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.

scholarly journals On the Injectivity of C1 Maps of the Real Plane

2002 ◽  
Vol 54 (6) ◽  
pp. 1187-1201 ◽  
Author(s):  
Milton Cobo ◽  
Carlos Gutierrez ◽  
Jaume Llibre
Keyword(s):  
Jacobian Conjecture ◽  
Real Plane ◽  
The Real ◽  
Complex Eigenvalues

AbstractLet X : → be a C1 map. Denote by Spec(X) the set of (complex) eigenvalues of DXp when p varies in . If there exists ∊ > 0 such that Spec(X) ∩ (− ∊, ∊) = ∅, then X is injective. Some applications of this result to the real Keller Jacobian conjecture are discussed.

scholarly journals A new qualitative proof of a result on the real jacobian conjecture

2015 ◽  
Vol 87 (3) ◽  
pp. 1519-1524 ◽  
Author(s):  
FRANCISCO BRAUN ◽  
JAUME LLIBRE
Keyword(s):  
Differential Equations ◽  
Qualitative Theory ◽  
Jacobian Conjecture ◽  
Homogeneous Part ◽  
The Real ◽  
Polynomial Map ◽  
Real Linear

Let F= (f, g) : R2 → R2be a polynomial map such that det DF(x) is different from zero for all x∈ R2. We assume that the degrees of fand gare equal. We denote by the homogeneous part of higher degree of f and g, respectively. In this note we provide a proof relied on qualitative theory of differential equations of the following result: If do not have real linear factors in common, then F is injective.

The Real Jacobian Conjecture for polynomials of degree 3

2001 ◽  
Vol 76 (1-2) ◽  
pp. 121-125 ◽  
Author(s):  
Janusz Gwoździewicz
Keyword(s):  
Jacobian Conjecture ◽  
The Real

scholarly journals A sufficient condition in order that the real Jacobian conjecture in R2 holds

2016 ◽  
Vol 260 (6) ◽  
pp. 5250-5258 ◽  
Author(s):  
Francisco Braun ◽  
Jaume Giné ◽  
Jaume Llibre
Keyword(s):  
Jacobian Conjecture ◽  
Sufficient Condition ◽  
The Real
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