On the asymptotic variance of the number of real roots of random polynomial systems

Diego Armentano, Jean-Marc Azaïs, Federico Dalmao, José R. León

Published 2017-03-23Version 1

We obtain the asymptotic variance, as the degree goes to infinity, of the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size. Our main tools are the Kac-Rice formula for the second factorial moment of the number of roots and a Hermite expansion of this random variable.