  We obtain a Central Limit Theorem for the normalized number of real roots of
a square Kostlan-Shub-Smale random polynomial system of any size as the degree
goes to infinity. A study of the asymptotic variance of the number of roots is
needed, this result was obtained in [2]. Afterwards we represent the number of
roots as an explicit non linear functional belonging to the It{\^o}-Wiener
chaos. This representation provides a tool for applying the Fourth Moment
Theorem and henceforth the asymptotic normality. MSC 2010 subject
classifications: Primary 60F05, 30C15, ; secondary 60G60, 65H10.
