A Numerical Algorithm for Zero Counting. I: Complexity and Accuracy

We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(n D kappa(f)) iterations where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials' degree, and kappa(f) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finite-precision arithmetic and a polynomial bound for the precision required to ensure the returned output is correct is exhibited. This bound is a major feature of our algorithm since it is in contrast with the exponential precision required by the existing (symbolic) algorithms for counting real zeros. The algorithm parallelizes well in the sense that each iteration can be computed in parallel polynomial time with an exponential number of processors.Comment: We made minor but necessary improvements in the presentatio

Computing the homology of basic semialgebraic sets in weak exponential time

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets which works in weak exponential time. That is, out of a set of exponentially small measure in the space of data the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity which is doubly exponential (and this is so for almost all data)