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Let X ⊂ (ℝⁿ,0) be a germ of a set at the origin. We suppose X is described by a subalgebra, Cₙ(M), of the algebra of germs of $C^{\infty }$ functions at the origin (see 2.1). This algebra is quasianalytic. We show that the germ X has almost all the properties of germs of semianalytic sets. Moreover, we study the projections of such germs and prove a version of Gabrielov’s theorem.

We give some structures without quantifier elimination but in which the closure, and hence the interior and the boundary, of a quantifier free definable set is also a quantifier free definable set.

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

Let $M\subset {ℝ}^n$ be a real-analytic submanifold and H(M) the algebra of real analytic functions on M. If K ⊂ M is a compact subset we consider $S_K=f\in H\left(M\right)\left|f\right(x)\ne 0forallx\in K$; $S_K$ is a multiplicative subset of $H\left(M\right)$. Let $S_K^{-1}H\left(M\right)$ be the localization of H(M) with respect to $S_K$. In this paper we prove, first, that $S_K^{-1}H\left(M\right)$ is a regular ring (hence noetherian) and use this result in two situations:    1) For each open subset $\Omega \subset {ℝ}^n$, we denote by O(Ω) the subalgebra of H(Ω) defined as follows: f ∈ O(Ω) if and only if for all x ∈ Ω, the germ of f at x, $f_x$, is algebraic...

The main result of this paper is the following: if the Weierstrass division theorem is valid in a quasianalytic differentiable system, then this system is contained in the system of analytic germs. This result has already been known for particular examples, such as the quasianalytic Denjoy-Carleman classes.

We give some examples of polynomially bounded o-minimal expansions of the ordered field of real numbers where the Weierstrass division theorem does not hold in the ring of germs, at the origin of ℝⁿ, of definable $C^{\infty }$ functions.