4 1. DIFFERENTIA L IDEAL S
To see that i t i s also linearl y independent , w e assume tha t som e non-trivia l F-
linear combinatio n o f Y , ..., Y wit h coefficient s Cj is zero . The n th e sam e
combination, withou t th e overbars , belong s t o / , givin g a relatio n o f th e for m
£-1 n
The analysi s o f
DnL
i n the preceding paragraph show s that th e coefficient o f F^
+")
on the right-hand sid e of this equation is bn, whil e it is 0 on the left-hand side , which
is a contradiction . Henc e ther e i s n o suc h linea r combination ; w e have establishe d
linear independenc e an d henc e tha t w e have a basis.
On th e strengt h o f Propositio n 1.3, w e make th e followin g definition :
DEFINITION
1.4. Le t L e F{Y} b e a moni c homogeneou s linea r differentia l
operator o f order £, and let / b e the ideal of F{ Y} generate d b y {D
1
L\i = 0,1,.. . }.
Then / i s called th e linea r differentia l idea l generate d b y L.
Proposition 1.3, o r more precisely its proof, actually establishes a stronger result :
THEOREM
1.5. Let L = Y
l
Xw= o
aiY^
be a linear homogeneous differential
operator in F{Y} of order £. Then {7 ( 0 ) ,... , Y {£-l),L,DL3D2L,... } is a basis for
F{ Y}\ . In particular, if I is the linear differential ideal generated by L, then F{ Y}/I
is isomorphic to the (ordinary) polynomial ring F[Y ,... , Y ] .
PROOF.
Th e proof of Proposition 1.3 shows that Y^
Dn~lL
plu s lower order
terms, provided n £, so that th e set in the statement o f Theorem 1.5 span s F{Y}\.
If som e linear combinatio n o f the se t is zero, then w e have a n equatio n o f the for m
that appeare d i n th e proo f o f Propositio n 1.3, s o th e coefficient s mus t al l b e zero .
Thus ou r se t i s indeed a basis .
When F{Y} i s regarde d a s th e ordinar y polynomia l rin g F[Y^°\ F ^ , . . . ] , a
change of basis in the homogeneous component of degree 1 (namely F{ Y}\) extend s
to a ring isomorphism s o that F{ Y} i s isomorphic to the polynomial ring in the new
basis. I f we apply thi s to the basis of Theorem 1.5, the ideal / i s carried t o th e idea l
generated b y the polynomia l indeterminate s {D 1 L\i = 0,1,.. . } , giving th e claime d
isomorphism o f th e theorem .
REMARK
1.6. W e note that Theore m 1.5 establishes that th e ideal / i s prime an d
that th e rin g F{Y}/I i s a Noetheria n integra l domain . Moreover , w e se e tha t th e
derivation D act s o n th e polynomia l rin g b y
D(Y(i)) = Y {i+l) i£;
7=0
Back i n Propositio n 1.1w , e considere d th e universa l propert y o f th e quotien t
F{Y}/I. wher e / i s the differentia l idea l generate d b y th e homogeneou s linea r dif -
ferential operato r L. I n fact, give n L, we could consider the polynomial algebr a in £
indeterminates and define a derivation of it by the formulas of Remark 1.6 directly and
then prov e that i t ha d th e universa l propert y o f Propositio n 1.1 . Althoug h w e have
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