The complexity and geometry of numerically solving polynomial systems
For researchers in numerical analysis and algebraic geometry, this is a survey summarizing existing work; it is incremental and does not introduce new methods or results.
This paper reviews the state-of-the-art in numerical methods for solving multivariate polynomial systems, highlighting the foundational contributions of Smale and Shub and recent advances by Beltran, Pardo, Buergisser, and Cucker. No new results or concrete numbers are presented.
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.