Computing the first Betti number and the connected components of semi-algebraic sets

Authors:
Saugata Basu;Richard Pollack;Marie-Françoise Roy
Affiliations:
Georgia Institute of Technology, Atlanta, GA;New York University, New York, NY;Université de Rennes, Rennes, Cedex, France
Venue:
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Year:
2005

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Abstract

In this paper we describe the first singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. We also describe algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set. Singly exponential algorithms for computing the zero-th 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 zero-th one.