Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets
<|reference_start|>Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets: In this paper we describe a singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti number, and the Euler-Poincar\'e characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti numbers other than the zero-th one. We also give algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set in single-exponential time improving on previous results.<|reference_end|>
@article{basu2006computing,
    title={Computing the First Betti Numberand Describing the Connected Components
  of Semi-algebraic Sets},
    author={Saugata Basu, Richard Pollack, Marie-Francoise Roy},
    journal={arXiv preprint arXiv:math/0603248},
    year={2006},
    archivePrefix={arXiv},
    eprint={math/0603248},
    primaryClass={math.AG cs.SC math.AT}
}