Sur la composition de séries formelles à croissance contrôlée

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 $F\left(0\right)=0.$ 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.

Détail

Classification : 13F25, 13J05, 32A05