Link between Noetherianity and the Weierstrass division theorem on some quasianalytic local rings (Q2846743)

Let \(\mathcal C = \{\mathcal C_n : n \in \mathbb N\}\) be a sequence of local quasianalytic rings \(\mathcal C_n\) of germs at \(0 \in \mathbb R^n\) of real-valued \(C^\infty\)-functions satisfying the following stability properties: \(\mathcal C\) contains all real polynomials and is closed under differentiation, composition, division by coordinates and the \(\mathcal C\) implicit function theorem holds. It is known by \textit{A. Elkhadiri} and \textit{H. Sfouli} [Stud. Math. 185, No. 1, 83--86 (2008; Zbl 1144.26032)] that the Weierstrass division theorem does not hold in \(\mathcal C\), unless the elements of \(\mathcal C\) are germs of real analytic functions. If the Weierstrass division theorem is available, then by standard arguments the rings \(\mathcal C_n\) are Noetherian. In this paper it is shown that, conversely, in a well-behaved Noetherian system \(\mathcal C\) the Weierstrass division theorem holds.NEWLINENEWLINEA system \(\mathcal C\) is called well-behaved if the local homomorphisms \(\mathcal C_n \to \mathcal C_n\), NEWLINENEWLINE\[NEWLINE f \mapsto f(x_1 x_2, x_2, \ldots, x_n) \quad \text{and} \quad f \mapsto f(x_1^d, x_2, \ldots, x_n), \quad d \in \mathbb N_{>0}, NEWLINE\]NEWLINE are strongly injective, i.e., the induced group homomorphisms \(\mathbb R[[x_1,\dots,x_n]]/\mathcal C_n \to \mathbb R[[x_1,\dots,x_n]]/\mathcal C_n\) are injective. Examples are germs of real analytic functions and germs of Nash functions. The author proves that in a well-behaved Noetherian system \(\mathcal C\) a local homomorphism \(\Phi : \mathcal C_n \to \mathcal C_k\) with generic rank \(n\) is strongly injective, which is a \(\mathcal C\) version of a result due to \textit{P. M. Eakin} and \textit{G. A. Harris} [Math. Ann. 229, 201--210 (1977; Zbl 0355.13010)]. As a consequence the Weierstrass division theorem holds, by the author's earlier work [Proc. Am. Math. Soc. 138, No. 4, 1433--1438 (2010; Zbl 1194.26041)].NEWLINENEWLINESince systems of germs of quasianalytic Denjoy-Carleman functions are not well-behaved, the question if the corresponding local rings are Noetherian remains open, see \textit{C. L. Childress} [Can. J. Math. 28, 938--953 (1976; Zbl 0355.32009)].