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
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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
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