> order_3_no_pole(-2*x,0,x);
The group of Dx^3-2*x*Dx is a semidirect product of C^2 by SL2.
The conjugation action of SL2 on C^2 is given as follows:
M.v = multiply( transpose(v), inverse(M) )
for M in SL2, v in C^2.
(multiplication of vector transpose by matrix inverse).

> # Note: The third argument, x, is the independent variable
> o3np(-2*x,0,x);
[C^2, SL2, vector_transpose_matrix_inverse]


125
> order_3_no_pole(2*x,1,x);
The group of Dx^3+2*x*Dx+1 is PSL2.

> o3np(2*x,1,x);
[0, PSL2, 0]




To use o3np.mpl, make sure that your copy of that file is
in a directory that™s accessible to MAPLE, then type

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( 33 .)



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