Jacobian Conjecture(mix)

[1] IntroductionThe 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 kThen 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 HistoryThe 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 ConceptsThe 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. The general condition:let F:C^n---->C^n be a polynomial map, i.e., F(x_1,x_2,...,x_n)=(f_1(x_1,x_2,...,x_n),f_2(x_1,x_2,...,x_n),...,f_n(x_1,x_2,...,x_n),) for certain polynomials f_i in C[x_1,x_2,...,x_n].If F is invertible, then its Jacobi determinant det(delta f_i /delta x_i)which is a polynomial over C vanishes nowhere and hence must be a non-zero constant.The Jacobian conjecture asserts the converse: every polynomial map F:C^n---->C^n whose Jacobi determinant is a non-zero constant is invertible.