A Numerical Algorithm for Zero Counting. III: Randomization and Condition
For researchers in numerical algebraic geometry, this work refines the probabilistic analysis of a previously proposed algorithm for counting real zeros.
This paper analyzes the condition number of polynomial systems as a random variable, providing bounds for its tail probability and expected logarithm under a probability measure on the space of systems.
In a recent paper (Cucker, Krick, Malajovich and Wschebor, A Numerical Algorithm for Zero Counting. I: Complexity and accuracy, J. Compl.,24:582-605, 2008) we analyzed a numerical algorithm for computing the number of real zeros of a polynomial system. The analysis relied on a condition number kappa(f) for the input system f. In this paper, we look at kappa(f) as a random variable derived from imposing a probability measure on the space of polynomial systems and give bounds for both the tail P{kappa(f) > a} and the expected value E(log kappa(f)).