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 with A² = 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 homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the following: for every homogeneous polynomial (of degree 4) the hypothesis for all m ≥ 1 implies that for all large m (Δ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-1, author = {Michiel de Bondt and Arno van den Essen}, title = {Recent progress on the Jacobian Conjecture}, journal = {Annales Polonici Mathematici}, volume = {85}, year = {2005}, pages = {1-11}, zbl = {1091.14019}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-1} }
Michiel de Bondt; Arno van den Essen. Recent progress on the Jacobian Conjecture. Annales Polonici Mathematici, Tome 85 (2005) pp. 1-11. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-1/