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Surjectivity of linear operators and semialgebraic global diffeomorphisms

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Abstract

We prove that a C∞ semialgebraic local diffeomorphism of ℝn with non-properness set having codimension greater than or equal to 2 is a global diffeomorphism if n − 1 suitable linear partial differential operators are surjective. Then we state a new analytic conjecture for a polynomial local diffeomorphism of ℝn. Our conjecture implies a very known conjecture of Z. Jelonek. We further relate the surjectivity of these operators with the fibration concept and state a general global injectivity theorem for semialgebraic map**s which turns out to unify and generalize previous results of the literature.

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Acknowledgment

We thank the referee for helpful comments and suggestions.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de São Carlos, 13565-905, São Carlos, São Paulo, Brazil

    Francisco Braun

  2. Faculdade de Matemática, Universidade Federal de Uberlândia, 38408-100, Uberlândia, Minas Gerais, Brazil

    Luis Renato Goncalves Dias & Jean Venato-Santos

Authors
  1. Francisco Braun
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  2. Luis Renato Goncalves Dias
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  3. Jean Venato-Santos
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Corresponding author

Correspondence to Luis Renato Goncalves Dias.

Additional information

Partial support provided by the grants 2019/07316-0 and 2020/14498-4 of the São Paulo Research Foundation (FAPESP).

Partial support provided by the Grant 304163/2017-1 of the National Council for Scientific and Technological Development—CNPq.

Support provided by the Grant APQ-02056-21 of Fapemig-Brazil.

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Braun, F., Dias, L.R.G. & Venato-Santos, J. Surjectivity of linear operators and semialgebraic global diffeomorphisms. JAMA 150, 789–802 (2023). https://doi.org/10.1007/s11854-023-0286-z

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  • DOI: https://doi.org/10.1007/s11854-023-0286-z

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