Bounds for quotients in rings of formal power series with growth constraints

Tom 151 / 2002

Vincent Thilliez Studia Mathematica 151 (2002), 49-65 MSC: 32B05, 13F25, 26E10. DOI: 10.4064/sm151-1-4


In rings $ {\mit \Gamma }_M $ of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence $ M=(M_l )_{ l \geq 0} $ (such as rings of Gevrey series), we find precise estimates for quotients $ F/{\mit \Phi }, $ where $ F $ and $ {\mit \Phi } $ are series in $ {\mit \Gamma }_M $ such that $ F $ is divisible by $ {\mit \Phi } $ in the usual ring of all power series. We give first a simple proof of the fact that $ F/{\mit \Phi } $ belongs also to $ {\mit \Gamma }_M, $ provided $ {\mit \Gamma }_M $ is stable under derivation. By a further development of the method, we obtain the main result of the paper, stating that the ideals generated by a given analytic germ in rings of ultradifferentiable germs are closed provided the generator is homogeneous and has an isolated singularity in $ {\mathbb R}^n. $ The result is valid under the aforementioned assumption of stability under derivation, and it does not involve (non-)quasianalyticity properties.


  • Vincent ThilliezCNRS–UMR 8524, Bâtiment M2
    Université des Sciences et Technologies de Lille
    59655 Villeneuve d'Ascq Cedex, France

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