 
Summary: On the roots of a random system of equations.
The theorem of Shub & Smale and some
extensions.
JeanMarc Aza¨is
, azais@cict.fr
Mario Wschebor
, wschebor@cmat.edu.uy
August 20, 2004
Abstract
We give a new proof of a theorem of Shub & Smale [9] on the
expectation of the number of roots of a system of m random polyno
mial equations in m real variables, having a special isotropic Gaussian
distribution. Further, we present a certain number of extensions, in
cluding the behaviour as m + of the variance of the number of
roots, when the system of equations is also stationary.
AMS subject classification: Primary 60G60, 14Q99. Secondary: 30C15.
Short Title: Random systems
Key words and phrases: Random polynomials, system of random equa
tions, Rice Formula.
1 Introduction
