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Triangularization properties of power linear maps and the Structural Conjecture

100%
de Bondt M., Yan D.
Annales Polonici Mathematici
|
2014
|
tom 112
|
nr 3
247-266
EN
We discuss several additional properties a power linear Keller map may have. The Structural Conjecture of Drużkowski (1983) asserts that certain two such properties are equivalent, but we show that one of them is stronger than the other. We even show that the property of linear triangularizability is strictly in between. Furthermore, we give some positive results for small dimensions and small Jacobian ranks.
2
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Recent progress on the Jacobian Conjecture

100%
de Bondt M., van den Essen A.
Annales Polonici Mathematici
|
2005
|
tom 87
|
nr 1
1-11
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.
3
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The Jacobian Conjecture for symmetric Drużkowski mappings

100%
de Bondt M., van den Essen A.
Annales Polonici Mathematici
|
2005
|
tom 86
|
nr 1
43-46
EN
Let k be an algebraically closed field of characteristic zero and $F:= x + (Ax)^{*d}: kⁿ → kⁿ$ a Drużkowski mapping of degree ≥ 2 with det JF = 1. We classify all such mappings whose Jacobian matrix JF is symmetric. It follows that the Jacobian Conjecture holds for these mappings.
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