scholarly journals New classes of polynomial maps satisfying the real Jacobian conjecture in ℝ2

2019 ◽  
Vol 91 (2) ◽  
Author(s):  
JACKSON ITIKAWA ◽  
JAUME LLIBRE
Keyword(s):  
Jacobian Conjecture ◽  
The Real ◽  
Polynomial Maps

Ont he monodromy representation of polynomial maps in n variables

2002 ◽  
Vol 39 (3-4) ◽  
pp. 361-367
Author(s):  
A. Némethi ◽  
I. Sigray
Keyword(s):  
Cohomology Group ◽  
Jacobian Conjecture ◽  
Natural Basis ◽  
Generic Fiber ◽  
Monodromy Representation ◽  
Polynomial Maps ◽  
Polynomial Map

For a   non-constant polynomial map f: Cn?Cn-1 we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of [2] or [10], where the case n=2 is solved. As  applications,  we verify the Jacobian conjecture for (f,g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of [5], [7], [8], [11] and [12], where the case  n=2 is treated.

scholarly journals The real jacobian conjecture on $\R^2$ is true when one of the components has degree 3

2010 ◽  
Vol 26 (1) ◽  
pp. 75-87 ◽  
Author(s):  
Francisco Braun ◽  
◽  
José Ruidival dos Santos Filho
Keyword(s):  
Jacobian Conjecture ◽  
The Real

A weight-homogenous condition to the real Jacobian conjecture in

2021 ◽  
pp. 1-9
Author(s):  
Francisco Braun ◽  
Claudia Valls
Keyword(s):  
Differential Equations ◽  
Vector Fields ◽  
Qualitative Theory ◽  
Hamiltonian Vector ◽  
Jacobian Conjecture ◽  
Local Diffeomorphism ◽  
The Real ◽  
Hamiltonian Vector Fields ◽  
Real Linear

Abstract It is known that a polynomial local diffeomorphism $(f,\, g): {\mathbb {R}}^{2} \to {\mathbb {R}}^{2}$ is a global diffeomorphism provided the higher homogeneous terms of $f f_x+g g_x$ and $f f_y+g g_y$ do not have real linear factors in common. Here, we give a weight-homogeneous framework of this result. Our approach uses qualitative theory of differential equations. In our reasoning, we obtain a result on polynomial Hamiltonian vector fields in the plane, generalization of a known fact.

An effective study of polynomial maps

2016 ◽  
Vol 16 (08) ◽  
pp. 1750141 ◽  
Author(s):  
Elżbieta Adamus ◽  
Paweł Bogdan ◽  
Teresa Crespo ◽  
Zbigniew Hajto
Keyword(s):  
Affine Space ◽  
Finite Sequence ◽  
Jacobian Conjecture ◽  
Equivalent Condition ◽  
Effective Algorithm ◽  
Polynomial Maps ◽  
Dimension Formula ◽  
Polynomial Map ◽  
Equivalent Statement

In this paper, using an effective algorithm, we obtain an equivalent statement to the Jacobian Conjecture. For a polynomial map [Formula: see text] on an affine space of dimension [Formula: see text] over a field of characteristic [Formula: see text], we define recursively a finite sequence of polynomial maps. We give an equivalent condition to the invertibility of [Formula: see text] as well as a formula for [Formula: see text] in terms of this finite sequence of polynomial maps. Some examples illustrate the effective aspects of our approach.

scholarly journals Irreducibility Properties of Keller Maps

2016 ◽  
Vol 23 (04) ◽  
pp. 663-680 ◽  
Author(s):  
Michiel de Bondt ◽  
Dan Yan
Keyword(s):  
Linear Combination ◽  
Characteristic Zero ◽  
Jacobian Matrix ◽  
Jacobian Conjecture ◽  
Extra Condition ◽  
Irreducible Polynomials ◽  
Polynomial Maps ◽  
Polynomial Map ◽  
Keller Map

Jędrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conjecture, we study irreducibility properties of several types of Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith, we generalize the result of Bakalarski that the components of cubic homogeneous Keller maps with a symmetric Jacobian matrix (over ℂ and hence any field of characteristic zero) are irreducible. Furthermore, we show that the Jacobian Conjecture can even be reduced to any of these types with the extra condition that each affinely linear combination of the components of the polynomial map is irreducible. This is somewhat similar to reducing the planar Jacobian Conjecture to the so-called (planar) weak Jacobian Conjecture by Kaliman.

scholarly journals Symmetric Jacobians

2014 ◽  
Vol 12 (6) ◽  
Author(s):  
Michiel Bondt
Keyword(s):  
Jacobian Conjecture ◽  
Arbitrary Degree ◽  
Polynomial Maps

AbstractThis 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.

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