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Stratifications

  • Jacek Bochnak
  • Michel Coste
  • Marie-Françoise Roy
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete / A Series of Modern Surveys in Mathematics book series (MATHE3, volume 36)

Abstract

In this chapter, we continue the study of semi-algebraic sets initiated in Chap. 2. In Section 1, we construct stratifications which have a cylindrical structure with respect to all successive projections R k R k-1, which is particularly useful in inductive arguments. In the second section, we prove that a closed and bounded semi-algebraic set can be semi-algebraically triangulated. In Section 3, we investigate the structure of continuous semi-algebraic mappings. As an application, we obtain the theorem of local conic structure of semi-algebraic sets. In Section 4 we prove that a continuous semi-algebraic function is triangulable. We obtain the finiteness of the number of topological types of polynomials in n variables of degree < d. Half-branches of algebraic curves are studied in Section 5. Semi-algebraic versions of Sard’s and Bertini’s theorems are contained in Section 6. The last section is devoted to Whitney’s conditions a and b.

Throughout this chapter, R denotes a real closed field.

Keywords

Algebraic Curve Algebraic Subset Nash Manifold Plane Algebraic Curve Nash Mapping 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Jacek Bochnak
    • 1
  • Michel Coste
    • 2
  • Marie-Françoise Roy
    • 1
  1. 1.Mathematisch InstituutVrije UniversiteitAmsterdamThe Netherlands
  2. 2.Institut Mathématique de RennesUniversité de Rennes 1Rennes cedexFrance

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