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

Mourad Berraho; Mourad Berraho

doi:10.3934/math.2020063

AIMS Mathematics

2020, Volume 5, Issue 2: 923-929. doi: 10.3934/math.2020063

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Research article

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

Mourad Berraho ^,

Department of Mathematics, University Ibn Tofail, Faculty of Sciences, Kenitra, Morocco

Received: 28 October 2019 Accepted: 02 January 2020 Published: 08 January 2020
MSC : 51M99, 14P20, 32C07

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}.
- Nash germs,
- splitting property,
- Borel mapping,
- quasianalytic,
- Denjoy-Carleman classes
Citation: Mourad Berraho. On a problem concerning the ring of Nash germs and the Borel mapping[J]. AIMS Mathematics, 2020, 5(2): 923-929. doi: 10.3934/math.2020063

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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}.

References

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