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Semi-Algebraic Sets

  • Saugata Basu
  • Richard Pollack
  • Marie-Francoise Roy
Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 10)

Abstract

We first define the topology of semi-algebraic sets and study connectedness in a general real closed field. In order to study the properties of closed and bounded semi-algebraic sets in Section 4, we introduce semi-algebraic germs in Section 3. The semi-algebraic germs over a real closed field constitute a real closed field containing infinitesimals, closely related to the field of Puiseux series, and play an important role throughout the whole book. We end the chapter with a section on semi-algebraic differentiable functions.

Keywords

Tangent Space Open Neighborhood Simple Root Implicit Function Theorem Finite Union 
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Bibliographical Notes

  1. 123.
    A. Tarski, Sur les ensembles définissables de nombres réels, Fund. Math. 17, 210–239 (1931).MATHGoogle Scholar
  2. 27.
    H. Brakhage, Topologische Eigenshaften algebraischer Gebilde über einen beliebigen reell-abgeschlossenen Konstantenkörper, Dissertation, Univ. Heidelberg (1954).Google Scholar
  3. 90.
    S. Lojasiewicz Ensembles semi-analytiques. Inst. Hautes Etudes Sci., (preprint) (1964).Google Scholar
  4. 91.
    S. Lojasiewicz Triangulation of semi-analytic sets. Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Mat. (3) 18, 449–474 (1964).MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Saugata Basu
    • 1
  • Richard Pollack
    • 2
  • Marie-Francoise Roy
    • 3
  1. 1.Georgia Institute of TechnologySchool of MathematicsAtlantaUSA
  2. 2.Courant Institute of Mathematical SciencesNew YorkUSA
  3. 3.IRMAR Campus de BeaulieuUniversité de Rennes IRennes cedexFrance

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