On a problem concerning the ring of Nash germs and the Borel mapping

Abstract

We denote by $\mathbb{R}$[[t]] the ring of formal power series with real coefficients. Let $\widehat{\mathcal{C}_{1}}\subset\mathbb{R}[[t]]$ be a subring. We say that $\widehat{\mathcal{C}_{1}}$ has the splitting property if for each $f\in\widehat{\mathcal{C}_{1}}$ and $A \cup B = \mathbb{N}$ such that $A\cap B = \emptyset$, if $f = G+H$ where $G = \displaystyle\sum_{w\in A}a_{w}t^{w}$ and $H = \displaystyle\sum_{w\in B}a_{w}t^{w}$ are formal power series, then $G\in\widehat{\mathcal{C}_{1}}$ and $H\in\widehat{\mathcal{C}_{1}}$. It is well known that the ring of convergent power series $\mathbb{R}${t} satisfies the splitting property. In this paper, we will examine this property for a subring of $\mathbb{R}${t} and for some local rings containing strictly $\mathbb{R}${t}.