Quantitative properties of real algebraic varieties

Lionel Alberti 1, 2
1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : The introduction (section 1) presents the general subject-matter of the thesis: the quantitative measurement of the geometric properties of real algebraic varieties, and especially their triangulation. Section 2 explains a fast and certified subdivision procedure triangulating an algebraic plane curve. The mathematical tools are the topological degree, and the Bernstein's polynomial basis. Section 3 is a copy of an article explaining the subdivision method for smooth surfaces in Rn. It includes a complexity analysis. Section 4 presents a quantitative version of Thom-Mather's topological triviality for singular semi-algebraic maps. Stem from it: A “metrically stable” version of the local conic structure theorem and of the existence of a “Milnor tube” around strata. A triangulation algorithm based on Vorono˘i partitions (not completely implementable because the effective estimation of transversality is not completely detailed). Section 5 is a copy of an article published in 2008 on a sweeping method to compute the topology of singular surfaces in R3. It is based on Thom-Mathers theorem. Section 6 presents a bound on the generic number of connected components in an affine section of a real analytic germ in terms of the multiplicity and of the dimension of the ambient space. The bound does not always apply and counter-examples are given in that case.
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Lionel Alberti. Quantitative properties of real algebraic varieties. Mathematics [math]. Université Nice Sophia Antipolis, 2008. English. ⟨tel-00449506⟩



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