8
SHREERAM S. ABHYANKAR AND ABDALLAH ASSI
8. Factorization of the Derivative
Now we are ready to state our derivative factorization as a Proposition; it
corresponds to Theorem (DF1) on page 152 of [AA].
PROPOSITION. Let T
=
T(F) where F E R \ k((X)) is devoid of multiple
factors in R. Then for the monic part (Fy )00 of Fy we have
(FY)oo
=
II
O'a(Fy).
BET\{Roo(T)}
and for every B
E
T we have
degyOs(Fy)
=
D"(B) and degyO'a(Fy)
=
D'(B).
NOTE. In the above factorization, the factor O'a(Fy) really occurs, i.e., its
Y-degree D'(B) is nonzero, if and only if either: (*) card(R*(B))
=
1 and for the
unique B' E R*(B) we have D*(B') D(B), or: (**) card(R*(B))
1.
Note that
in the irreducible case, i.e., when F
=fER~,
(*) is always satisfied. To illustrate
matters further, in the nontrivial irreducible case, i.e., when F
=fER~
with
degy
f =
n 1, using the notation of Section 3, if c1
fl.
Z then let us put:
h
=
h,
and ci
=
Ci and ri
=
ri for 1 ::::;
i ::::;
h,
and
(:4 =
di for 1 ::::;
i ::::;
h
+
1, whereas if
c1
E
Z then let us put:
h =
h-
1, and C;
=
Ci+l and ri
=
ri+l for 1 ::::;
i::::; h,
and
(:4 =
di+1 for 1 ::::;
i::::;
h
+
1.
Then his a positive integer, c1 c2
· · · eli
are in
Q \ Z, and n
=
d1 d2
· · ·
dh+l
=
1 are integers with
(:4
=
0 (mod (;4+1) for
1 ::::;
i ::::;
h.
Let Bo
=
(CT(Bo), .A(Bo)) E R~ with CT(Bo)
=
{!}
and .A(Bo)
=
-oo.
For 1 ::::;
i ::::;
h
let Bi
=
(CT(Bi), .A(Bi)) E R'r with CT(Bi)
=
{!}
and .A(Bi)
=
ci.
Then T
=
T(f)
=
{Bo,Bl, ... , Bfi} with Roo(T)
=
Bo B1
· ..
Bli, and for
1 ::::;
i ::::;
h
we have S(Bi)
=
((hri)/n
2
and D(Bi)
=
nj(h. For 1 ::::;
i ::::;
h,
upon
letting R*(Bi)
=
{Ba, we also have D*(BD
=
nf(h+l· Now
(fy )oo
=
(1/n)Jy
=
II
O'ai (fy)
l~i~h
and for 1 ::::;
i ::::;
h
we have
degyO'ai(Jy)
=
D'(Bi)
=
-D(Bi)
+
D*(BD
=
(nf(h+l)- (nj(h) 0.
Let us factor fy into irreducible factors by writing
fy
=
n
II
f(j) with f(j)
E R~
l~j~x
and for 1 ::::;
i ::::;
h
let us put
i*
=
{j
E
{1, ... , x} : noc(f, f(j))
=
C;}. Then
{1, ... , x}
=
lJ1ih
i*
is a partition into pairwise disjoint nonempty sets, and for
1 ::::;
i ::::;
h
we ha;e-
O'ai (fy)
=
II
f(j) with 0 degy f(j) E (nj(h)Z for all
j
E
i*
jEi*
and
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