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Computing the First Betti Number of a Semi-Algebraic Set

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Abstract

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 zeroth Betti number, and the Euler–Poincaré characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti numbers other than the zeroth one. As a consequence we also obtain algorithms for computing semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set in singly exponential time, which improves on the complexity of the previously published algorithms for this problem.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Saugata Basu

  2. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Richard Pollack

  3. IRMAR (URA CNRS 305), Université de Rennes, Campus de Beaulieu, 35042, Rennes cedex, France

    Marie-Françoise Roy

Authors
  1. Saugata Basu
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  2. Richard Pollack
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  3. Marie-Françoise Roy
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Correspondence to Saugata Basu.

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Communicated by Felipe Cucker

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Basu, S., Pollack, R. & Roy, MF. Computing the First Betti Number of a Semi-Algebraic Set. Found Comput Math 8, 97–136 (2008). https://doi.org/10.1007/s10208-007-9001-1

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  • DOI: https://doi.org/10.1007/s10208-007-9001-1

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