·

(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

[Ber90] Daniel Bertrand. Extensions de d-modules et groupes de Galois di¬´rentiels. In

e

F.B. et al., editor, P-adic analysis (Trento, 1989), volume 1454 of Lecture Notes in

Mathematics, pages 125“141, Berlin, 1990. Springer.

[Ber92] Daniel Bertrand. Un analogue di¬´rentiel de la th´orie de kummer. In P. Philippon,

e e

editor, Approximations Diophantiennes et Nombres Transcendents, Luminy 1990,

pages 39“49, Berlin, 1992. Walter de Gruyter.

[Bro96] Manuel Bronstein. Symbolic Integration I: Transcendental Functions. Algorithms

and Computation in Mathematics. Springer, Berlin, 1996.

[BS99] Peter Berman and Michael Singer. Calculating the Galois group of L1 (L2 (y)) =

0, L1 , L2 completely reducible operators. Journal of Pure and Applied Algebra,

139:3“23, 1999.

[CS99] Elie Compoint and Michael Singer. Calculating Galois groups of completely re-

ducible linear operators. Journal of Symbolic Computation, 28(4-5):473“494, 1999.

[FH91] W. Fulton and J. Harris. Representation Theory: A First Course. Readings in

Mathematics. Springer, New York, 1991.

[Gri90] D. Y. Grigoriev. Complexity of factoring and calculating the gcd of linear ordinary

di¬erential operators. Journal of Symbolic Computation, 10(1):7“38, 1990.

[Hae87] A. Hae¬‚iger. Local theory of meromorphic connections in dimension one (Fuchs

theory). In Borel et al., editor, Algebraic D-Modules, pages 129“149. Academic

Press, 1987.

[HK71] K. Ho¬man and R. Kunze. Linear Algebra. Prentice Hall, Englewood Cli¬s, New

Jersey, second edition, 1971.

121

[Hum81] James Humphreys. Linear Algebraic Groups. Graduate Texts in Mathematics.

Springer, New York, 1981.

[Kap76] I. Kaplansky. An Introduction To Di¬erential Algebra, 2nd ed. Hermann, Paris,

1976.

[Kat87] Nicholas Katz. A simple algorithm for cyclic vectors. Amer. J. Math, 109:65“70,

1987.

[Kol68] E. Kolchin. Algebraic groups and algebraic dependence. Amer. J. Math, 90:1151“

1164, 1968.

[Lan84] S. Lang. Algebra. Addison-Wesley, Menlo Park, California, second edition, 1984.

[Mag94] Andy R. Magid. Lectures on Di¬erential Galois Theory, volume 7 of University

Lecture Series. American Mathematical Society, Providence, Rhode Island, 1994.

[Mos56] G. D. Mostow. Fully reducible subgroups of algebraic groups. Amer. J. Math,

78:211“264, 1956.

[MS96] C. Mitschi and M. Singer. Connected linear groups as di¬erential Galois groups.

Jour. Alg., 184:333“361, 1996.

[dP98a] M. van der Put. Recent work in di¬erential Galois theory. In S´minaire Bourbaki:

e

volume 1997/1998, Ast´risque. Soci´t´ Math´matique de France, Paris, 1998.

e ee e

[dP98b] M. van der Put. Symbolic analysis of di¬erential equations. In Cohen et. al., editor,

Some Tapas of Computer Algebra. Springer, 1998.

[dPS] M. van der Put and M.F. Singer. Di¬erential Galois Theory. Manuscript, 2001.

[Sin96] Michael Singer. Testing reducibility of linear di¬erential operators: a group theoretic

perspective. Appl. Algebra Eng. Commun. Comput., 7:77“104, 1996.

[Sin99] Michael Singer. Direct and inverse problems in di¬erential Galois theory. In Cassidy

Bass, Buium, editor, Selected Works of Ellis Kolchin with Commentary, pages 527“

554. American Mathematical Society, 1999.

[SU93] Michael Singer and Felix Ulmer. Galois groups of second and third order linear

di¬erential equations. Journal of Symbolic Computation, 16:9“36, 1993.

122

[Tsa96] S. P. Tsarev. An algorithm for complete enumeration of all factorizations of a

linear ordinary di¬erential operator. In L.Y.N., editor, Proc. 1996 Internat. Symp.

on Symbolic and Algebraic Computation, pages 226“231, New York, 1996.

[dW53] B. L. van der Waerden. Modern Algebra. Frederick Ungar Publishing Co., New

York, second edition, 1953.

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