Recent progress on the Jacobian Conjecture

In this paper 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 = 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 − ∇f , with f ∈ k homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the following: for every homogeneous polynomial f ∈ k (of degree 4) the hypothesis ∆(f) = 0 for all m ≥ 1 implies that ∆m−1(fm) = 0 for all large m (∆ is the Laplace operator). In the last section we descibe Kumar’s formulation of the JC in terms of smoothness of a certain family of hypersurfaces. 1