### [1] Introduction

#### The Jacobian Conjecture in its simplest form is the following:

#### Jacobian Conjecture for two variables:

####
Given two polynomials f(x,y), g(x,y) in two variables over
a field k of characteristic 0, suppose that the following
Jacobian condition is satisfied,

J_{x,y}(f(x,y),g(x,y))=non-zero constant in k
Then we have k[x,y]=k[f,g].
The above mentioned Jacobian conjecture has been open since
1939 . Many interesting theorems follow if the Jacobian
Conjecture is true. For instance, one can deduce the
automorphism theorem of the plane quickly. Certainly there
are even harder Jacobian conjectures for three or more
variables. However, there is hardly any evidence for them
to be true! Some ten years ago, in collaboration with
A. Sathaye we used computer software to compute the three
variable case with the restriction that all three polynomials
involved are of degree less than or equal to three. The
result was affirmative, and a print-out of about one hundred
pages was generated.

One simple and useful result from the Jacobian conjecture is
that by a simple reduction argument we know if there is a
counter example to the Jacobian conjecture, then there must
be a counter example $f, g$ with the degrees of $f, g$
non-divisible by each other. In the following we usually
assume that the pair $f, g$ in our discussion are with
degrees non-divisible.

### [2] A Brief History

####
The history of the Jacobian conjecture is well-known, over a
hundred papers had been published on it. Originally the
conjecture was formulated by Keller as a problem associated
with the ``Ganze Cremona--Transformationen." In the late 60's,
we were informed about this problem by the late Professor
Zariski. Since the kernel of the problem is about the Jacobian
of a transformation, we decided to call it the Jacobian
Conjecture. Abhyankar was one of the main movers of this
conjecture and motivated research on the subject. This
conjecture could be understood by anyone with a background
in Calculus and hence it was studied by mathematicians in
many disciplines, especially Algebra, Analysis, and Complex
Geometry. From 1971 to 1978, we concentrated on this conjecture
and established several results. Among them we proved that
if the degrees of the two polynomials in two variables are
less than or equal to one hundred, then the conjecture is true.
We summarized those results in an article in 1983. The article
of Bass, Connell and Wright is indispensable reading. The
authors presented many equivalent forms of the conjecture
and discussed many lines of research. Their K-theoretic
approaches and S.S.S.Wang's result about arbitrary dimensional
result of quadratic equations are especially significant.

Recently, the activities of many good mathematicians, among
them A.Sathaye, D.T. Le, Friedland, Kaliman and others aroused
our interest again.

### [3] Basic Concepts

####
The approaches used by analysts and geometers are beyond the
scope of this presentation. The algebraic approaches are
essentially the following.

### Approaches (1)

the K-theoretic method or the stable method, had been
developed by Bass-Connell-Wright. In this method, one trades
the coefficients of the polynomials with the degrees of the
polynomials and the dimension of the space. Eventually, they
showed that if the dimension of the space could be allowed
to be arbitrarily large, then the degrees of the polynomials
could be restricted to three. Note that S.S.S.Wang showed
that the Jacobian conjecture is true for quadratic equations
for any dimension. There is a gap of degree three and two
which could not be bridged for the past ten years.

### Approach (2)

the classical Jacobian criteria for power series, implies that
$k[[f(x,y),g(x,y) ]]=k[[x,y]]$. Thus we have
x=F(f,g), y=G(f,g)
as power series. By the uniqueness of expressions, if the
Jacobian conjecture is true, then $F,G$ must be polynomials.
To prove the Jacobian conjecture, it suffices to show that F,G
are polynomials. This is one approach started by Abhyankar-Bass.
Thus they consider the ``Inverse degree".

### Approach (3)

It is the study of the two curves $f=0, g=0$ over the field k
and the curve $F(x,f,g)=0$ over the field $k(x)$, especially
the singularities of them at infinity. This was an approach of
Abhyankar-Moh, and was partially done in Abhyankar's work and
completely finished in our work. Many concrete results were
established. We will explain more about this approach in the
following section.

### [4] Our research direction

The analysis of the singularities at the infinity of the three
curves involved is equivalent to the study of the
desingularization processes at the infinity. Thus the
possible singularities distributions, what we called the
``tree data" at infinity are completely known. Moreover, there
are deep numerical relations between the singularities at
infinity of the three curves involved . Those numerical
relations assume the form of a system of Diophantine equations
and thus could be checked by computers. From mathematical
reasoning we were able to show that the condition of the
Jacobian being non-zero constant is very restrictive and with
the help of a computing program, to deduce that the conjecture
is true for polynomials of degrees less than or equal to one
hundred. Certainly the number 100 is artificial and can be
increased. Indeed, once we used a computer program to filter
all pairs up to 1,000, and find only some 40 cases which have
to be treated.
The final approach should be to put all the scattering data
together. For this we will to study the defining equation
F(x,f,g)=0. To be more precise, we want to express the
defining equation F(x,f,g)=0 in terms of its approximate roots
T_i^\psi (f,g)$ and study the totality of all singularities at
infinity for each $T_i^\psi (f,g)$ and the constraints of the
defining equation on all of them.