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Connected Linear Groups as Differential Galois Groups over $C(x)$.

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HAL Id: hal-00129541

https://hal.archives-ouvertes.fr/hal-00129541

Preprint submitted on 8 Feb 2007

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Connected Linear Groups as Differential Galois Groups over C (x).

Claude Mitschi, Michael F. Singer

To cite this version:

Claude Mitschi, Michael F. Singer. Connected Linear Groups as Differential Galois Groups overC(x)..

1995. �hal-00129541�

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`ba6è6V;×0çYP WBî ƒ7¹, Š¼ ^‚½~¾À¿ ;J.(7:9k7 ¾T¼ ¥?5+6¬ ^ 01=5L>7l­69k50£:5GL2LÌ·£:LD1k7l¬6±¥À7o,

 ¼ ^ ð ^dcü

R e b

9 e ‰ & f e%g

¼ 9Ápb!ȕ^

T

3 7?02¬67l+¯,/0F·°,/.671À027?5L>7l­69k5:DF ^ ;J0A,¯.

R 9 h

9 9 f

hȕ^

T

5G+6¬C,¯.671À027?5L>7:­(9k5P<‰DF ^dc ;J0A,¯.

R i 9

9 a i f i

ȕ^

T

(11)

3 7,/.67l9k7lFKD9k71k7l7:¡’7:L7:®C7l+/,¯1 ÆML°¼

9 9

5G+6¬ ÆN>!¼ i

9 a i

hRÁ

i

Èq^‚½~¾À¿

1E 6£b.‹,¯.65l,

¥l±_,/.67l9k7q01x5[FK (L2L ^jc 2M369k0®C0A,¯0A@67µ7uQR,/7:+(1k0D+ Š½97I¿MÁQ7 ¼ e 9

9 e ‰ &

;J.67:9E7

åÛæ ½=76¿’¼ i 9

9 a i

±Ì„.(021J021J7pxR 60Ì@5L7:+¯,³,¯D#¬67l®C5+6¬60+6>‹,¯.65l,?,/.(7‚»J5LD021

>9kD (3dDGF

Š½

e ¿ Db@67:9

Š

01

^dc

;J.67l9k7

e

Â8¼

i e ±

±_,/.67l9k7‹7vQŸ021Y,/15G+7lL27l®‹7l+¯,

6 ¼ ¥ b

9 ¥

;J0A,¯.{å D E æ

½96ƒ¿J¼ ÆPL

§1k 6£b.d,/.65l,

,/.671M67lL2¬dDFƒ£:D+61Y,/5+¯,¯1&DF

Š½96¨ÁQ7I¿

021

^

±…¤=Dl,¯7_,/.65l,

å D E æ

½=6ƒ¿ ¼ e 9

9 e ‰ &

å½æ e 9

9 e ‰ & ‰ & ¥ b

9 ¥ e 9

9 e ‰ & e 9

9 e ‰ & ‰ &

¼ e 9

9 e ‰ &

å½æ

¥ e ‰ S b

9 ¥ e 9

9 e ‰ & ‰ &

¼ e 9

9 e ‰ & 9 ½e ‰ S

bH¿Â

9 9 e 9

9 e ‰ & ‰ &

¼ 9 e S ½e ‰ S

bH¿

Â

9 9

¼ 9 b8ÂÈa i b

9 9

„.67:9E7:FKD9E7„,¯.67?£:D+(¬60A,¯02DG+}å D E æ ½96H¿Ã¼ ÆPL 02147xi (0A@5L7:+¯,›,/D bŸÂÈa i

b!¼kh ±

~±_,/.67’®C53 SÌàá e 9

9 e ‰ & ‰ & ¥ b 9 ¥ ‰ &

Sý

¥ b

9 ¥ ¿ e 9

9 e ‰ &

0145

^ 2M021ED®CD9k36.(021E® FK9kD® »B5GL½|Š½

¥ b

9 ¥ e 9

9 e ‰ & ¿ Š½

e

¿k¿ D+¯,/D˜$ ½^?¿ ±

„.6021&L25ª,k,/7l9&£lD+6¬60Ì,/0D+Ž01›7pxR 60Ì@(5GL27l+/,=,/DJ,¯.67J£:D+6¬(0A,¯02D+‹,/.65l,4,¯.67³®C53 S}àá

