Errata for Computing the Top Betti Numbers of Semialgebraic Sets Defined by Quadratic Inequalities in Polynomial Time

Abstract

For any l>0, we present an algorithm which takes as input a semi-algebraic set, S, defined by P 1≤0,…,P s ≤0, where each P i ∈R[X 1,…,X k ] has degree≤2, and computes the top l Betti numbers of S, b k−1(S),…,b k−l (S), in polynomial time. The complexity of the algorithm, stated more precisely, is $\sum_{i=0}^{\ell+2}{s\choose i}k^{2^{o(\min(\ell,s))}}$. For fixed l, the complexity of the algorithm can be expressed as $s^{\ell+2}+k^{2^{O(\ell)}}$, which is polynomial in the input parameters s and k. To our knowledge this is the first polynomial time algorithm for computing nontrivial topological invariants of semialgebraic sets in R k 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 $k^{2^{O(S)}}$.