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.
