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

Abdelhafed Elkhadiri¹

¹ Department of Mathematics Faculty of Sciences University Ibn Tofail B.P. 133, Kénitra, Morocco

Annales Polonici Mathematici, Tome 100 (2011) no. 3, pp. 261-275.

Résumé

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~$\mathcal{A}_{R,n}$.

Détail

DOI : 10.4064/ap100-3-4

Keywords: real closed field denote mathcal ring germs origin infty functions neighborhood each mathbb construct quasianalytic subring mathcal subset mathcal natural properties prove each mathbb mathcal noetherian ring mathbb field real numbers mathcal mathbb mathcal where mathcal ring germs origin mathbb real analytic functions finally prove real nullstellensatz solve hilberts problem ring mathcal