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

Résumé

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 $f ∈ k^{[n]}$ homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the following: for every homogeneous polynomial $f ∈ k^{[n]}$ (of degree 4) the hypothesis $Δ^m(f^m) = 0$ for all m ≥ 1 implies that $Δ^{m-1}(f^m) = 0$ 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.

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Publié le : 2005-01-01

Zbl 1091.14019

EUDML-ID : urn:eudml:doc:280480