Sur la composition de séries formelles à croissance contrôlée

Augustin Mouze

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2002)

  • Volume: 1, Issue: 1, page 73-92
  • ISSN: 0391-173X

Abstract

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Let F be a holomorphic map from ℂ s to ℂ s defined in a neighborhood of zero such that F ( 0 ) = 0 . If the jacobian determinant of F is not identically zero, P. M. Eakin and G. A. Harris proved the following result: any formal power series 𝒜 such that 𝒜 ∘ F is analytic is itself analytic. If the jacobian determinant of F is identically zero, they proved that the previous conclusion is no more true. J. Chaumat and A.-M. Chollet extended this result in the case of formal power series satisfying growth conditions, of Gevrey type for instance. The author gets similar results when the map F is no more holomorphic. The loss of regularity on 𝒜 is optimal.

How to cite

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Mouze, Augustin. "Sur la composition de séries formelles à croissance contrôlée." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 1.1 (2002): 73-92. <http://eudml.org/doc/84468>.

@article{Mouze2002,
author = {Mouze, Augustin},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {fre},
number = {1},
pages = {73-92},
publisher = {Scuola normale superiore},
title = {Sur la composition de séries formelles à croissance contrôlée},
url = {http://eudml.org/doc/84468},
volume = {1},
year = {2002},
}

TY - JOUR
AU - Mouze, Augustin
TI - Sur la composition de séries formelles à croissance contrôlée
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2002
PB - Scuola normale superiore
VL - 1
IS - 1
SP - 73
EP - 92
LA - fre
UR - http://eudml.org/doc/84468
ER -

References

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  1. [1] E. Bierstone, Control of radii of convergence and extension of subanalytic functions, Preprint, University of Toronto (2001). Zbl1082.32002MR2045414
  2. [2] E. Bierstone – P. D. Milman, Geometric and differential properties of subanalytic sets, Ann. of Math. 147 (1998), 731-785. Zbl0912.32006MR1637671
  3. [3] J. Chaumat – A. M. Chollet, Propriétés de l’intersection des classes de Gevrey et de certaines autres classes, Bull. Sci. Math. 122 (1998), 455-485. Zbl0930.26013MR1649410
  4. [4] J. Chaumat – A. M. Chollet, On composite formal power series, Trans. Amer. Math. Soc. 353 (2001), 1691-1703. Zbl0965.13015MR1806723
  5. [5] P. M. Eakin – G. A. Harris, When Φ ( f ) convergent implies f is convergent, Math. Ann. 229 (1977), 201-210. Zbl0355.13010MR444651
  6. [6] A. M. Gabrielov, Formal Relations Between Analytic Functions, Math. USSR-Iz. 7 (1973), 1056-1088. Zbl0297.32007MR346184
  7. [7] G. Glaeser, Fonctions composées différentiables, Ann. of Math. (1963), 193-209. Zbl0106.31302MR143058
  8. [8] B. Malgrange, Sur le théorème de Maillet, Asymptotic Anal. 2 (1989), 1-4. Zbl0693.34004MR991413
  9. [9] A. Mouze, Un théorème d’Artin pour des anneaux de séries formelles à croissance contrôlée, C. R. Acad. Sci. (Paris) 330 (2000), 15-20. Zbl0969.13007MR1741161
  10. [10] A. Mouze, Anneaux de séries formelles à croissance contrôlée, Thèse, Université de Lille, juin 2000. 
  11. [11] R. Moussu – J. C. Tougeron, Fonctions composées analytiques et différentiables, C. R. Acad. Sci. (Paris) 282 (1976), 1237-1240. Zbl0334.32012MR409876
  12. [12] V. Thilliez, Sur les fonctions composées ultradifférentiables, J. Math. Pures Appl. (1997), 499-524. Zbl0878.58008MR1465608
  13. [13] J. C. Tougeron, “Idéaux de fonctions différentiables”, Springer Verlag, Berlin (1972). Zbl0251.58001MR440598

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