Betti numbers of semialgebraic sets defined by quantifier-free formulae

Reference:

Gabrielov, A. and Vorobjov, N., 2005. Betti numbers of semialgebraic sets defined by quantifier-free formulae. Discrete & Computational Geometry, 33 (3), pp. 395-401.

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Abstract

Let X be a semialgebraic set in R-n defined by a Boolean combination of atomic formulae of the kind h * 0 where * is an element of { >, greater than or equal to, = }, deg(h) < d, and the number of distinct polynomials h is k. We prove that the sum of Betti numbers of X is less than O(k(2)d)(n).

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