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Abstract

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