LINEAR IDEAL S 5
not chose n thi s path, preferrin g t o begin with the universal property, i t is convenient
to tak e the polynomial rin g version a s the definition o f a universal solutio n algebra .
DEFINITION
1.7. Le t L = Y £ Yfi=o ai ^^ ^ e a n n e a r homogeneou s differentia l
operator i n F{Y}/I. Th e (ordinary ) polynomia l rin g R = F\yo- ... -yi-\] wit h
derivation
DR{yi) = yt+\9 i £;
£-\
is calle d th e universal solutio n algebr a fo r (th e differential equation ) L ( = 0 ) (ab-
breviated USA-L) .
By Proposition 1.1 , if S i s any i^-algebra i n which L = 0 has a solution y, the n
there i s a uniqu e differentia l F-algebr a homomorphis m fro m th e univeral solutio n
algebra R t o S sendin g yo to y.
We return to the discussion of linear differential ideal s and establish the converse
of Propositio n 1.3.
PROPOSITION 1.8. Let I C F{Y} be a linear differential ideal of dimension £.
Then there is a monic homogeneous linear differential operator L of order £ in I such
that I is the linear differential ideal generated by L. Moreover, L is the unique monic
homogeneous linear differential operator in I of order £, and this is the minimal order
for a homogeneous linear differential operator in I.
PROOF.
Le t an overba r denot e imag e i n F{Y}\/(I n F{Y}\) a s in the proo f
of Propositio n 1.3, an d cal l thi s degre e on e quotient V. Choos e k maxima l suc h
that Y ,... , Y ar e linearly independent. (O f course k £ 1.) Then ther e is an
element L o f / o f the for m
L= Y^-Y,aiY
{i)-
Let J b e the (linear) differentia l idea l generate d b y L, and let W denot e the degree
one quotien t F{Y}\/(J f]F{Y}\). Sinc e J C I, W map s surjectivel y ont o V. B y
assumption, V ha s dimension £ an d by Proposition 1.3 W ha s dimension k - f 1.
Thus surjectivit y implie s tha t k + \ £; since w e already kno w tha t k + 1 £ we
have tha t k + 1 = i, fro m whic h i t follows tha t / n F{ Y}
{
= J n F{ Y} {. Sinc e /
and J ar e linear, an d hence generate d b y their intersection s wit h F{Y}\, i t follow s
that I = J.
Now le t M b e any monic homogeneou s linea r differentia l operato r i n /, sa y
A 7 - 1
M = Y
{n)
-^biY
{i).
/=o
The (linear ) differentia l idea l generate d b y M wil l have dimension n by Proposition
1.3 and be contained i n /, s o as above the surjectivity o n degree on e quotients wil l
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