Recent progress on the Jacobian Conjecture

Michiel de Bondt, and Arno van den Essen. "Recent progress on the Jacobian Conjecture." Annales Polonici Mathematici 87.1 (2005): 1-11. <http://eudml.org/doc/280480>.

@article{MichieldeBondt2005,
abstract = {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.},
author = {Michiel de Bondt, Arno van den Essen},
journal = {Annales Polonici Mathematici},
keywords = {Jacobian Conjecture; Hessian Conjecture; Laplace operator},
language = {eng},
number = {1},
pages = {1-11},
title = {Recent progress on the Jacobian Conjecture},
url = {http://eudml.org/doc/280480},
volume = {87},
year = {2005},
}

TY - JOUR
AU - Michiel de Bondt
AU - Arno van den Essen
TI - Recent progress on the Jacobian Conjecture
JO - Annales Polonici Mathematici
PY - 2005
VL - 87
IS - 1
SP - 1
EP - 11
AB - 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.
LA - eng
KW - Jacobian Conjecture; Hessian Conjecture; Laplace operator
UR - http://eudml.org/doc/280480
ER -