Real roots of polynomials in the context of group theory

By the probability of a real root of a random real polynomial of degree $n$ we mean the average number of its real roots divided by $n$. This probability tends to zero with the increasing degree of the polynomial. The transition from ordinary polynomials to Laurent ones yields an unexpected result: the probability that a root is real tends to $1/\sqrt 3$, rather than to zero. A similar effect has also been described for systems of $n$ Laurent polynomials of $n$ variables. Regarding Laurent polynomials as functions connected with torus representations, we describe a similar phenomenon for representations of an arbitrary compact Lie group. In the case when the group is simple, a formula for the above limit of the probability is presented. Bibliography: 24 titles.