Abstract. We show a Condition Number Theorem for the condition number of zero counting for real polynomial systems. That is, we show that this condition number equals the inverse of the normalized distance to the set of ill-posed systems (i.e., those having multiple real zeros). As a consequence, a smoothed analysis of this condition number follows. 1

Abstract. In a recent paper [9] we analyzed a numerical algorithm for computing the number of real zeros of a polynomial system. The analysis relied on a condition number κ(f) for the input system f. In this paper we continue this analysis by looking at κ(f) as a random variable derived from imposing a probability measure on the space of polynomial systems. We give bounds for both the tail P{κ(f)> a} and the expected value E(log κ(f)). 1