Computing the Top Betti Numbers of Semi-algebraic Sets Defined by Quadratic Inequalities in Polynomial Time

For any l > 0, we present an algorithm which takes as input a semi-algebraic set, S, defined by P1 ≤ 0, . . . , Ps ≤ 0, where each Pi ∈ R[X1, . . . ,Xk ] has degree ≤ 2, and computes the top l Betti numbers of S, bk−1(S), . . . , bk−l(S), in polynomial time. The complexity of the algorithm, stated more precisely, is ∑l+2 i=0 ( s i ) k2 O(min(l,s)) . For fixed l, the complexity of the algorithm can be expressed as sl+2k2 O(l) , which is polynomial in the input parameters s and k. To our knowledge this is the first polynomial time algorithm for computing non-trivial topological invariants of semi-algebraic sets in R defined by polynomial inequalities, where the number of inequalities is not fixed and the polynomials are allowed to have degree greater than one. For fixed s, we obtain by letting l = k, an algorithm for computing all the Betti numbers of S whose complexity is k2 O(s) .