FACTORING THE JACOBIAN
9
9. Factorization of the Jacobian
Finally we are ready to state our jacobian factorization as a Theorem; it cor-
responds to Theorem (JF1) on page 160 of [AA].
THEOREM.
LetT
=
T(FG) where F
E
R \ k( (X)) is devoid of multiple factors
in R, and 0
=I=
GER. Then we have the following.
(I) If BET is such that nB(G)
=
1 and S(G, B)
=I=
0 then we have
degyf2B(J(F,G))
=
degyf2B((FG)y)
=
degyf2B(Fy)
=
D"(B)
and
degyf2~(J(F,G))
=
degyf2~((FG)y)
=
degyf2~(Fy)
=
D'(B)
and
nB(J(F,G))
=
n~(J(F,G))
II
n~,(J(F,G))
B1 E1r(T,B)
where for every B'
E
rr(T, B) we have nB'(G)
=
1 and S(G, B')
=I=
0 and
and
and
and
degyf2
8
,(J(F,G))
=
degyf2
8
,((FG)y)
=
degyf2
8
,(Fy)
=
D"(B')
degyf2~,(J(F,G))
=
degyn~,((FG)y)
=
degyf2~,(Fy)
=
D'(B').
{II) If BET is such that f2B(G)
=
1 then we have
degyf2B((FG)y)
=
degyf2B(Fy)
=
D"(B)
degyf2~((FG)y)
=
degyf2~(Fy)
=
D'(B)
nB((FG)y)
=
n~((FG)y)
II
n~,((FG)y)
B'E1r(T,B)
where for every B'
E
rr(T, B) we have nB' (G)
=
1 and
degyf2B'((FG)y)
=
degyf2B'(Fy)
=
D"(B')
and
degyf2~,((FG)y) = degyf2~,(Fy) =
D'(B').
NOTE.
Upon letting
H
=
J(F, G), let us factor
H
into irreducible components
as in Section 7. Let B
E
T
=
T(FG). Then a(B) is a set of certain irreducible
components of FG. Moreover f2B(H) is the product of those irreducible components
of
H
whose normalized contact with every member of a(B) is at least .A(B); amongst
these, only those irreducible components are kept in
n~(H)
whose normalized
contact with every member of a(B) is exactly .A( B). Finally, the NOTE in Section
7 tells us how to calculate the relevant intersection multiplicities.
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