Recent progress on the Jacobian Conjecture
Michiel de Bondt ; Arno van den Essen
Annales Polonici Mathematici, Tome 85 (2005), p. 1-11 / Harvested from The Polish Digital Mathematics Library

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 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 fk[n] homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the following: for every homogeneous polynomial fk[n] (of degree 4) the hypothesis Δm(fm)=0 for all m ≥ 1 implies that Δm-1(fm)=0 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.

Publié le : 2005-01-01
EUDML-ID : urn:eudml:doc:280480
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     year = {2005},
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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/