Sý”bƒ¿Eaib 02145 ^ 2M021ED®CD9k3(.6021E® DFƒ»B5GL½|Š½”b

e ¿ Š½

e

¿E¿ D+¯,/D ^ ±

„.67l9k7lFKD9k7‘,/D9k7:5GL20¶:7  51’5°»B5GL2D01‘>9ED 63ŽDM@67:9

^½s¾À¿

§Ÿ;=7-®³ (1|,ÑM6+6¬

hRÁ

i È

^‚½~¾À¿ 5+(¬ ½Ûb¨Á

e ¿ 1E 6£b.‹,¯.65l,

e  a i e ¼ 9

(12)

i

5G+6¬Ž1k (£b.,/.65l, Š½

e ¿ 021&5ž02£l59k¬Ë2—¡›7:1E1k0Dl,C7uQi,¯7:+61E02DG+xDGF Š ;J0Ì,/.Ž»J5LD021&>9kDG 63 ^jc

5G+6¬ Š½Ûb

e ¿ 021&5ž0£:59E¬!2—¡›7l1k1k0Dl,7vQR,/7l+61k0D+!DF Š½

e ¿ ;J0A,¯.¸»J5L2DG021›>9kDG 63 ^ ±’*FŸ;=7

L7o,

i ¼ ¾ 5+6¬ hx¼ ¥J,¯.67:+

e ¼ Pmlonqp

S

5+(¬ bf¼›Pmlon P·‰Ylon ±}3 7J;J02LLw1E.6DM; 0+

,¯.67d+67vQR,!1E7:£¹,/0D+x,¯.65l,‹,/.601‚1E7:L7:£¹,/0D+ ®C7:7¹,/1DG 69‚+(7:7:¬(15+6¬ ¬601k£: (1k1.6DM; ,¯.601

£b.(D02£l7_;‘51„®C5¬(7:±“¤=Dl,¯7º,/.(5l,?0+-,¯.601„7vQŸ5®C36L7„,¯.67’9k7l369k7l1k7l+/,¯5l,/0D+"[d021„>0Ì@I7l+#­¯·

[ ½e

¿Ã¼

e S ±

¢?DM@5£:0£01³5­6L7B,¯DŽ9k7M6+67ž9kDG3ŸD1E0A,¯02DG+{±zŽ02+d,/.67FKDGL2LDM;J02+6>‹;‘5M·±;602+(£:7X

01?9k7l¬6 6£o,¯0A@67J;=7‚®C5M·Ž;J9E0A,¯7L$ 5G1J5z1E 6® DF029E9k7:¬( 6£:0­6L7L2b®CDi¬6 6L7:1k±z»J9kD (3602+(>

01kDG®‹DG9k36.60£C£:D3(027l1k§;‘7‘;J9k0Ì,/7$ ¼ $sr

%

&ì'

?+?+?

' $ r ,

-

§;J.67l9k7‘,/.(7Ñ$E/„59E7-+6D+601kD2

®CD9E36.60£.…2M®CD¬6 (L27l1k±¢À7¹,"[Ã/1“ á »Z ½ $E/ ¿ ­87B,/.67°9E7:369E7:1E7:+¯,/5l,¯02DG+ DFÏDG+

7l5£b.µ1k0®‹3(L27z®CD¬( 6L27°5G+6¬ N [Ã/†t< á >L½ $0/ ¿ ­Ÿ7³,¯.67z9k7l369k7l1k7:+¯,¯5l,/0D+61‹D+¸,/.(7:09

