  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.
