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(N) = d where N denotes the number of roots of the system. Under the condition that d does not grow very fast, we prove that lim supm→+∞ V ar( N dm/2 ) ≤ 1. Moreover, if d ≥ 3 then V ar( N X dm/2 )→ 0 as m→ +∞, which implies N X dm/2 → 1 in probability. AMS subject classification: Primary 60G60, 14Q99. Secondary: 30C15. Short Title: Random polynomial systems