Computing the First Few Betti Numbers of Semi-algebraic Sets in Single Exponential Time

For every fixed l > 0, we describe a singly exponential algorithm for computing the first l Betti number of a given semi-algebraic set. More precisely, we describe an algorithm that given a semi-algebraic set S ⊂ Rk a semi-algebraic set defined by a Boolean formula with atoms of the form P > 0, P < 0, P = 0 for P ∈ P ⊂ R[X1, . . . ,Xk], computes b0(S), . . . , bl(S). The complexity of the algorithm is (sd) O(l) , where where s = #(P) and d = maxP∈P deg(P ). Previously, singly exponential time algorithms were known only for computing the Euler-Poincaré characteristic, the zero-th and the first Betti numbers.