Recent progress on the Jacobian Conjecture
Volume 87 / 2005
Abstract
We describe some recent developments concerning the Jacobian Conjecture (JC). First we describe Drużkowski's result in [6] which asserts that it suffices to study the JC for Drużkowski mappings of the form $x+(Ax)^{*3}$ with $A^2=0$. Then we describe the authors' result of [2] which asserts that it suffices to study the JC for so-called gradient mappings, i.e. mappings of the form $x-\nabla f$, with $f\in k^{[n]}$ homogeneous of degree $4$. Using this result we explain Zhao's reformulation of the JC which asserts the following: for every homogeneous polynomial $f\in k^{[n]}$ (of degree $4$) the hypothesis ${\mit\Delta} ^m(f^m)=0$ for all $m\geq 1$ implies that ${\mit\Delta} ^{m-1}(f^m)=0$ for all large $m$ (${\mit\Delta} $ is the Laplace operator). In the last section we describe Kumar's formulation of the JC in terms of smoothness of a certain family of hypersurfaces.
