Computing the euler–poincaré characteristics of sign conditions

Abstract

Computing various topological invariants of semi-algebraic sets in single exponential time is an active area of research. Several algorithms are known for deciding emptiness, computing the number of connected components of semi-algebraic sets in single exponential time etc. However, an algorithm for computing all the Betti numbers of a given semi-algebraic set in single exponential time is still lacking. In this paper we describe a new, improved algorithm for computing the Euler–Poincaré characteristic (which is the alternating sum of the Betti numbers) of the realization of each realizable sign condition of a family of polynomials restricted to a real variety. The complexity of the algorithm is $$s^{k'+1}O(d)^{k} + s^{k'}((k' log_{2}(s) + k log_{2}(d))d)^{O(k)},$$ where s is the number of polynomials, k the number of variables, d a bound on the degrees, and k0 the real dimension of the variety. A consequence of our result is that the Euler–Poincaré characteristic of any locally closed semi-algebraic set can be computed with the same complexity. The best previously known single exponential time algorithm for computing the Euler–Poincaré characteristic of semi-algebraic sets worked only for a more restricted class of closed semi-algebraic sets and had a complexity of (ksd)O(k).