The complexity and geometry of numerically solving polynomial systems

TL;DR

This paper reviews the development of numerical methods for solving systems of multivariate polynomial equations, highlighting key contributions from Smale, Shub, Beltran, Pardo, Buergisser, and Cucker.

Contribution

It provides an overview of the evolution and current state of efficient numerical algorithms for polynomial system solving, emphasizing foundational and recent advances.

Findings

Summarizes key historical developments in polynomial system solving.

Highlights recent advances and contributions by leading researchers.

Provides insights into the efficiency and effectiveness of current methods.

Abstract

These pages contain a short overview on the state of the art of efficient numerical analysis methods that solve systems of multivariate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on the collaboration between Stephen Smale and Michael Shub, which set the foundations of this approach to polynomial system--solving, culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo, Peter Buergisser and Felipe Cucker.