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About certain subsets of the Euclidean space. (Sur certains sous-ensembles de l’espace euclidien.) (French) Zbl 0744.14036
Let $$\tilde{\mathcal A}_ m$$ be the algebra of functions on $$\mathbb{R}^ n$$ generated by polynomial functions and exponentials of linear forms. The subset $$S$$ in $$\mathbb{R}^ n$$ belongs to $${\mathcal P}_ n$$ if and only if there exist $$m$$ and $$F$$ in $$\tilde{\mathcal A}_{n+m}$$ for which $$S$$ is the image of the zerosubset of $$F$$ by the canonical projection of $$\mathbb{R}^{n+m}$$ onto $$\mathbb{R}^ n$$. Let $$\tilde{\mathcal P}_ n$$ be the smallest subset of parts in $$\mathbb{R}^ n$$ which contains $${\mathcal P}_ n$$, their closures and the images by the canonical projection of the elements in $$\tilde{\mathcal P}_{n+m}$$. — This family of sets is defined by an induction in two steps. The main goal of this article is to prove that $$\tilde{\mathcal P}_{n+m}$$ contains the complementary part of each element in $$\tilde{\mathcal P}_{n+m}$$, the union and the intersection of every finite family in $$\tilde{\mathcal P}_{n+m}$$. The key results for the proof are theorems by A. G. Khovanskij on the set of the solutions of a Pfaff system on a Pfaff manifold.

##### MSC:
 14P20 Nash functions and manifolds 46E99 Linear function spaces and their duals 58A17 Pfaffian systems 58A07 Real-analytic and Nash manifolds 14A05 Relevant commutative algebra
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