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