Sur la composition de séries formelles à croissance contrôlée
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 1 (2002) no. 1, p. 73-92

Let F be a holomorphic map from s to 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 𝒜F 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.

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},
     publisher = {Scuola normale superiore},
     volume = {5e s{\'e}rie, 1},
     number = {1},
     year = {2002},
     pages = {73-92},
     zbl = {pre02216748},
     mrnumber = {1994802},
     language = {fr},
     url = {http://www.numdam.org/item/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, Serie 5, Volume 1 (2002) no. 1, pp. 73-92. http://www.numdam.org/item/ASNSP_2002_5_1_1_73_0/

[1] E. Bierstone, Control of radii of convergence and extension of subanalytic functions, Preprint, University of Toronto (2001). | MR 2045414 | Zbl 1082.32002

[2] E. Bierstone - P. D. Milman, Geometric and differential properties of subanalytic sets, Ann. of Math. 147 (1998), 731-785. | MR 1637671 | Zbl 0912.32006

[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 1649410 | Zbl 0930.26013

[4] J. Chaumat - A. M. Chollet, On composite formal power series, Trans. Amer. Math. Soc. 353 (2001), 1691-1703. | MR 1806723 | Zbl 0965.13015

[5] P. M. Eakin - G. A. Harris, When Φ(f) convergent implies f is convergent, Math. Ann. 229 (1977), 201-210. | MR 444651 | Zbl 0355.13010

[6] A. M. Gabrielov, Formal Relations Between Analytic Functions, Math. USSR-Iz. 7 (1973), 1056-1088. | MR 346184 | Zbl 0297.32007

[7] G. Glaeser, Fonctions composées différentiables, Ann. of Math. (1963), 193-209. | MR 143058 | Zbl 0106.31302

[8] B. Malgrange, Sur le théorème de Maillet, Asymptotic Anal. 2 (1989), 1-4. | MR 991413 | Zbl 0693.34004

[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 1741161 | Zbl 0969.13007

[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 409876 | Zbl 0334.32012

[12] V. Thilliez, Sur les fonctions composées ultradifférentiables, J. Math. Pures Appl. (1997), 499-524. | MR 1465608 | Zbl 0878.58008

[13] J. C. Tougeron, “Idéaux de fonctions différentiables”, Springer Verlag, Berlin (1972). | MR 440598 | Zbl 0251.58001