A Numerical Algorithm for Zero Counting. III: Randomization and Condition

TL;DR

This paper studies the behavior of a condition number in a polynomial zero counting algorithm when the input systems are randomly distributed, providing probabilistic bounds and expected values for the condition number.

Contribution

It introduces a probabilistic analysis of the condition number for polynomial systems, extending previous deterministic complexity results to a stochastic setting.

Findings

Bounds for the tail probability P{kappa(f) > a}

Expected value E(log kappa(f)) derived

Enhanced understanding of the condition number's distribution

Abstract

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)).