Recent progress on the Jacobian Conjecture

• Autorzy: Michiel de Bondt , Arno van den Essen
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 14R15: Jacobian problem

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2005
• Tom: 87
• Numer: 1
• Strony: 1-11

Daty

wydano

2005

Twórcy

autor

Michiel de Bondt

• Department of Mathematics, University of Nijmegen, Postbus 9010, 6500 GL Nijmegen, The Netherlands

autor

Arno van den Essen

• Department of Mathematics, University of Nijmegen, Postbus 9010, 6500 GL Nijmegen, The Netherlands

Identyfikatory

DOI

10.4064/ap87-0-1