Pseudo-immersions
Annales de l'Institut Fourier, Volume 37 (1987) no. 2, pp. 195-221.

Let f:(R m ,0)(R n ,0) be a 𝒞 -germ. f is said to be a pseudo-immersion (noted fΨ n,m ) if for continuous germ g:(R,0)(R m ,0), fg𝒞 implies g𝒞 . Ψ n,1 , is completely determined, for each natural n,Ψ 2,2 is shown to coincide with Diff 2 . If K=R or C and g:KK is such that g 2 and g 3 are in 𝒞 . If K=H (field of Hamiltonians), a counter-exemple shows that this implication is no more valid.

Si f est un germe 𝒞 de (R n ,0), on dira que f est une pseudo-immersion (on notera fΨ n,m ) si tous les germes continus g de (R,0) dans (R m ,0), tels que fg𝒞 sont eux-mêmes 𝒞 . On détermine complètement Ψ n,1 , et on montre que Ψ 2,2 = Diff 2 . Par ailleurs, si K=R ou C et si g est une application de K dans K telle que g 2 et g 3 sont 𝒞 , alors g est aussi 𝒞 . Si K=H (corps des hamiloniens) alors cette implication n’est plus vraie.

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     title = {Pseudo-immersions},
     journal = {Annales de l'Institut Fourier},
     pages = {195--221},
     publisher = {Institut Fourier},
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     volume = {37},
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Joris, Henri; Preissmann, Emmanuel. Pseudo-immersions. Annales de l'Institut Fourier, Volume 37 (1987) no. 2, pp. 195-221. doi : 10.5802/aif.1092. http://www.numdam.org/articles/10.5802/aif.1092/

[1] R. Narasimhan, Analysis on Real and Complex Manifolds, Second edition, Masson, Paris, 1973.

[2] H. Joris, Une C∞-application non-immersive qui possède la propriété universelle des immersions, Archiv der Mathematik, 39 (1982), 269-277. | MR | Zbl

[3] J. Boman, Differentiability of a function and of its compositions with functions of one variable, Math. Scand., 20 (1967), 249-268. | MR | Zbl

[4] J. Duncan, S. G. Krantz, H. R. Parks, Non-linear Conditions for Differentiability of Functions, Journal d'Analyse Math., 45 (1985), 46-68. | MR | Zbl

[5] C. G. Gibson, Singular Points of Smooth Mappings, Pitman, London, 1979. | MR | Zbl

[6] S. S. Abhyankar, Lectures on Expansion Techniques in Algebraic Geometry, Tata Institute, Bombay, 1977. | MR | Zbl

[7] O. Zariski, P. Samuel, Commutative Algebra, Vol. II, Van Nostrand, Princeton 1960. | MR | Zbl

[8] N. Bourbaki, Algèbre Commutative, Chap. 7, Hermann, Paris, 1965. | Zbl

[9] R. Narasimhan, Complex Analysis in One Variable, Birkhäuser, Boston, 1985. | MR | Zbl

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