Théorèmes de préparation Gevrey et étude de certaines applications formelles

Volume 81 / 2003

Augustin Mouze Annales Polonici Mathematici 81 (2003), 139-156 MSC: 13F25, 32B05, 14B12, 13J05. DOI: 10.4064/ap81-2-5


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.


  • Augustin MouzeLaboratoire de Mathématiques, UMR 8524
    Université des Sciences et Technologies de Lille
    59650 Villeneuve d'Ascq Cedex, France

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