2 SHREERAM S. ABHYANKAR AND ABDALLAH ASSI
J(F,
G) Jcx,Y)
(F,
G)
=
FxGy - FyG x
of
F
and
G
with respect to
X
and
Y,
where subscripts denote partial derivatives. In this paper, we shall describe a
factorization of the jacobian
J(F,
G) in terms of the contact structure of the mero-
morphic curves
F
and
G.
Referring to our longer paper [AA] for proofs and details,
here we shall give a general exposition of the matter. It may be hoped that this
factorization will give a better understanding of the bivariate jacobian conjecture
which predicts that ifF and G are polynomials in X andY with 0
'#
J(F,
G) E k,
then
k[F,
G] =
k[X, Y].
Note that by taking
G =-X
we get
J(F,
G)=
Fy;
in this
situation our results generalize the work of Delgado [De], Kuo-Lu [KL], and Merle
[Me],
who dealt with the derivative
Fy
in the analytic case when
F
is a power
series in
X
and
Y.
To elucidate our jacobian factorization, we shall describe the
resulting derivative factorization in greater detail.
Merle's work gives the factorization of
Fy
in the analytic irreducible case, i.e.,
when
F
E
k[[X, Y]]
is irreducible, by using the arithmetic of the semigroup
ofF;
namely the factorization of the intersection multiplicity of
F
with
Fy
in the semi-
group of
F
yields the factorization of
Fy
into packages of irreducible components.
The arithmetic of the semigroup
ofF
does not seem to suffice in the reducible case.
Delgado's work generalized the idea of Merle in the case of two branches; his work
is rather technical and, as he points out, his method is not applicable to the case
of more than two branches.
Our method, based on the simple idea of deformations, generalizes all this
previous work not only to the meromorphic reducible case but also to a factorization
of the jacobian. More specifically, we get a factorization of the jacobian
J(F,
G) in
terms of the contact structure of the meromorphic curves
F
and
G,
which we codify
in their tree
T(FG).
We are also interested in the converse problem of getting some
information on the tree in terms of the jacobian. In particular, we would like to
characterize the tree when the jacobian depends only on the variable X, i.e., when
it belongs to
k((X)).
In case of polynomials F and G, reversing the roles of
X
and
Y,
this could eventually lead us to knowing when their jacobian belongs to both
the rings
k((X))
and
k((Y)),
and hence to
k,
which indeed is the jacobian problem.
At any rate, our trees are of interest in the study of maps of the affine plane into
itself. We propose to explore these issues in a forthcoming paper.
2. Deformations
To fix notation, let
R
be the ring of all polynomials in
Y
over
k((X)),
let R~
be the set of all monic polynomials in Y over
k((X)),
and let
R~
be the set of all
irreducible monic polynomi_als in Y over k((X)).
As said above, the aim of this paper is to describe a factorization of the jacobian
J(F,
G) with
F
and
G
in
R.
The proof (for details see our paper [AA]) is based
on the method of deformations as developed in Abhyankar's 1977 Kyoto paper
[Al].
This method can be illustrated very briefly thus. We would like to study a
meromorphic curve, i.e., a member
H
of
R,
by comparing it with various irreducible
meromorphic curves, i.e., members
f
of
R~.
To do this, we take a root
off,
deform it
by replacing one of its coefficients by an indeterminate, substitute this deformation
in
H,
and then look at the initial form of the resulting incarnation of
H.
Here
we take initial forms relative to
X.
Thus, for 0
'#
H(X, Y)
=
:E
H[i]
(Y)Xi
with
H[il(Y)
E
k[Y],
the smallest
i
with H[il(Y)
'#
0 is the X-order ordxH(X,
Y)
of
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