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4. Semialgebraic and Tame Sets

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1834)

Abstract

We prove in this chapter a classical result: the number of connected components of a plane section \(\mathrm{P}\cap \mathrm{A}\) of a semialgebraic set \(\mathrm{A}\) is uniformly bounded with respect to \(\mathrm{P}\). An explicit bound is given in terms of the diagram of \(\mathrm{A}\) and the dimension of \(\mathrm{P}\). We give a construction which provides a semialgebraic section of bounded complexity for any polynomial mapping of semialgebraic sets. In particular, any two points in a connected semialgebraic set can be joined by a semialgebraic curve of bounded complexity. We also give the definition of an o-minimal structure on the real field and show that in such a category the uniform bound for the number of connected components of plane sections holds.

Keywords

  • Plane Section
  • Algebraic Curve
  • Nonempty Interior
  • Real Analytic Manifold
  • Semialgebraic Subset

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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Correspondence to Yosef Yomdin .

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© 2004 Springer-Verlag

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Yomdin, Y., Comte, G. (2004). 4. Semialgebraic and Tame Sets. In: Tame Geometry with Application in Smooth Analysis. Lecture Notes in Mathematics, vol 1834. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-40960-1_4

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  • DOI: https://doi.org/10.1007/978-3-540-40960-1_4

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20612-5

  • Online ISBN: 978-3-540-40960-1

  • eBook Packages: Springer Book Archive