A Geometric Solution to the Jacobian Problem

Abstract

In this article given a geometric solution to the well-known Jacobian problem. The twodimensional polynomial Keller map is considered in four-dimensional Euclidean space R 4 . Used the concept of parallel. A well-known example of Vitushkin is also considered. Earlier it was known that Vitushkin’s map has a nonzero constant Jacobian and it is not injective. We will show that the Vitushkin map is not surjective and moreover it has two inverse maps in the domain of its definition

Keywords

• Geometric Solution,Jacobian Problem

References

1. [1]Newman D. T., One-one polynomial maps, Proc. Amer. Math. Soc., 11, 1960, 867–870.
2. [2]Bialynicki-Birula A., Rosenlicht V., Injective morphisms of real algebraic varieties, Proc.of the AMS., 13, 1962 , 200–203.
3. [3]Yagzhev A. V., On Keller’s problem, Siberian Math.J., 21, 1980, 747–754.
4. [4]Bass H., Connel E., Wright D., The Jacobian Conjecture: Reduction of Degree and Formal Expansion of the Inverse, Bulletin of the AMS, №7 (1982), 287–330.
5. [5]S. Cynk and K. Rusek, Injective endomorphisms of algebraic and analytic sets, Annales Polonici Mathematici, 56 № 1 (1991), 29–35 .
6. [6]Aro van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Progress in Mathematics, 2000, 77–79 .