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

TL;DR

The paper constructs two principal families of non-Noetherian quasianalytic rings: a shifted Denjoy–Carleman framework, where unions of shifted classes $C_{M_p}$ (and a carefully chosen $\tilde{M}$) yield a non-Noetherian system strictly containing the analytic germs, and a model-theoretic framework using rings of smooth definable germs $\mathcal{D}_n$ in a polynomially bounded o-minimal structure, which likewise becomes non-Noetherian when it contains strictly analytic functions. It then develops the notion of a well-behaved quasianalytic system, proving strong injectivity for the structural maps $e$ and $r_d$ (via Glaeser’s composite function theorem and related analytic tools), and showing that such a system cannot be Noetherian if it strictly contains the analytic system. Consequently, the paper demonstrates concrete non-Noetherian quasianalytic rings and provides a broader methodological path—via shifted Denjoy–Carleman classes and o-minimal definability—to exhibit non-Noetherian behavior in quasianalytic contexts. These results illuminate the limits of Noetherian properties, Weierstrass-type phenomena, and approximation theorems within quasianalytic and definable-function frameworks, with implications for both analysis and model theory.

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 $\{ \mathcal{D}_n\,/\, n\in\mathbb{N}^*\}$ is not noetherian, i.e. there exists $m\in\mathbb{N}$, $m > 1$, such that the ring $\mathcal{D}_m$ is not noetherian.

Theorems & Definitions (30)

• Definition 3.1
• Remark 3.2
• Lemma 3.3
• Proof 3.4
• Lemma 3.5
• Proof 3.6
• Remark 3.7
• Proposition 3.8
• Proof 3.9
• Lemma 4.1