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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.

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
Full Text: DOI Numdam EuDML
[1] N. BOURBAKI, Fonctions d’une variable réelle, Chapitres 4, 5, 6, 7, Hermann, Paris, 1961. · Zbl 0131.05001
[2] J.-Y. CHARBONNEL, Méthode des orbites. Applications exponentielles et cônes polyédraux, preprint.
[3] A.M. GABRIELOV, Sur LES projections d’ensembles semi-analytiques, Analyse fonctionnelle et ses applications, tome 2, n°4 (1968).
[4] E.A. GORIN, Asymptotic properties of polynomials and algebraic functions of severable variables, Russian mathematical surveys, Vol. 16, n°1 (1961), 93-119. · Zbl 0102.25401
[5] H. HIRONAKA, Introduction aux sous-ensembles sous-analytiques, Colloque sur les singularités en Géométrie Analytique (Cargèse 1972), Astérisque 7-8 (1973). · Zbl 0287.14005
[6] A.G. KHOVANSKH, Sur une classe de systèmes d’équations transcendantes, Doklady Academii Nauk, tome 255, n°4, 804-807.
[7] A.G. KHOVANSKH, Variétés analytiques réelles ayant une propriété de finitude et intégrales abéliennes complexes, Analyse fonctionnelle et ses applications, tome 18, n°2 (1984).
[8] S. LOJASIEWICZ, Ensembles semi-analytiques, Preprint I.H.E.S, (1965).
[9] VAN DEN DRIES, Tarski’s problem and Pfaffian functions, Logic colloquium’84, Studies in Logic and the foundations of Mathematics, North-Holland, 1986, p. 59-90. · Zbl 0616.03018
[10] A.N. VARCHENKO, Estimations du nombre des zéros d’une intégrale abélienne, dépendant d’un paramètre, et cycles limites, Analyse fonctionnelle et ses applications, tome 18, n°2 (1984).
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