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

### Tom 151 / 2002

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

#### Streszczenie

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.

#### Autorzy

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

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