Efficient polynomial system-solving by numerical methods

Abstract

These pages contain a short overview on the state of the art of efficient Numerical Analysis methods that solve systems of multi-variate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on the collaboration between Steve Smale and Mike Shub, which set the foundations of this approach to polynomial systemsolving.

Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems

“…Following a line of research opened in the 20th century by Smale (1985Smale ( , 1986, Renegar (1987Renegar ( , 1989, Demmel (1988), Shub (1993), Malajovich (1994), and Shub and Smale (1993a,b,c, 1994, 1996 and developped in the 21st century by Armentano et al (2016Armentano et al ( , 2018, Bates et al (2013), Beltrán (2011), Beltrán and Pardo (2008, 2009a,b, 2011, Beltrán andShub (2009), Briquel et al (2014), Cucker (2011, 2013), Hauenstein and Liddell (2016), Hauenstein and Sottile (2012), Lairez (2017), and Malajovich (2018), to name a few, I am interested in the number of elementary operations that one needs to compute one zero of a polynomial system in a numerical setting. On this topic, Smale's question is a landmark: "Can a zero of n complex polynomial equations in n unknowns be found approximately, on average, in polynomial time with a uniform algorithm?"…”