THE BOREL MAPPING OVER SOME QUASIANALYTIC LOCAL RINGS

Abstract

Let $M={(M_j)}_j$ be an increasing sequence of positive real numbers with $M_0=1$ such that the sequence $M_{j+1}/M_j$ increases and let ${\mathcal{E}}_n(M)$ be the Denjoy-Carleman class associated to this sequence. Let ${\hat{\mathcal{E}}}_n(M)$ denote the Taylor expansion at the origin of all elements that belong to the ring ${\mathcal{E}}_n(M)$. We say that ${\hat{\mathcal{E}}}_n(M)$ satisfies the splitting property if for each $f\in {\hat{\mathcal{E}}}_n(M)$ and $A\cup B={\mathbb{N}}^n$ a partition of ${\mathbb{N}}^n$, when $G=\sum _{w\in A}{a}_w{x}^w$ and $H=\sum _{w\in B}{a}_w{x}^w$ are formal power series with $f=G+H$, then $G\in {\hat{\mathcal{E}}}_n(M)$ and $H\in {\hat{\mathcal{E}}}_n(M)$. Our first goal is to show that if the Borel mapping ${}^{\wedge }:{\mathcal{E}}_1(M)\to \mathbb{R}[[x_1]]$ is a homeomorphism onto its range for the inductive topologies, then the ring ${\mathcal{E}}_1(M)$ coincides with the ring of real analytic germs. Secondly, we will give a negative answer to the splitting property for the quasianalytic local rings ${\mathcal{E}}_n(M)$.In the last section, we will show that the ring of smooth germs that are definable in the polynomially bounded o-minimal structure of the real field expanded by all restricted functions in some Denjoy-Carleman rings does not satisfy the splitting property in general.