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

Published online by Cambridge University Press:  19 November 2021

Francisco Braun
Departamento de Matemática, Universidade Federal de São Carlos, 13565–905 São Carlos, São Paulo, Brazil (
Claudia Valls
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049–001Lisboa, Portugal (


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.

Research Article
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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Braun, F. and dos Santos Filho, J. R., The real Jacobian conjecture on $ {\mathbb {R}}^{2}$ is true when one of the components has degree $3$, Discrete Contin. Dyn. Syst. 26 (2010), 7587.CrossRefGoogle Scholar
Braun, F. and Llibre, J., On the connection between global centers and global injectivity in the plane, arXiv:1706.02643 [math.DS]Google Scholar
Braun, F. and Oréfice-Okamoto, B., On polynomial submersions of degree $4$ and the real Jacobian conjecture in $ {\mathbb {R}}^{2}$, J. Math. Anal. Appl. 443 (2016), 688706.10.1016/j.jmaa.2016.05.048CrossRefGoogle Scholar
Braun, F., Giné, J. and Llibre, J., A sufficient condition in order that the real Jacobian conjecture in $ {\mathbb {R}}^{2}$ holds, J. Differ. Equ. 260 (2016), 52505258.CrossRefGoogle Scholar
Cima, A., Gasull, A. and Mañosas, F., On polynomial Hamiltonian planar vector fields, J. Differ. Equ. 106 (1993), 367383.CrossRefGoogle Scholar
Cima, A., Gasull, A. and Mañosas, F., Injectivity of polynomial local homeomorphisms of $ {\mathbb {R}}^{n}$, Nonlinear Anal. 26 (1996), 877885.10.1016/0362-546X(94)00328-FCrossRefGoogle Scholar
Cobo, M., Gutierrez, C. and Llibre, J., On the injectivity of $C^{1}$–maps of the real plane, Canad. J. Math. 54 (2002), 11871201.10.4153/CJM-2002-045-0CrossRefGoogle Scholar
Dumortier, F., Llibre, J. and Artés, J. C., Qualitative theory of planar differential systems, Universitext (Springer-Verlag, Berlin, 2006).Google Scholar
van den Essen, A., Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, Volume 190 (Birkhäuser Verlag, Basel, 2000).10.1007/978-3-0348-8440-2CrossRefGoogle Scholar
Fernandes, A., Gutierrez, C. and Rabanal, R., Global asymptotic stability for differentiable vector fields of $ {\mathbb {R}}^{2}$, J. Differ. Equ. 206 (2004), 470482.CrossRefGoogle Scholar
Gavrilov, L., Isochronicity of plane polynomial Hamiltonian systems, Nonlinearity 10 (1997), 433448.10.1088/0951-7715/10/2/008CrossRefGoogle Scholar
González-Velasco, E. A., Generic properties of polynomial vector fields at infinity, Trans. Amer. Math. Soc. 143 (1969), 201222.CrossRefGoogle Scholar
Gwoździewicz, J., The real Jacobian conjecture for polynomials of degree 3, Ann. Polon. Math. 76 (2001), 121125.10.4064/ap76-1-12CrossRefGoogle Scholar
Mello, L. F. and Xavier, F., Dynamics, points and places at infinity, and the inversion of polynomial self-maps of $ {\mathbb {R}}^{2}$, Expo. Math 38 (2020), 365376.10.1016/j.exmath.2019.03.001CrossRefGoogle Scholar
Pinchuk, S., A counterexample to the strong real Jacobian conjecture, Math. Z. 217 (1994), 14.10.1007/BF02571929CrossRefGoogle Scholar
Randall, J. D., The real Jacobian problem, Singularities, Part 2 (Arcata, Calif., 1981), pp. 411–414, Proc. Sympos. Pure Math., Volume 40 (Amer. Math. Soc., Providence, RI, 1983).CrossRefGoogle Scholar
Sabatini, M., A connection between isochronous Hamiltonian centres and the Jacobian conjecture, Nonlinear Anal. 34 (1998), 829838.10.1016/S0362-546X(97)00604-4CrossRefGoogle Scholar