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
Previous Page Next Page