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