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Acta Mathematica Hungarica

, Volume 138, Issue 1–2, pp 85–101 | Cite as

Unimodularity of zeros of self-inversive polynomials

  • Matilde N. Lalín
  • Chris J. Smyth
Article

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 as well as by 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.

Key words and phrases

reciprocal polynomial self-inversive polynomial Ramanujan polynomial unit circle 

Mathematics Subject Classification

26C10 11B68 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2012

Authors and Affiliations

  1. 1.Département de mathématiques et de statistiqueUniversité de MontréalMontreal, QCCanada
  2. 2.School of Mathematics and Maxwell Institute for Mathematical SciencesUniversity of EdinburghEdinburghScotland

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