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

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-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.

• Type: Conference
• Year: 2005
• Where: STOC
• Authors: Saugata Basu, Richard Pollack, Marie-Françoise Roy
• Keywords: Algorithms, Betti Number, Individual Betti Numbers, Semi-algebraically Connected Components, STOC 2005