6

SHREERAM S. ABHYANKAR AND ABDALLAH ASSI

with the understanding that if R*(B) is an infinite set then D'(B) = oo. For any

BE

R'p

we also define the doubly primitive degree D"(B) of B by putting

{

-D(B)

+

L:fEu(B)

degy

f

if BE R'p

D"(B)

= .

'p

-1

+

L:fEu(B)

degy

f

If BE R=

with the understanding that if a(B) is an infinite set then D"(B) = oo.

NOTE. Basically, the stem a(B) of a bud B = (a(B), -.(B)) is a nonempty

set of irreducible meromorphic curves f(X, Y) = 0 whose fractional meromorphic

roots mutually coincide up to

X

.(B), and its flower r(B) is the set of all irre-

ducible meromorphic curves whose fractional meromorphic roots coincide with the

fractional meromorphic roots of members of a(B) up to X .(B).

6. Trees

Let fiU be the set of all trees in R, where by a tree we mean a subset T of

R'p

such

that T contains an improper bud, and for any B' #Bin T with -.(B') =-.(B) we

have r(B') nr(B) =

0;

note that then Tis partially ordered by defining B';::::: B

in T to mean -.(B')

2::

-.(B) and r(B')

C

r(B); hence in particular T has a unique

improper bud; we call this improper bud the root ofT and denote it by R=(T);

note that here the superscript in fiU is a "sharp" as opposed to that in

RQ,

at the top

of Section 2, which was a "natural." For any tree T, we put A(T) ={-.(B): BET}

and we call A(T) the level set ofT. For any generalized bud Bin any tree T, we

put 1r(T, B) = {B' E T : B' B} and we call 1r(T, B) the B-preroof ofT, and

we put p(T, B)= {B' E 1r(T, B) :there is noB" E 1r(T, B) with B' B"} and we

call p(T, B) the B-roof of T.

A

tree T is said to be strict if for every ).. E A(T) we have a(R=(T)) =

UBET-JO'(B) where T(.) is the set of all B E T with -.(B) = A. Given any ).. E

Q

U {- oo}, we see that

f

"'.

f'

gives an equivalence relation on RQ where

f

"'.

f'

means noc(/, /') ;::::: A. It follows that, given any

a

C

RQ and A

C

Q, there is a unique

strict tree T(a, A) with A(T(a, A)) = {

-00}

u

A such that a(ROC!(T(a, A))) =a or

{Y} according as a is nonempty or empty; we call T(a, A) the A-tree of a; note

that, if a is nonempty then, for every ).. E A, the stems of the buds of T(a, A) of

level ).. are the equivalence classes of

a

under

"'.j

likewise,

if

a

is empty then, for

every ).. E A, the stem of the unique bud of T(a, A) of level ).. is {Y}.

Note that a tree Tis finite iff its level set A(T) is finite and T has at most a

finite number of generalized buds of any given level. We let RU denote the set of all

finite trees in R, and we let RU* denote the set of all finite strict trees in R; again

note that here the superscript in RU is a "sharp" as opposed to that in RQ, at the

top of Section 2, which was a "natural." For any FER, with its monic irreducible

factors F1, ... , Fx(F) as in Section 4, we put T(F) = T( {F1, ... , Fx(F)}, C(F)) and

we call T(F) the tree ofF, and we note that then T(F) E RU*.

NOTE. We may say that a tree T' is a subtree of a tree T if for every B' E T'

there exists some (and hence a unique) BET such that a(B')

c

a( B) and -.(B') =

-.(B). Then every tree is clearly a subtree of the universal tree T(RQ, Q), which