À07‘5L>7:­(9k51E±43 7‘¬(7:+6Dl,¯7‘­¯·u[ r(v

/

5+6¬ N [ rQv

/

,¯.67‘9k7l369k7l1k7l+/,¯5l,/0D+614D+J,/.67l029_38DM;‘7l9k1E±

Y&1³­87:FKD9E7:§H;=71E.65LLº0¬67:+¯,¯02F· 7:5£b.$E/›5+6¬•0Ì,/1Šƒ075L>7l­69k5.:G/Ã;J0A,¯.q1kD®C7 ^ Ë v ±

¢?DM@5£:0£?1k.6DM;J1 ½ ¤¥l©«§ž9kD38D1E0A,¯02D+¥?I ¿ 

Ö SUQB×ÃQjØpÙÛÚpÙ|QBÜi[HX w ZH\/]  `b\zcxnbr:eIeƒ\bn/]\bu!gmKeƒ\bc:j¸cghi\/`kjlc:mon³h/jlr:…U†fu\|vƒeƒ\bu!r:ˆi\kj

c:e cghi\/`kjlc:monbcgKgp¸n/gsr:t\bu›vw\/gsu^ c:eƒuzgs\/]HŠ‡`b\Cczu:mŒ=\kjl\keƒ]~mocgvw\/gsu

m

mo]s{nbr:eIt]c:eƒ]st

^‚œ÷›tbtb…¸^d\!]s{8c]

 ¼ ½

$sr

%

&('

?+?+?

' $ r ,

- ¿ ð

c:txc`br:ˆi\/œ‡ZH\/]›ÆN>È

<“Þ

c:eƒu

Æ / È[½:brQv

/ ¿ Þ `b\=tb…™nk{ ]s{8c]

Ý œ³]~{8\kjl\C\kš/mKt]~t‚c:e \/gs\_^d\keƒ]87°È m

mo]~{ å½æ ½97I¿w¼ ÆN>Étb…™nk{]~{8c]ƒŠ½=76¿’mKtc&yo…™gKg

—†8j}mµ^-mo]smKˆi\³\kši]|\keItbmor:e‡roy&Š(€

ä œ³]~{8\kjl\-\kš/mKt]st‹c:eµ\/gs\_^d\keƒ]~t6 / È $ rQv

/ m

mo]~{H68Â/ acN [ r(v

/

½|Æ>ƒ¿A6 / ¼fÆ / €Ãtb…™nk{x]~{8c]

]~{8\vw\/gsuroyJnbr:eIt]c:eƒ]st‹roy›Š½96 / Á7I¿&mKt’^J€

R œ³]~{8\4^dco†.S<àá [ rQv

/

½=7

‰ & ¿ ?“½ASý96 /¿ ax6 / ¿HmKt›cw^J—mKtrp^dr:j †8{mKtÉ^‡yoj:rp^ »J5L ½Š½=6 /7I¿ Š½97I¿k¿

r:eƒ]r $ r(v

/

½^?¿k¿/œ

]8{8\keCy:r:j6!¼G6

& F

)*)*)

F 6 - Á å½æ ½=687I¿Ã¼ÇÆ

& F

)p)*)

F Æ - F Æ>fc:eƒuBŠ½=687I¿›mKtCc4yo…™gKg

 —†8j}mµ^-mo]~mKˆi\³\kši]\keItbmor:eƒœ

û†üCy z1ÿ

, ÿ

){}|+N~{}

À7¹,

 ¼ $Ñð 51’5­8Db@67:±13 7-1E.65LLGM69k1Y,‹1k.6DM; ,/.65l,=,/.67-0+¯@I7:9E1k7‹369kDG­6L27l® £l5+

­87=9E7:¬6 (£:7:¬³,¯D›,¯.67=£l51k7º;J.(7:9k75 01w5B¬609k7:£¹,?369ED¬6 (£o,

= ð A

;J.67:9E7

=

01w5›,¯D9k (1

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