The Jacobian Conjecture: New Equivalences

2021 ◽  
pp. 91-111
Author(s):  
Arno van den Essen ◽  
Shigeru Kuroda ◽  
Anthony J. Crachiola
Keyword(s):  
Jacobian Conjecture

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 Polynomial Retracts and the Jacobian Conjecture

1999 ◽  
Vol 352 (1) ◽  
pp. 477-484 ◽  
Author(s):  
Vladimir Shpilrain ◽  
Jie-Tai Yu
Keyword(s):  
Jacobian Conjecture

Polynomial remainders and plane automorphisms

2003 ◽  
Vol 68 (1) ◽  
pp. 73-79
Author(s):  
Takis Sakkalis
Keyword(s):  
Jacobian Conjecture ◽  
Real Polynomial ◽  
Polynomial Automorphisms ◽  
Polynomial Map

This note relates polynomial remainders with polynomial automorphisms of the plane. It also formulates a conjecture, equivalent to the famous Jacobian Conjecture. The latter provides an algorithm for checking when a polynomial map is an automorphism. In addition, a criterion is presented for a real polynomial map to be bijective.

scholarly journals Plane Jacobian conjecture for simple polynomials

2008 ◽  
Vol 93 (3) ◽  
pp. 247-251
Author(s):  
Nguyen Van Chau
Keyword(s):  
Jacobian Conjecture

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.

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