Newton iteration, conditioning and zero counting

TL;DR

This paper discusses methods for counting real roots of polynomial systems, focusing on Newton iteration, alpha-theory, and an inclusion-exclusion algorithm, along with complexity analysis of these numerical techniques.

Contribution

It introduces a novel inclusion-exclusion algorithm for real polynomial root counting and analyzes its complexity using advanced numerical tools.

Findings

The inclusion-exclusion algorithm effectively counts real roots.

Complexity analysis provides bounds for the algorithm's performance.

Newton iteration and alpha-theory underpin the root-counting approach.

Abstract

Those lectures revolve around the following problem: given a system of n real polynomials in n variables, count the number of real roots. The first lecture is a course on Newton iteration and alpha-theory. The second describes an inclusion-exclusion algorithm for real polynomials, developed by Felipe Cucker, Teresa Krick, Mario Wschebor and myself. The third lecture introduces tools for complexity analysis of numerical algorithms, and uses those tools to analyze our root-counting algorithm.