In rings of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in such that F is divisible by Φ in the usual ring of all power series. We give first a simple proof of the fact that F/Φ belongs also to , provided 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 ℝⁿ. The result is valid under the aforementioned assumption of stability under derivation, and it does not involve (non-)quasianalyticity properties.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm151-1-4,
author = {Vincent Thilliez},
title = {Bounds for quotients in rings of formal power series with growth constraints},
journal = {Studia Mathematica},
volume = {151},
year = {2002},
pages = {49-65},
zbl = {0998.32004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm151-1-4}
}
Vincent Thilliez. Bounds for quotients in rings of formal power series with growth constraints. Studia Mathematica, Tome 151 (2002) pp. 49-65. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm151-1-4/