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Arbres de contact des singularités quasi-ordinaires et graphes d'adjacence pour les 3-variétés réelles

Abstract : A germ of equidimensional and reduced analytic space is called quasi-ordinary if it admits a finite projection onto a smooth space, whose discriminant locus is a divisor with normal crossings. The subject of this work is the generalization to quasi-ordinary germs of the known relations between various invariants of germs of plane curves. In the first chapter we present a global view of the concept of approximate root of a polynomial. We insist on the applications to the study of germs of plane curves, and we show that for most of these applications, the more general concept of \textit(semi-roots) is sufficient. At the beginning of the second chapter we use toric geometry to construct a normalization of quasi-ordinary germs. For irreducible germs of dimension 2 and embedding dimension 3, we give an explicit algorithm of normalization and we associate them intrinsically a semigroup. We deduce then a new proof of the invariance of normalized characteristic exponents. The concept of semiroot is essential for our method. In the third chapter we give a theorem of structure for the derivative of a quasi-ordinary polynomial, when it is itself quasi-ordinary. This generalizes a known theorem on the structure of polar curves of germs of plane curves. In order to formulate it, we introduce the Eggers-Wall tree, which allows to factorize the comparable germs following their contact with the studied germ. In the last chapter we interpret topologically the Eggers-Wall tree and the factorization of comparable germs in the case of germs of plane curves. In order to do this, we prove a general theorem on the localization after isotopy of isolable and sedentary knots in compact, orientable and irreducible 3-manifolds whose boundaries contain only tori.
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Submitted on : Monday, May 5, 2003 - 4:45:24 PM
Last modification on : Monday, May 5, 2003 - 4:45:24 PM
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Patrick Popescu-Pampu. Arbres de contact des singularités quasi-ordinaires et graphes d'adjacence pour les 3-variétés réelles. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2001. Français. ⟨tel-00002800⟩

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