Contemporary Mathematics
Volume 266, 2000
Factoring the Jacobian
Shreeram S. Abhyankar and Abdallah Assi
ABSTRACT. A meromorphic curve is a generalization of an analytic curve. It
is a polynomial in one variable whose coefficients are meromorphic series in
another variable over an algebraically closed ground field of characteristic zero.
The contact structure of two meromorphic curves leads to a factorization of
their jacobian. This provides a useful technique to tackle the bivariate jacobian
conjecture.
1.
Introduction
The technique of Newton Polygon was put forth by Newton to prove Newton's
Theorem on fractional power series expansion. This technique has many powerful
off-shoots. For instance,
as
discussed in Abhyankar's 1977 Kyoto paper
[Al),
it
gives rise to the method of deformations. In turn deformations lead to a factoriza-
tion of the jacobian of two meromorphic curves. By a meromorphic (plane) curve
we mean a polynomial
F(X, Y)
in
Y
with coefficients in the (formal) meromor-
phic series field k((X)) over an algebraically closed ground field k of characteristic
zero.
If
the said coefficients are in the (formal) power series ring k[[X]), then the
meromorphic curve is reduced to an analytic curve. Indeed, by the Weierstrass
Preparation Theorem, given any analytic curve, i.e., a member of the power series
ring k[[X, Y]], after making a linear transformation and then multiplying it by a
suitable unit in k[[X,
Y]],
it can be converted into a monic polynomial in
Y
with
coefficients in k[[X]]. An algebraic curve is given by a member of the polynomial
ring k[X, Y], and it may be regarded
as
an analytic curve around any point. As
indicated in
[Al],
meromorphic curves are the proper generalizations of algebraic
and analytic curves, suitable for effectively handling questions of affine geometry
such
as
the epimorphism theorem and the automorphism theorem. Likewise for the
jacobian problem.
So let
F
=
F(X, Y)
and
G
=
G(X, Y)
be meromorphic curves over an alge-
braically closed ground field
k
of characteristic zero, i.e., let
F
and
G
be polynomi-
als in Y over the (formal) meromorphic series field k((X)). Consider the jacobian
1991 Mathematics Subject Classification. 12F10, 14H30, 20006, 20E22.
Key words and phmses. Newton polygon, Deformations, Jacobian of meromorphic curves.
Abhyankar's work was partly supported by NSF Grant DMS 97-32592 and NSA grant MDA
904-99-1-0019.
©
2000 American Mathematical Society
1
http://dx.doi.org/10.1090/conm/266/04287
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