Some non noetherian $C^{\infty }$ quasianalytic local rings

Abstract

We give an example of a non-noetherian quasi-analytic ring constructed using a quasi-analytic Denjoy-Carleman class. If we denote by ${\mathcal{D}}_n$ the ring of those $C^{\infty }$ quasianalytic function germs at $0\in {\mathbb{R}}^n$ which are definable in a polynomially bounded o-minimal structure. We show that the system $\{{\mathcal{D}}_n/n\in {\mathbb{N}}^{\ast }\}$ is not noetherian, i.e. there exists $m\in \mathbb{N}$, $m>1$, such that the ring ${\mathcal{D}}_m$ is not noetherian.