On the Kostlan-Shub-Smale model for random polynomial systems. Variance of the number of roots

We consider a random polynomial system with m equations and m real unknowns. Assume all equations have the same degree d and the law on the coefficients satisfies the Kostlan-Shub-Smale hypotheses. It is known that E(NX)=dm/2 where NX denotes the number of roots of the system. Under the condition that d does not grow very fast, we prove that limsupm→+∞VarNXdm/2⩽1. Moreover, if d⩾3 then VarNXdm/2→0 as m→+∞, which implies NXdm/2→1 in probability.