Bounds for quotients in rings of formal power series with growth constraints
Volume 151 / 2002
Abstract
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.
