FACTORING THE JACOBIAN
7
is a strict tree of infinite height. This universal tree is like the ASHWATTHA
TREE of the Bhagwad-Gita. The stem of its root contains the embryos of all the
past, present and future creatures in nascent form. Its trunks travel upwards first
comprised of large tribes and then of smaller and smaller clans. Its "ultimate"
shoots reaching heaven are the individual souls eager to embrace their maker.
7. Slices
Given any H
=
H(X, Y) E R of Y-degree 0, we can write
H=
II
Hj
os;is;x(H)
where Ho
=
Ho(X) E K((X)) and Hi
=
Hj(X, Y) E
R~
is of Y-degree Oi for
1
~
j
~
x.(H), and x.(H) is a nonnegative integer such that: if H E k((X)) then
x.(H)
=
0, whereas
if
H
¢
k((X)) then x.(H) equals the number of irreducible
components of H (counted with multiplicities), i.e., irreducible factors of H in R.
We put
Hoo
=
II
Hj
Is;js;x(H)
and we call this the monic part of
H.
For any BE
flo
we put
II
ls;js;x(H) with HjEr(B)
and we call
O.B(H)
the B-slice of
H,
and we put
n~(H) =
II
ls;js;x(H) with HjEr'(B)
and we call
n~(H)
the primitive B-slice of
H.
Recall that the intersection multiplicity int(!,g) off E
R~
with g E R is
defined by putting int(!,g)
=
ordxResy(f,g), where Resy(f,g) denotes the Y-
resultant off and g. For any B E
flo,
we define the B-strength S(H, B) of H by
putting
{
ordx Ho(X)
+
L:
1--
3·x(H)
OiS(R*(Hi,
B))
if BE R"
S(H,B)
=
ordxHo(X)
+
L:
1s;is;x(H)
ordxHj(X, Y) if BE R"oo
with the understanding that if H
=
0 then S(H, B)
=
oo.
NOTE. By Generalized Newton Polygon (6) on page 347 of [Al], which was
reproduced as (GNP5) to (GNP7) on page 125 of [AA], we see that for every BE R"
and
f
E
a(B)
we have
int(!,O.~(H))
=
nS(B)degyO.~(H)
where degy
f =
n.
Previous Page Next Page