On real roots of polynomial systems of equations in the context of group theory

The probability that a zero of a random real polynomial of increasing degree is real tends to zero. However, passing from polynomials to Laurent polynomials yields a surprising result: the probability that a root is real tends not to zero, but to $1\mathrm{/}\sqrt{3}$. A similar phenomenon has also been observed for systems of Laurent polynomials in several variables. By considering Laurent polynomials as functions associated with torus representations, we describe an analogous phenomenon for representations of any reductive linear group. In the case of a simple group, we provide a formula for the aforementioned limiting probability.