Let be a holomorphic map from to defined in a neighborhood of zero such that If the jacobian determinant of is not identically zero, P. M. Eakin and G. A. Harris proved the following result: any formal power series such that is analytic is itself analytic. If the jacobian determinant of 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 is no more holomorphic. The loss of regularity on is optimal.
@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}, zbl = {pre02216748}, mrnumber = {1994802}, language = {fr}, url = {archive.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, Série 5, Tome 1 (2002) no. 1, pp. 73-92. http://archive.numdam.org/item/ASNSP_2002_5_1_1_73_0/
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