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Content available remote

Symmetric Jacobians

100%
Bondt M.
Open Mathematics
|
2014
|
tom 12
|
nr 6
787-800
EN
This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such maps. For instance, we show that it suffices to prove the Jacobian conjecture for polynomial maps x + H over ℂ such that satisfies all symmetries of the square, where H is homogeneous of arbitrary degree d ≥ 3.
2
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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.
3
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On the formal inverse of polynomial endomorphisms

80%
Ossowski P.
Colloquium Mathematicum
|
1998
|
tom 78
|
nr 1
97-104
4
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The Jacobian Conjecture in case of "non-negative coefficients"

80%
Drużkowski L.
Annales Polonici Mathematici
|
1997
|
tom 66
|
nr 1
67-75
EN
It is known that it is sufficient to consider in the Jacobian Conjecture only polynomial mappings of the form $F(x₁,...,x_n) = x - H(x) := (x₁ - H₁(x₁,...,x_n),...,x_n - H_n(x₁,...,x_n))$, where $H_j$ are homogeneous polynomials of degree 3 with real coefficients (or $H_j = 0$), j = 1,...,n and H'(x) is a nilpotent matrix for each $x = (x₁,...,x_n) ∈ ℝ^n$. We give another proof of Yu's theorem that in the case of non-negative coefficients of H the mapping F is a polynomial automorphism, and we moreover prove that in that case $deg F^{-1} ≤ (deg F)^{ind F - 1}$, where $ind F := max{ind H'(x): x ∈ ℝ^n}$. Note that the above inequality is not true when the coefficients of H are arbitrary real numbers; cf. [E3].
5
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Global attractor of a differentiable autonomous system on the plane

80%
Nguyen C.
Annales Polonici Mathematici
|
1995
|
tom 62
|
nr 2
143-154
EN
We study the structure of a differentiable autonomous system on the plane with non-positive divergence outside a bounded set. It is shown that under certain conditions such a system has a global attractor. The main result here can be seen as an improvement of the results of Olech and Meisters in [7,9] concerning the global asymptotic stability conjecture of Markus and Yamabe and the Jacobian Conjecture.
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