# Jacobian Conjecture

## T.T.Moh

### 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.

###  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.