FACTORING THE JACOBIAN
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.
Given any H
H(X, Y) E R of Y-degree 0, we can write
Ho(X) E K((X)) and Hi
Hj(X, Y) E
is of Y-degree Oi for
x.(H), and x.(H) is a nonnegative integer such that: if H E k((X)) then
k((X)) then x.(H) equals the number of irreducible
components of H (counted with multiplicities), i.e., irreducible factors of H in R.
and we call this the monic part of
For any BE
ls;js;x(H) with HjEr(B)
and we call
the B-slice of
and we put
ls;js;x(H) with HjEr'(B)
and we call
the primitive B-slice of
Recall that the intersection multiplicity int(!,g) off E
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
we define the B-strength S(H, B) of H by
if BE R"
ordxHj(X, Y) if BE R"oo
with the understanding that if H
0 then S(H, B)
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"