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

TL;DR

This paper investigates the probability of real roots in polynomial systems, revealing a surprising limiting probability of 1/√3 for Laurent polynomials and extending this phenomenon to group representations.

Contribution

It introduces a novel phenomenon where the probability of real roots approaches 1/√3 for Laurent polynomials and generalizes this to reductive group representations.

Findings

Probability of real roots in Laurent polynomials tends to 1/√3.

Derived a formula for the limiting probability in simple group cases.

Extended the phenomenon to systems associated with reductive linear groups.

Abstract

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 $1/\sqrt{3}$. 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.