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A Generalization of the Poincaré Compactification and the Real Jacobian Conjecture

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Abstract

The Poincaré compactification is an extension of a polynomial vector field to a compact manifold. We generalize this construction to weight-homogeneous vector fields with weight exponent \((1,\ell )\) and different weight-degrees. Then we apply this generalization of the Poincaré compactification to obtain new developments in the real Jacobian conjecture.

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References

  1. Bass, H., Connell, E.H., Wright, D.: The Jacobian conjecture: reduction of degree and formal expansion of the inverse. Bull. Am. Math. Soc. 7, 287–330 (1982)

    Article  MathSciNet  Google Scholar 

  2. Blows, T.R., Lloyd, N.G.: The number of limit cycles of certain polynomial differential equations. Proc. Roy. Soc. Edinburgh 98A, 215–239 (1984)

    Article  MathSciNet  Google Scholar 

  3. Braun, F., Llibre, J.: On the connection between global centers and global injectivity, preprint, (2017)

  4. Braun, F., Oréfice-Okamoto, B.: On polynomial submersions of degree \(4\) and the real Jacobian conjecture in \({\mathbb{R}}^2\). J. Math. Anal. Appl. 43, 688–706 (2006)

    Google Scholar 

  5. Braun, F., 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, 75–87 (2010)

    Article  MathSciNet  Google Scholar 

  6. Braun, F., Valls, C.: A weight-homogeneous condition to the real Jacobian conjecture in \({\mathbb{R}}^2\). Proc. Edinb. Math. Soc. 64(4), 1028–1036 (2021)

    Article  MathSciNet  Google Scholar 

  7. Cima, A., Gasull, A., Llibre, J., Mañosas, F.: Global injectivity of polynomial maps via vector fields, in: Automor-phisms of Affine Spaces, Cura cao, 1994, Kluwer Academic Publishing, Dordrecht, pp. 105–123 (1995)

  8. Cima, A., Gasull, A., Mañosas, F.: Injectivity of polynomial local homeomorphisms of \({\mathbb{R}}^n\). Nonlinear Anal. 26, 877–885 (1996)

    Article  MathSciNet  Google Scholar 

  9. Cima, A., Llibre, J.: Bounded polynomial vector fields. Trans. Am. Math. Soc. 318, 557–579 (1990)

    Article  MathSciNet  Google Scholar 

  10. Cobo, M., Gutierrez, C., Llibre, J.: On the injectivity of \(C^1\)-maps of the real plane. Canad. J. Math. 54, 1187–1201 (2002)

    Article  MathSciNet  Google Scholar 

  11. Coll, B., Gasull, A., Prohens, R.: Differential Equations defined by the sum of two quasi-homogeneous vector fields. Can. J. Math. 49, 212–231 (1997)

    Article  MathSciNet  Google Scholar 

  12. Garcia, A., Pérez-Chavela, E., Susin, A.: A generalization of the Poincaré compactification. Arch. Rational Mech. Anal. 179, 285–302 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  13. Delgado, J., Lacomba, E.A., Llibre, J., Pérez, E.: Poincaré compactification of the Kepler and collinear \(3\)-body problem, Proceedings of the Second Workshop on Dynamical Systems of the Euler Institute, Saint Petersburg, Russian (1993)

  14. Delgado, J., Lacomba, E.A., Llibre, J., Pérez, E.: Poincaré compactification of polynomial Hamiltonian vector fields, Hamiltonain Dynamical Systems, History, Theory and Applications. The IMA Volumes in Mathematics and its Applications, Volume 63, Springer-Verlag, 99–115, (1994)

  15. Fernandes, A., Gutierrez, C., Rabanal, R.: Global asymptotic stability for differentiable vector fields of \({\mathbb{R}}^2\). J. Differ. Equ. 206, 470–482 (2004)

    Article  ADS  Google Scholar 

  16. Gavrilov, L.: Isocronicity of plane polynomial Hamiltonian systems. Nonlinearity 10, 433–448 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  17. Jiang, Q., Llibre, J.: Qualitative classification of singular points. Qual. Theor. Dyn. Syst. 6, 87–167 (2005)

    Article  MathSciNet  Google Scholar 

  18. Llibre, J., Valls, C.: A sufficient condition for the real Jacobian conjecture in \({\mathbb{R}}^2\). Nonlinear Anal. Real World Appl. 60, 103298 (2021)

    Article  Google Scholar 

  19. Milnor, J.: Topology from a Differentiable Viewpoint. University of Press of Virginia, Charlottesville (1965)

    Google Scholar 

  20. Pinchuk, S.: A counterexample to the strong real Jacobian conjecture. Math. Z. 217, 1–4 (1994)

    Article  MathSciNet  Google Scholar 

  21. Plastock, R.: Homeomorphisms between Banach spaces. Trans. Am. Math. Soc. 200, 169–183 (1974)

    Article  MathSciNet  Google Scholar 

  22. Poincare, H.: Sur les courbes définies par une équation différentielle, J. Math. Pures et Appl., Série III 7, 375–422 (1881)

  23. Poincare, H.: Sur les courbes définies par une équation différentielle. J. Math. Pures et Appl., Série III 8, 251–296 (1882)

    Google Scholar 

  24. Poincare, H.: Sur les courbes définies par une équation différentielle. J. Math. Pures et Appl., Série IV 1, 167–244 (1885)

    Google Scholar 

  25. Poincare, H.: Sur les courbes définies par une équation différentielle. J. Math. Pures et Appl., Série IV 2, 151–217 (1886)

    Google Scholar 

  26. Sabatini, M.: A connection between isochronous Hamiltonian centres and the Jacobian conjecture. Nonlinear Anal. 34, 829–838 (1998)

    Article  MathSciNet  Google Scholar 

  27. van den Essen, A.: Polynomial Automorphisms and the Jacobian Conjecture, Progress in Mathematics, vol. 190. Birkhäuser Verlag, Basel (2000)

    Google Scholar 

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  1. Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001, Lisboa, Portugal

    Claudia Valls

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  1. Claudia Valls
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Correspondence to Claudia Valls.

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Partially supported by FCT/Portugal through the Project UID/MAT/04459/2013.

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Valls, C. A Generalization of the Poincaré Compactification and the Real Jacobian Conjecture. J Dyn Diff Equat 36, 619–631 (2024). https://doi.org/10.1007/s10884-022-10149-y

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  • DOI: https://doi.org/10.1007/s10884-022-10149-y

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