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A weight-homogenous condition to the real Jacobian conjecture in $ {\mathbb {R}}^{2}$

Part of: General first-order equations and systems Affine geometry Partial differential equations

Published online by Cambridge University Press:  19 November 2021

Francisco Braun Open the ORCID record for Francisco Braun [Opens in a new window]  and
Claudia Valls
Francisco Braun
Affiliation:
Departamento de Matemática, Universidade Federal de São Carlos, 13565–905 São Carlos, São Paulo, Brazil (franciscobraun@dm.ufscar.br)
Claudia Valls
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049–001Lisboa, Portugal (cvalls@math.ist.utl.pt)
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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.

Keywords

Real Jacobian conjectureglobal injectivityweight-homogeneous polynomials

MSC classification

Primary: 14R15: Jacobian problem
Secondary: 35F05: Linear first-order equations 35A30: Geometric theory, characteristics, transformations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 4 , November 2021 , pp. 1028 - 1036
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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