Théorèmes de préparation Gevrey et étude de certaines applications formelles

Augustin Mouze¹

¹ Laboratoire de Mathématiques, UMR 8524 Université des Sciences et Technologies de Lille 59650 Villeneuve d'Ascq Cedex, France

Annales Polonici Mathematici, Tome 81 (2003) no. 2, pp. 139-156.

Résumé

We consider subrings $A$ of the ring of formal power series. They are defined by growth conditions on coefficients such as, for instance, Gevrey conditions. We prove preparation theorems of Malgrange type in these rings. As a consequence we study maps $F$ from ${{\mathbb C}}^{s}$ to ${{\mathbb C}}^{p}$ without constant term such that the rank of the Jacobian matrix of $F$ is equal to 1. Let ${\cal A}$ be a formal power series. If $F$ is a holomorphic map, the following result is well known: ${\cal A}\circ F$ is analytic implies there exists a convergent power series $\, \widetilde { \!{\cal A}\, }$ such that ${\cal A}\circ F= \, \widetilde {\! {\cal A}\, }\circ F.$ We get similar results when the map $F$ is no longer holomorphic.

Détail

DOI : 10.4064/ap81-2-5

Mots-clés : consider subrings ring formal power series defined growth conditions coefficients instance gevrey conditions prove preparation theorems malgrange type these rings consequence study maps mathbb mathbb without constant term rank jacobian matrix equal cal formal power series holomorphic map following result known cal circ analytic implies there exists convergent power series widetilde cal cal circ widetilde cal circ get similar results map longer holomorphic