x PREFAC E
PROOF.
Sinc e J ? i s a Noetheria n ring , maxima l prim e differentia l ideal s exist .
Then R/P ha s no non-zer o prim e differential ideals , hence by the theorem Q(R/P)
has no new constants. Sinc e R i s differentially generate d over F b y solutions of L = 0,
so ar e R/P an d E.
To guarante e th e thir d conditio n fo r Picard-Vessio t extensions , w e need t o us e
Wronskian determinants :
DEFINITION.
Le t y\,... ,y
s
b e element s o f th e differentia l fiel d E. The n
... ^ 0 )
w = w(y\
9
...,ySi
„(°)
y\
„(0
y\
„2)
y\
' (-v-l )
y\
„(°)
yi
„(0
yi
J2)
yi
' (.v-l )
y\
«(°)
?3
i,(0
^3
„(2)
J3
'
(.v -
^3
»(2)
" .. . y ^
Just lik e th e classica l Wronskia n determinan t use d i n analysis , th e Wronskia n
determines linea r independence :
PROPOSITION. The elements y\,... ,y
s
of the differential field E are linearly inde-
pendent over the constants of E if and only ifw(y\,... ,y
s
) ^ 0 .
As in analysis , thi s propositio n implie s tha t th e dimensio n o f the spac e o f solu -
tions o f L = 0 is limited :
COROLLARY. L 0 has at most £ solutions in E linearly independent over constants.
PROOF.
Suppos e ther e ar e s = £ + 1 elements y,, 1 i s, wit h L(yt) = 0 .
Then i n th e determinan t w{y\,... ,y
s
) th e las t ro w i s a linea r combinatio n o f th e
preceding ones , an d henc e th e determinan t i s 0. Bu t the n b y th e propositio n th e y\
are linearl y dependen t ove r th e constants .
As a consequence, we now have a constructive proo f o f the-existence o f Picard -
Vessiot extensions :
THEOREM
2 . Let P C F\yij][w(y
L0
- -ye.o)"1] = i?[w -1 ] = S be a maximal
prime differential ideal. Then E = Q(S/P) D F (E is the fraction field of S/P) is a
Picard-Vessiot extension of F for L.
PROOF.
B y Theorem 1, E ha s no new constants. Le t y{ denot e th e image of j/. o
in E. The n w{y\,.. .
9
yi) ^ 0 sinc e i t i s th e imag e o f w, whic h i s a uni t i n S an d
hence i n S/P. Thi s mean s tha t th e y
f
ar e linearl y independen t ove r C . Also , E i s
generated ove r F a s a differentia l fiel d b y th e y
t
b y construction .
Next, w e note th e followin g normalit y propert y o f Picard-Vessio t extensions :
NORMALITY.
Suppos e E\ D F , i = 1,2, ar e Picard-Vessio t extension s o f F fo r
L an d tha t E D F i s a n o ne w constant s extension . Suppos e tha t o\ : E\ » E ar e
differential embedding s which are the identity on F. Le t V\ be the solutions of L 0
in Ej, an d let V be the solutions in E. The n aj(Vj) C V an d a dimension count based
on condition (3 ) for Picard-Vessio t extension s an d th e corollary preceding Theore m
2 establishes tha t 0 7 (F/) = F . I t the n follow s fro m conditio n (1) fo r Picard-Vessio t
extensions tha t o\{E\) = 02(Ei).
Previous Page Next Page