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

100%
Elkhadiri A.
Annales Polonici Mathematici
|
2011
|
tom 100
|
nr 3
261-275
EN
Let R be a real closed field, and denote by $𝓔_{R,n}$ the ring of germs, at the origin of Rⁿ, of $C^∞$ functions in a neighborhood of 0 ∈ Rⁿ. For each n ∈ ℕ, we construct a quasianalytic subring $𝓐_{R,n} ⊂ 𝓔_{R,n}$ with some natural properties. We prove that, for each n ∈ ℕ, $𝓐_{R,n}$ is a noetherian ring and if R = ℝ (the field of real numbers), then $𝓐_{ℝ,n} = 𝓗ₙ$, where 𝓗ₙ 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 $𝓐_{R,n}$.
2
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The theorem of the complement for a quasi subanalytic set

100%
Elkhadiri A.
Studia Mathematica
|
2004
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tom 161
|
nr 3
225-247
EN
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 $C^{∞}$ 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.
3
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Noethérianité de certaines algèbres de fonctions analytiques et applications

64%
Elkhadiri A., Hlal M.
Annales Polonici Mathematici
|
2000
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tom 75
|
nr 3
247-256
EN
Let $M ⊂ ℝ^{n}$ 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 $S_{K}={f ∈ H(M)| f(x) ≠ 0 for all x ∈ K}$; $S_{K}$ is a multiplicative subset of $H(M)$. Let $S_{K}^{-1}H(M)$ be the localization of H(M) with respect to $S_{K}$. In this paper we prove, first, that $S_{K}^{-1}H(M)$ is a regular ring (hence noetherian) and use this result in two situations:    1) For each open subset $Ω ⊂ ℝ^{n}$, 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 on $H(ℝ^{n})$. We prove that if Ω is a bounded subanalytic subset, then O(Ω) is a regular ring (hence noetherian).    2) Let $M ⊂ ℝ^{n}$ be a Nash submanifold and N(M) the ring of Nash functions on M; we have an injection N(M) → H(M). In [2] it was proved that every prime ideal p of N(M) generates a prime ideal of analytic functions pH(M) if M or V(p) is compact. We use our Theorem 1 to give another proof in the situation where V(p) is compact. Finally we show that this result holds in some particular situation where M and V(p) are not assumed to be compact.
4
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Weierstrass division theorem in quasianalytic local rings

64%
Elkhadiri A., Sfouli H.
Studia Mathematica
|
2008
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tom 185
|
nr 1
83-86
EN
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.
5
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Weierstrass division theorem in definable $C^{∞}$ germs in a polynomially bounded o-minimal structure

64%
Elkhadiri A., Sfouli H.
Annales Polonici Mathematici
|
2006
|
tom 89
|
nr 2
127-137
EN
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 $C^{∞}$ functions.
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