Unimodularity of zeros of self-inversive polynomials

TL;DR

This paper extends Cohn's condition for zeros of self-inversive polynomials to lie on the unit circle, applies it to Ramanujan-related polynomials, and proves their zeros are on the unit circle, confirming a conjecture.

Contribution

It generalizes a key condition for zeros on the unit circle and applies it to a new polynomial family related to Ramanujan polynomials, proving all zeros lie on the unit circle.

Findings

Generalized Cohn's condition for self-inversive polynomials

Proved all polynomials in the studied family have zeros on the unit circle

Confirmed a conjecture by Lalín and Rogers

Abstract

We generalise a necessary and sufficient condition given by Cohn for all the zeros of a self-inversive polynomial to be on the unit circle. Our theorem implies some sufficient conditions found by Lakatos, Losonczi and Schinzel. We apply our result to the study of a polynomial family closely related to Ramanujan polynomials, recently introduced by Gun, Murty and Rath, and studied by Murty, Smyth and Wang and Lal\'in and Rogers. We prove that all polynomials in this family have their zeros on the unit circle, a result conjectured by Lal\'in and Rogers on computational evidence.