FACTORING THE JACOBIAN
3
H(X, Y) and, upon letting 'Y
=
ordx H(X, Y), the X-initial form infox H(X, Y) of
H(X, Y) is its smallest X-order term Hbl(Y)X'Y.
NOTE. For example, by Newton's Theorem we have
J(Xn, Y)
=
II
[Y- Zj(X)]
l~j~n
where n is the Y -degree of
f
and
Zj(X)
=
L:j[i]Xi E k((X)) with Zj[i] E k.
iEZ
Deforming z1(X) at
X~-'
we get the deformation
where Z is an indeterminate, and substituting it back in
f
we get
f(Xn, ((X, Z))
=
L
~i(Z)Xi
i~v
where
~i(Z)
E k[Z] with
~,.,(Z)
=f
0. Now infoxf(Xn,((X,Z))
=
~,.,(Z)X"'
and,
with the normalized characteristic sequence (cihih off and the (J.t/n)-strength
S(f,J.t/n) off as defined below, it is shown in
(A-1]
that
vjn2
=
S(f,J.t/n) and
moreover:
~,.,(Y)
has more than one root in k
=?
J.t/n is a noninteger member of
the sequence
(cih~i~h·
Actually we could write Y for Z and then f(Xn,((X, Y))
is obtained from f(Xn, Y) by the linear k((X))-automorphism
Y
1---+
X~-'Y
+ (z1(X)-
zl[J.t]X~-')
of k( (X) )[Y]. For the significance of the set of initial forms info
x
H (
xn, (
(X, Z))
where H(X, Y) is any meromorphic curve and J.t varies over all integers, see
[Al].
3. Characteristic Sequences
As said above, given any f
=
f(X, Y) E
R~
of Y-degree n, by Newton's
Theorem we can write f(Xn, Y)
=
f1
1
~j~n[Y-
Zj(X)] where Zj = Zj(X) =
L:iEZ Zj[i]Xi
E
k((X)) with Zj[i]
E
k. Let Suppxzj be the X-support of Zj, i.e., the
set of all integers i for which Zj [i]
=f
0, and note that this is independent of
j.
The
said independence follows from the fact (cf.
[Al])
that any Zj(X) can be converted
into any other Zj'
(X)
by multiplying
X
by a suitable n-th root of
1.
Let m
0
= n.
Let m 1
· · ·
mh be the sequence of integers augumented by mli+l = oo and de-
fined by putting m1 = min(Suppxzl) and mi = min(Suppxz1 \moZ+· · ·+mi-lZ)
for 2
~
i
~
h+
1.
Let dh+2 = oo and for 0
~
i
~
h+ 1let di = GCD(mo, ... , mi-l)·
The sequence m
=
(mi)o~i~h+l
is called the newtonian sequence of charac-
teristic exponents off relative ton, and the sequenced=
(di)o~i9+ 2
is called
the GCD-sequence of m. For i = 0, 1,
h
+ 1 let qi = mi and for 2
~
i
~
h
let qi = mi -mi-l· For i = 0,
h
+ 1 let ri = Si = qi and for 1
~
i
~
h
let
si = q1d1 + · · · + qidi and ri = sddi. The sequence q =
(qi)o~i~h+l
is called
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