On the Roots of a Random System of Equations. The Theorem of Shub and Smale and Some Extensions

Abstract

We give a new proof of a theorem of Shub and Smale on the expectation of the number of roots of a system of m random polynomial equations in m real variables, having a special isotropic Gaussian distribution. Further, we present a certain number of extensions, including the behavior as m → +∞ of the variance of the number of roots, when the system of equations is also stationary.