Palindromic random trigonometric polynomials

TL;DR

This paper demonstrates that reversing coefficients of certain trigonometric polynomials relates the number of real roots, leading to the conclusion that random trigonometric polynomials with i.i.d. coefficients have at least half of their zeros real on average.

Contribution

It establishes a novel relationship between a polynomial and its reversed coefficient version, providing a lower bound on the expected fraction of real roots for random trigonometric polynomials.

Findings

Reversing coefficients links the number of real roots in related polynomials.

Expected fraction of real zeros is at least 50% for i.i.d. coefficient polynomials.

The 50% bound is proven to be optimal.

Abstract

We show that if a real trigonometric polynomial has few real roots, then the trigonometric polynomial obtained by writing the coefficients in reverse order must have many real roots. This is used to show that a class of random trigonometric polynomials has, on average, many real roots. In the case that the coefficients of a real trigonometric polynomial are independently and identically distributed, but with no other assumptions on the distribution, the expected fraction of real zeros is at least one-half. This result is best possible.