Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets

Details Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets

2006-03-10

arxiv.org/abs/math/0603248v1

Mathematics

Algebraic Geometry

Scientific paper

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.

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