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A note on the divergence-free Jacobian Conjecture in R2

100%

Sabatini M.

2009

tom Vol. 57, no 2

109-115

We give a shorter proof to a recent result by Neuberger [Rocky Mountain J. Math. 36 (2006)], in the real case. Our result is essentially an application of the global asymptotic stability Jacobian Conjecture. We also extend some of the results of Neuberger's paper.

Symmetric Jacobians

100%

Bondt M.

nr 6

787-800

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.

A counterexample to a conjecture of Drużkowski and Rusek

100%

van den Essen A.

1995

tom 62

nr 2

173-176

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.

Injectivity of non-singular planar maps with one convex component

80%

Sabatini M.

Bulletin of the Polish Academy of Sciences. Mathematics

2021

tom Vol. 69, no. 2

107--113

We prove that if a non-singular planar map Λ∈C2(R2,R2) has a convex component, then it is injective. We do not assume strict convexity.

The Jacobian Conjecture in case of "non-negative coefficients"

80%

Drużkowski L.

1997

tom 66

nr 1

67-75

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].

A geometry of Pinchuk's map

80%

Goździewicz G.

2000

tom Vol. 48, no 1

69--75

Sergey Pinchuk found a polynomial map from the real plane to itself which is a local diffeomorphism but is not one-to-one. The aim of this paper is to give a geometric description of Pinchuk's map.

An effective approach to Picard-Vessiot theory and the Jacobian Conjecture

80%

Bogdan P. , Hajto Z. , Adamus E.

tom Vol. 26

49--59

In this paper we present a theorem concerning an equivalent statement of the Jacobian Conjecture in terms of Picard-Vessiot extensions. Our theorem completes the earlier work of T. Crespo and Z. Hajto which suggested an effective criterion for detecting polynomial automorphisms of affine spaces. We show a simplified criterion and give a bound on the number of wronskians determinants which we need to consider in order to check if a given polynomial mapping with non-zero constant Jacobian determinant is a polynomial automorphism. Our method is specially efficient with cubic homogeneous mappings introduced and studied in fundamental papers by H. Bass, E. Connell, D.Wright and L. Drużkowski.

On the formal inverse of polynomial endomorphisms

80%

Ossowski P.

Colloquium Mathematicum

1998

tom 78

nr 1

97-104

A geometry of polynomial transformations of the real plane

70%

Jelonek Z.

2000

tom Vol. 48, no 1

57--62

In this paper we study the set of points at which a real polynomial mapping of the plane is not proper.

Some remarks to the Jacobian conjecture

60%

Lara-Dziembek S. , Biernat G. , Pawlak E.

Journal of Applied Mathematics and Computational Mechanics

2017

tom Vol. 16, nr 1

87--96

This work is related to the Jacobian Conjecture. It contains the formulas concerning algebraic dependence of the polynomial mappings having two zeros at infinity and the constant Jacobian. These relations mean that such mappings are non-invertible. They reduce the Jacobian Conjecture only to the case of mappings having one zero at infinity. This case is already solved by Abhyankar. The formulas presented in the paper were illustrated by the large example.