FACTORING THE JACOBIAN 5
with z~(X)
E
k((X)), and we define the contact set C(F) ofF by putting
C(F)
=
{(1/ N~)ordx [z~ (X) - z~, (X)] : 1 ::;
j
j' ::;
N with z~ (X)
=I
z} (X)}.
Note that then C(F) consists of the noninteger members of the normalized charac-
teristic sequences of Ft. ... ,
Fx(F),
together with noc(Fj, Fj') for all
Fj
=I
Fj'.
NOTE. In
R~,
the isosceles triangle property says that:
noc(f, /")
:2:
min(noc(f, /'), noc(f', /"))
and: if noc(f, f')
=I
noc(f', f") then
noc(f,
f")
=
min(noc(f,
J'),
noc(f', /")).
5. Buds
Let Rb be the set of all buds in R, where by a bud we mean a pair B
=
(C7(B), ..(B)) with
0
=I
C7(B)
C
R~
and ..(B)
E
Q such that noc(J, f')
:2:
..(B) for
all
f
and
f'
in C7(B); we call C7(B) the stem of B, and ..(B) the level of B; we also
put r(B)
=
{!
E
R~
: noc(f, f')
:2:
..(B) for all
f'
E
C7(B)}, and we call r(B) the
flower of B. Let Rb* be the set of all strict buds in R, where by a strict bud we
mean a bud B such that for all
f
and f' in C7(B) we have noc(f, f') ..(B). By
allowing the level ..(B) of a bud B to be -oo we get the set
R~
of all improper
buds; for any improper bud we have ..(B)
=
-oo and r(B)
=
R~.
We put Rb
=
R'r U R~ and we call a member of Rb a generalized bud. For any generalized
bud B we put r*(B)
=
{!
E
r(B) : noc(f,
f')
..(B) for some
f'
E.
C7(B)} and
r'(B)
=
r(B) \ r*(B); we call r*(B) the strict flower of B, and we call r'(B) the
primitive flower of B.
For any B
E
Rb we define the strength S(B) of B by putting S(B)
=
S(f, ..(B)) for some
f
E
r(B) where we note that S(J, ..(B)) is independent of
which
f
in r(B) we take. For any
f
E
R~ and B
E
Rb, we define the strict B-
friend R* (!,B) of B to be the unique member of Rb* U
R~
with 0'( R* (!,B))
=
{!}
such that ..(R*(f, B))
=
min{noc(f,
J') : f'
E
r(B)}. For any B
E
Rb we de-
fine R*(B) to be the set of all B'
E
Rb*
U
R~
such that ..(B')
=
..(B) and
C7(B')
=
r*(B')
n
C7(B), and we call members R*(B) strict friends of B. For
any B
E
Rb we define the degree D(B) of B by putting D(B)
=
D(f, ..(B)) for
some
f
E
r(B) where we note that D(J, ..(B)) is independent of which fin r(B)
we take. For any B
E
R'r* we define the strict degree D*(B) of B by putting
D*(B)
=
D*(f, ..(B)) for some
f
E
r*(B) where we again note that D*(f, ..(B)) is
independent of which fin r*(B) we take. For any BE Rb we define the primitive
degree D'(B) of B by putting
{
-D(B)
+
EB'ER*(B)
D*(B') if BE R'p
D'(B)
=
-1
+
EB'ER*(B) 1
if BE R~
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