A Numerical Algorithm for Zero Counting. II: Distance to Ill-posedness and Smoothed Analysis

TL;DR

This paper establishes a condition number theorem linking zero counting for real polynomial systems to the inverse of the normalized distance to ill-posed systems, enabling smoothed analysis of the problem.

Contribution

It introduces a condition number theorem for zero counting, connecting it to the distance to ill-posed systems, and applies smoothed analysis to this condition number.

Findings

Condition number equals inverse of normalized distance to ill-posed systems

Smoothed analysis of the condition number is derived

Provides theoretical foundation for numerical stability in zero counting

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.