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

Vincent Thilliez. "Bounds for quotients in rings of formal power series with growth constraints." Studia Mathematica 151.1 (2002): 49-65. <http://eudml.org/doc/284416>.

@article{VincentThilliez2002,
abstract = {In rings $Γ_\{M\}$ of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence $M = (M_\{l\})_\{l≥0\}$ (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in $Γ_\{M\}$ 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 $Γ_\{M\}$, provided $Γ_\{M\}$ 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 ℝⁿ. The result is valid under the aforementioned assumption of stability under derivation, and it does not involve (non-)quasianalyticity properties.},
author = {Vincent Thilliez},
journal = {Studia Mathematica},
keywords = {formal power series; division theorems; analytic and ultradifferentiable function germs},
language = {eng},
number = {1},
pages = {49-65},
title = {Bounds for quotients in rings of formal power series with growth constraints},
url = {http://eudml.org/doc/284416},
volume = {151},
year = {2002},
}

TY - JOUR
AU - Vincent Thilliez
TI - Bounds for quotients in rings of formal power series with growth constraints
JO - Studia Mathematica
PY - 2002
VL - 151
IS - 1
SP - 49
EP - 65
AB - In rings $Γ_{M}$ of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence $M = (M_{l})_{l≥0}$ (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in $Γ_{M}$ 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 $Γ_{M}$, provided $Γ_{M}$ 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 ℝⁿ. The result is valid under the aforementioned assumption of stability under derivation, and it does not involve (non-)quasianalyticity properties.
LA - eng
KW - formal power series; division theorems; analytic and ultradifferentiable function germs
UR - http://eudml.org/doc/284416
ER -