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

D. Armentano , Jean-Marc Azaïs (1) , F. Dalmao , J. Leon (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 L2C - Laboratoire Charles Coulomb

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.

Domaines: Probabilités [math.PR]

D. Armentano, Jean-Marc Azaïs, F. Dalmao, J. Leon. On the asymptotic variance of the number of real roots of random polynomial systems. Proceedings of the American Mathematical Society, 2018, 146 (12), pp.5437-5449. ⟨10.1090/proc/14215⟩. ⟨hal-01980703⟩