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1

Injectivity of non-singular planar maps with one convex component

Sabatini Marco

Bulletin of the Polish Academy of Sciences. Mathematics

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2021

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Vol. 69, no. 2

107--113

EN

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.

2

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

Bogdan P., Hajto Z., Adamus E.

Schedae Informaticae

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2017

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Vol. 26

49--59

EN

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.

3

Some remarks to the Jacobian conjecture

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

Journal of Applied Mathematics and Computational Mechanics

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2017

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Vol. 16, nr 1

87--96

EN

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.

4

A note on the divergence-free Jacobian Conjecture in R2

Sabatini M.

Bulletin of the Polish Academy of Sciences. Mathematics

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2009

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Vol. 57, no 2

109-115

EN

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.

5

On the global asymptotic stability problem and the Jacobian conjecture

Drużkowski L. M.

Control and Cybernetics

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2005

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Vol. 34, no 3

747-762

EN

In this survey, we recall the formulation of the problems and give a review of some nontrivial results in the area. Let F = (F1,...,Fn] : R^n - --> R^n be a C^1 map and let F'(x) and Jac F(x) = det F'(x) denote the Jacobian matrix and the jacobian of F at a point x belongs to R^n, respectively. The Global Asymptotic Stability Problem (GASP) reads as follows: Assume that F(0] = 0 and at any point x belongs to R^n all eigenvalues of F'(x) have negative real parts. Then consider the associated system of differential equations x'j(t] = Fj(x1(t), ...,Xn(t)), j = 1,...,n. The question is whether the solution x[t] = 0 is globally asymptotically stable. If n > 2, then the answer is negative (even if F is a a polynomial automorphism), so from now on (GASP) denotes (GASP) restricted to R^2. In 1963, Olech showed that under the (GASP) assumption (i. e., Jac F[x) > 0 and Trace F'(x) = [...] < 0 for any x belong to R^2) the conclusion of (GASP) is equivalent to the injectivity of F. In 1994, Fessler, and independently Gutierrez, proved the injectivity of F and, due to the above mentioned Olech's equivalence, gave the affirmative answer to the two-dimensional (GASP). Let K denote R or C, n > 1. The Jacobian Conjecture can be formulated as follows: If F = (F1, ... ,Fn) : K^n --> K^n is a polynomial map with a constant nonzero jacobian, then F is a polynomial automorphism (i.e. there exists F^-1 and F^-1 is also a polynomial map). Although the Jacobian Conjecture is still unsolved even in the case of n = 2, it is convenient, to consider the so called Generalized Jacobian Conjecture (for short (GJC)): the Jacobian Conjecture holds for every n > 1. We give a review of some interesting conditions equivalent to the Jacobian Conjecture, including Meisters and Olech's result on the existence of a poly-flow solution of the associated Ważewski equation x'(t) = [F'(x(t))]^-1 (a). We also present, a reduction of (GJC) to the case of F of degree 3 and of special forms, then some partial results, and (JC)'s relations with other problems.

6

A geometry of polynomial transformations of the real plane

Jelonek Z.

Bulletin of the Polish Academy of Sciences. Mathematics

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2000

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Vol. 48, no 1

57-62

EN

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

7

A geometry of Pinchuk's map

Goździewicz G.

Bulletin of the Polish Academy of Sciences. Mathematics

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2000

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Vol. 48, no 1

69-75

EN

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.