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On Bounding the Betti Numbers and Computing the Euler Characteristic of Semi-Algebraic Sets

1999
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Discrete & Computational Geometry
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In this paper we prove new bounds on the sum of the Betti numbers of closed semi-algebraic sets and also give the first single exponential time algorithm for computing the Euler characteristic of arbitrary closed semi-algebraic sets. Given a closed semi-algebraic set S ⊂ R k defined as the intersection of a real variety, Q = 0, deg(Q) ≤ d, whose real dimension is k , with a set defined by a quantifierfree Boolean formula with no negations with atoms of the form P i = 0, P i ≥ 0, P i ≤ 0, deg(P

doi:10.1007/pl00009443
fatcat:odogbws56vb6tmffbdtlj6mb5u