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