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A simple solution of Hilbert's fourteenth problem in dimension five

100%
van den Essen A.
Colloquium Mathematicum
|
2006
|
tom 105
|
nr 1
167-170
EN
We give a short proof of a counterexample (due to Daigle and Freudenburg) to Hilbert's fourteenth problem in dimension five.
2
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The sixtieth anniversary of the Jacobian Conjecture: a new approach

100%
van den Essen A.
Annales Polonici Mathematici
|
2001
|
tom 76
|
nr 1-2
77-87
EN
We investigate an approach of Bass to study the Jacobian Conjecture via the degree of the inverse of a polynomial automorphism over an arbitrary ℚ-algebra.
3
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A counterexample to a conjecture of Drużkowski and Rusek

100%
van den Essen A.
Annales Polonici Mathematici
|
1995
|
tom 62
|
nr 2
173-176
EN
Let F = X + H be a cubic homogeneous polynomial automorphism from $ℂ^n$ to $ℂ^n$. Let $p$ be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that $deg F^{-1} ≤ 3^{p-1}$. We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.
4
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The solution of the Tame Generators Conjecture according to Shestakov and Umirbaev

100%
van den Essen A.
Colloquium Mathematicum
|
2004
|
tom 100
|
nr 2
181-194
EN
The tame generators problem asked if every invertible polynomial map is tame, i.e. a finite composition of so-called elementary maps. Recently in [8] it was shown that the classical Nagata automorphism in dimension 3 is not tame. The proof is long and very technical. The aim of this paper is to present the main ideas of that proof.
5
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A class of counterexamples to the Cancellation Problem for arbitrary rings

64%
van den Essen A., van Rossum P.
Annales Polonici Mathematici
|
2001
|
tom 76
|
nr 1-2
89-93
EN
We present a class of counterexamples to the Cancellation Problem over arbitrary commutative rings, using non-free stably free modules and locally nilpotent derivations.
6
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Recent progress on the Jacobian Conjecture

64%
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.
7
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The Jacobian Conjecture for symmetric Drużkowski mappings

64%
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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