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Parametrization of semialgebraic sets

Author

Listed:
  • González-López, M.J.
  • Recio, T.
  • Santos, F.

Abstract

In this paper we consider the problem of the algorithmic parametrization of a d-dimensional semialgebraic subset S of Rn (n > d) by a semialgebraic and continuous mapping from a subset of Rd. Using the Cylindrical Algebraic Decomposition algorithm we easily obtain semialgebraic, bijective parametrizations of any given semialgebraic set; but in this way some topological properties of S (such as being connected) do not necessarily hold on the domain of the so-constructed parametrization. If the set S is connected and of dimension one, then the Euler condition on the associated graph characterizes the existence of an almost everywhere injective, finite-to-one parametrization of S with connected domain. On the other hand, for any locally closed semialgebraic set S of dimension d > 1 and connected in dimension l (i.e. such that there exists an l-dimensional path among any two points in S) we can always algorithmically obtain a bijective parametrization of S with connected in dimension l domain. Our techniques are mainly combinatorial, relying on the algorithmic triangulation of semialgebraic sets.

Suggested Citation

  • González-López, M.J. & Recio, T. & Santos, F., 1996. "Parametrization of semialgebraic sets," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 42(4), pages 353-362.
    • Handle: RePEc:eee:matcom:v:42:y:1996:i:4:p:353-362
    DOI: 10.1016/S0378-4754(96)00009-2
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