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A finite linear combination of matrix elements of a finite dimensionalrepresentation $\pi$ of a Lie group $K$ is said to be a $\pi$-polynomial on$K$. If $K$ is compact then any $\pi$-polynomial uniquely extends to aholomorphic function on the complexification $K_\C$ of $K$. For a system of $n$$\pi_i$-polynomials, where $n=\dim(K)$, we consider the proportion of realroots, that is the ratio of roots number to the number of roots in $K_\C$. Itturns out that for growing representations $\pi_i$ and random system of$\pi_i$-polynomials, the expected proportion of real roots converges not to$0$, but to a nonzero constant. The limit is calculated in terms of the volumesof some compact convex sets that determine the growth of representation $\pi$.For a $1$-dimensional torus $K$ the limit is $1/\sqrt 3$.

How many roots of a random polynomial system on a Lie group are real?