| Geometry © 2004 Springer-Verlag New York, LLC Betti Numbers of Semialgebraic Sets Defined by Quantifier-Free Formulae ∗ (2008) | |||||||||||||
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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 ∗∈{>, ≥, =}, 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. Let an algebraic set X ⊂ Rn be defined by polynomial equations of degrees less than d. The well-known results of Oleinik, Petrovskii [8], [9], Milnor [6], and Thom [12] provide the upper bound b(X) ≤ d(2d − 1) n−1 for the sum of Betti numbers b(X) of X (with respect to the singular homology). In a more general case of a set X defined by a system of k non-strict polynomial inequalities of degrees less than d, the sum of Betti numbers does not exceed O(kd) n. These results were later extended and refined. Basu [1] proved that if a semialgebraic set X is basic (i.e., X is defined by a system of equations and strict inequalities), or is defined by a Boolean combination (with no negations) of only non-strict or of only strict inequalities, then b(X) ≤ O(kd) n, | |||||||||||||
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