Unimodularity of zeros of self-inversive polynomials

2012/01/03 by Matilde Laĺın, Lalin, Matilde N, Chris Smyth +1 · 1 citation
Mathematics · #11B68 #26C10 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1201.0774

openalex publication_date 2012/01/03 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

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 Lalín and Rogers. We prove that all polynomials in this family have their zeros on the unit circle, a result conjectured by Lalín and Rogers on computational evidence.

Cited by

• Variant of Ramanujan polynomials and a conjecture of Maji & Sarkar

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