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