# Wydawnictwa / Czasopisma IMPAN / Annales Polonici Mathematici / Wszystkie zeszyty

## On some noetherian rings of $C^{\infty}$ germs on a real closed field

### Tom 100 / 2011

Annales Polonici Mathematici 100 (2011), 261-275 MSC: Primary 26E10, 13F25, 32B05, 32B20; Secondary 03C10. DOI: 10.4064/ap100-3-4

#### Streszczenie

Let $R$ be a real closed field, and denote by $\mathcal{E}_{R,n}$ the ring of germs, at the origin of $R^n$, of $C^\infty$ functions in a neighborhood of $0\in R^n$. For each $n\in\mathbb{N}$, we construct a quasianalytic subring $\mathcal{A}_{R,n}\subset\mathcal{E}_{R,n}$ with some natural properties. We prove that, for each $n\in\mathbb{N}$, $\mathcal{A}_{R,n}$ is a noetherian ring and if $R=\mathbb{R}$ (the field of real numbers), then $\mathcal{A}_{\mathbb{R},n}=\mathcal{H}_n$, where $\mathcal{H}_n$ is the ring of germs, at the origin of $\mathbb{R}^n$, of real analytic functions. Finally, we prove the Real Nullstellensatz and solve Hilbert's 17th Problem for the ring~$\mathcal{A}_{R,n}$.