Surjectivity of linear operators and semialgebraic global diffeomorphisms
We prove that a C ∞ semialgebraic local diffeomorphism of ℝ n 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 diffeo...
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| Published in | Journal d'analyse mathématique (Jerusalem) Vol. 150; no. 2; pp. 789 - 802 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Jerusalem
The Hebrew University Magnes Press
01.09.2023
Springer Nature B.V |
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| Abstract | We prove that a
C
∞ semialgebraic local diffeomorphism of ℝ
n
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 ℝ
n
. 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 mappings which turns out to unify and generalize previous results of the literature. |
|---|---|
| AbstractList | We prove that a
C
∞ semialgebraic local diffeomorphism of ℝ
n
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 ℝ
n
. 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 mappings which turns out to unify and generalize previous results of the literature. 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 mappings which turns out to unify and generalize previous results of the literature. |
| Author | Dias, Luis Renato Goncalves Braun, Francisco Venato-Santos, Jean |
| Author_xml | – sequence: 1 givenname: Francisco surname: Braun fullname: Braun, Francisco organization: Departamento de Matemática, Universidade Federal de São Carlos – sequence: 2 givenname: Luis Renato Goncalves surname: Dias fullname: Dias, Luis Renato Goncalves email: lrgdias@ufu.br organization: Faculdade de Matemática, Universidade Federal de Uberlândia – sequence: 3 givenname: Jean surname: Venato-Santos fullname: Venato-Santos, Jean organization: Faculdade de Matemática, Universidade Federal de Uberlândia |
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| References | Jelonek (CR17) 2002; 239 Campbell (CR8) 2011; 24 Braun, Dias, Venato-Santos (CR3) 2021; 58 Jelonek (CR16) 1999; 315 Pinchuk (CR19) 1994; 217 Dimca (CR11) 2004 Tibăr, Zaharia (CR23) 1999; 99 Braun, dos Santos Filho, Teixeira (CR5) 2022; 119 Duistermaat, Hörmander (CR12) 1972; 128 Gwoździewicz (CR15) 2000; 48 Fernandes, Maquera, Venato-Santos (CR14) 2014; 157 Byrnes, Lindquist (CR6) 2008; 136 Białynicki-Birula, Rosenlicht (CR1) 1962; 13 Costa, Gascon, González-López (CR10) 1988; 93 Krasiński, Spodzieja (CR18) 1992 Stein (CR20) 1989; 57 Stein (CR21) 1989; 52 Stein (CR22) 1990; 54 Camacho, Lins Neto (CR7) 1985 Campbell (CR9) 2013; 51 Braun, dos Santos Filho (CR4) 2016; 204 Fernandes, Gutierrez, Rabanal (CR13) 2004; 206 Bochnak, Coste, Roy (CR2) 1998 L A Campbell (286_CR8) 2011; 24 L A Campbell (286_CR9) 2013; 51 Y Stein (286_CR22) 1990; 54 M Tibăr (286_CR23) 1999; 99 S Pinchuk (286_CR19) 1994; 217 J Bochnak (286_CR2) 1998 A Fernandes (286_CR14) 2014; 157 F Braun (286_CR4) 2016; 204 C Byrnes (286_CR6) 2008; 136 A Białynicki-Birula (286_CR1) 1962; 13 Y Stein (286_CR21) 1989; 52 T Krasiński (286_CR18) 1992 Y Stein (286_CR20) 1989; 57 Z Jelonek (286_CR17) 2002; 239 F Braun (286_CR3) 2021; 58 J J Duistermaat (286_CR12) 1972; 128 A F Costa (286_CR10) 1988; 93 A Fernandes (286_CR13) 2004; 206 F Braun (286_CR5) 2022; 119 Z Jelonek (286_CR16) 1999; 315 C Camacho (286_CR7) 1985 A Dimca (286_CR11) 2004 J Gwoździewicz (286_CR15) 2000; 48 |
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Appl. doi: 10.1016/j.topol.2016.03.010 contributor: fullname: F Braun – volume: 315 start-page: 62 year: 1999 ident: 286_CR16 publication-title: Math. Ann. doi: 10.1007/s002080050316 contributor: fullname: Z Jelonek – volume: 52 start-page: 62 year: 1989 ident: 286_CR21 publication-title: J. Anal. Math. contributor: fullname: Y Stein – volume-title: Real Algebraic Geometry year: 1998 ident: 286_CR2 doi: 10.1007/978-3-662-03718-8 contributor: fullname: J Bochnak – volume: 217 start-page: 62 year: 1994 ident: 286_CR19 publication-title: Math. Z. doi: 10.1007/BF02571929 contributor: fullname: S Pinchuk – volume: 136 start-page: 62 year: 2008 ident: 286_CR6 publication-title: Proc. Amer. Math. Soc. doi: 10.1090/S0002-9939-08-09245-9 contributor: fullname: C Byrnes – volume: 93 start-page: 62 year: 1988 ident: 286_CR10 publication-title: Invent. Math. doi: 10.1007/BF01410199 contributor: fullname: A F Costa – volume-title: Sheaves in Topology year: 2004 ident: 286_CR11 doi: 10.1007/978-3-642-18868-8 contributor: fullname: A Dimca – volume: 48 start-page: 62 year: 2000 ident: 286_CR15 publication-title: Bull. Polish Acad. Sci. Math. contributor: fullname: J Gwoździewicz – volume: 24 start-page: 62 year: 2011 ident: 286_CR8 publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2010.08.015 contributor: fullname: L A Campbell – start-page: 308 volume-title: Real Algebraic Geometry (Rennes, 1991) year: 1992 ident: 286_CR18 doi: 10.1007/BFb0084629 contributor: fullname: T Krasiński – volume: 54 start-page: 62 year: 1990 ident: 286_CR22 publication-title: J. Anal. Math. doi: 10.1007/BF02796150 contributor: fullname: Y Stein – volume: 157 start-page: 62 year: 2014 ident: 286_CR14 publication-title: Math. Proc. Cambridge Philos. Soc. doi: 10.1017/S0305004114000279 contributor: fullname: A Fernandes – volume: 239 start-page: 62 year: 2002 ident: 286_CR17 publication-title: Math. Z. doi: 10.1007/s002090100298 contributor: fullname: Z Jelonek – volume: 13 start-page: 62 year: 1962 ident: 286_CR1 publication-title: Proc. Amer. Math. Soc. doi: 10.1090/S0002-9939-1962-0140516-5 contributor: fullname: A Białynicki-Birula |
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| Snippet | We prove that a
C
∞ semialgebraic local diffeomorphism of ℝ
n
with non-properness set having codimension greater than or equal to 2 is a global diffeomorphism... 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... |
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| SubjectTerms | Abstract Harmonic Analysis Analysis Differential equations Dynamical Systems and Ergodic Theory Functional Analysis Isomorphism Linear operators Mathematics Mathematics and Statistics Operators (mathematics) Partial Differential Equations Polynomials |
| Title | Surjectivity of linear operators and semialgebraic global diffeomorphisms |
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