How many roots of a system of random trigonometric polynomials are real?

TL;DR

This paper investigates the number of real roots in systems of random trigonometric polynomials, revealing that a positive fraction of roots remain real as degree increases, contrasting with polynomial cases.

Contribution

It extends the understanding of real roots in random systems by deriving a formula for their expected count and comparing it with known results, showing the persistence of real roots.

Findings

The average fraction of real zeros of random trigonometric polynomials converges to 1/√3.

The expected number of roots relates to the mixed volume of certain ellipsoids.

Nonzero fraction of real roots persists in systems of random trigonometric polynomials.

Abstract

The expected number of zeros of a random real polynomial of degree $N$ asymptotically equals $\frac{2}{\pi}\log N$. On the other hand, the average fraction of real zeros of a random trigonometric polynomial of increasing degree $N$ converges to not $0$ but to $1/\sqrt 3$. An average number of roots of a system of random trigonometric polynomials in several variables is equal to the mixed volume of some ellipsoids depending on the degrees of polynomials. Comparing this formula with Theorem BKK we prove that the phenomenon of nonzero fraction of real roots remains valid.