Sur la composition de séries formelles à croissance contrôlée

Mouze, Augustin

Mouze, Augustin

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 1, pp. 73-92.

Résumé

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 $𝒜 \circ F$ 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.

MR | 1 citation dans Numdam

Classification : 13F25, 13J05, 32A05

@article{ASNSP_2002_5_1_1_73_0,
     author = {Mouze, Augustin},
     title = {Sur la composition de s\'eries formelles \`a croissance contr\^ol\'ee},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {73--92},
     publisher = {Scuola normale superiore},
     volume = {5e s{\'e}rie, 1},
     number = {1},
     year = {2002},
     mrnumber = {1994802},
     language = {fr},
     url = {http://www.numdam.org/item/ASNSP_2002_5_1_1_73_0/}
}

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
SP  - 73
EP  - 92
VL  - 1
IS  - 1
PB  - Scuola normale superiore
UR  - http://www.numdam.org/item/ASNSP_2002_5_1_1_73_0/
LA  - fr
ID  - ASNSP_2002_5_1_1_73_0
ER  -

%0 Journal Article
%A Mouze, Augustin
%T Sur la composition de séries formelles à croissance contrôlée
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2002
%P 73-92
%V 1
%N 1
%I Scuola normale superiore
%U http://www.numdam.org/item/ASNSP_2002_5_1_1_73_0/
%G fr
%F ASNSP_2002_5_1_1_73_0

Mouze, Augustin. Sur la composition de séries formelles à croissance contrôlée. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 1, pp. 73-92. http://www.numdam.org/item/ASNSP_2002_5_1_1_73_0/

Bibliographie
Cité par

[1] E. Bierstone, Control of radii of convergence and extension of subanalytic functions, Preprint, University of Toronto (2001). | MR | Zbl

[2] E. Bierstone - P. D. Milman, Geometric and differential properties of subanalytic sets, Ann. of Math. 147 (1998), 731-785. | MR | Zbl

[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. | MR | Zbl

[4] J. Chaumat - A. M. Chollet, On composite formal power series, Trans. Amer. Math. Soc. 353 (2001), 1691-1703. | MR | Zbl

[5] P. M. Eakin - G. A. Harris, When $Φ (f)$ convergent implies $f$ is convergent, Math. Ann. 229 (1977), 201-210. | MR | Zbl

[6] A. M. Gabrielov, Formal Relations Between Analytic Functions, Math. USSR-Iz. 7 (1973), 1056-1088. | MR | Zbl

[7] G. Glaeser, Fonctions composées différentiables, Ann. of Math. (1963), 193-209. | MR | Zbl

[8] B. Malgrange, Sur le théorème de Maillet, Asymptotic Anal. 2 (1989), 1-4. | MR | Zbl

[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. | MR | Zbl

[10] A. Mouze, Anneaux de séries formelles à croissance contrôlée, Thèse, Université de Lille, juin 2000.

[11] R. Moussu - J. C. Tougeron, Fonctions composées analytiques et différentiables, C. R. Acad. Sci. (Paris) 282 (1976), 1237-1240. | MR | Zbl

[12] V. Thilliez, Sur les fonctions composées ultradifférentiables, J. Math. Pures Appl. (1997), 499-524. | MR | Zbl

[13] J. C. Tougeron, “Idéaux de fonctions différentiables”, Springer Verlag, Berlin (1972). | MR | Zbl