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The degree at infinity of the gradient of a polynomial in two real variables

Maciej Sękalski (2005)

Annales Polonici Mathematici

Let f:ℝ² → ℝ be a polynomial mapping with a finite number of critical points. We express the degree at infinity of the gradient ∇f in terms of the real branches at infinity of the level curves {f(x,y) = λ} for some λ ∈ ℝ. The formula obtained is a counterpart at infinity of the local formula due to Arnold.

The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms

Satoshi Koike, Laurentiu Paunescu (2009)

Annales de l’institut Fourier

Let A n be a set-germ at 0 n such that 0 A ¯ . We say that r S n - 1 is a direction of A at 0 n if there is a sequence of points { x i } A { 0 } tending to 0 n such that x i x i r as i . Let D ( A ) denote the set of all directions of A at 0 n .Let A , B n be subanalytic set-germs at 0 n such that 0 A ¯ B ¯ . We study the problem of whether the dimension of the common direction set, dim ( D ( A ) D ( B ) ) is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of A and B are also subanalytic. In particular if two subanalytic...

The theorem of the complement for a quasi subanalytic set

Abdelhafed Elkhadiri (2004)

Studia Mathematica

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.

Théorème de préparation pour les fonctions logarithmico-exponentielles

Jean-Marie Lion, Jean-Philippe Rolin (1997)

Annales de l'institut Fourier

Nous donnons une preuve géométrique du théorème d’élimination des quantificateurs pour les fonctions logarithmico-exponentielles prouvé initialement par van den Dries, Macintyre et Marker. Notre démonstration n’utilise pas de Théorie des Modèles. Elle repose sur un théorème de préparation pour les fonctions sous-analytiques.

Topological invariants of analytic sets associated with Noetherian families

Aleksandra Nowel (2005)

Annales de l’institut Fourier

Let Ω n be a compact semianalytic set and let be a collection of real analytic functions defined in some neighbourhood of Ω . Let Y ω be the germ at ω of the set f f - 1 ( 0 ) . Then there exist analytic functions v 1 , v 2 , ... , v s defined in a neighbourhood of Ω such that 1 2 χ ( lk ( ω , Y ω ) ) = i = 1 s sgn v i ( ω ) , for all ω Ω .

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