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

Vincent Thilliez¹

¹ CNRS–UMR 8524, Bâtiment M2 Université des Sciences et Technologies de Lille 59655 Villeneuve d'Ascq Cedex, France

Studia Mathematica, Tome 151 (2002) no. 1, pp. 49-65

Résumé

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.

Détail

DOI : 10.4064/sm151-1-4

Keywords: rings mit gamma formal power series several variables whose growth coefficients controlled suitable sequence geq rings gevrey series precise estimates quotients mit phi where mit phi series mit gamma divisible mit phi usual ring power series first simple proof mit phi belongs mit gamma provided mit gamma stable under derivation further development method obtain main result paper stating ideals generated given analytic germ rings ultradifferentiable germs closed provided generator homogeneous has isolated singularity mathbb result valid under aforementioned assumption stability under derivation does involve non quasianalyticity properties