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Integers


Mathematical Citation Quotient (MCQ) 2016: 0.40

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1867-0660
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Newman Polynomials, Reducibility, and Roots on the Unit Circle

Idris Mercer
Published Online: 2012-08-01 | DOI: https://doi.org/10.1515/integers-2011-0120

Abstract.

A length k Newman polynomial is any polynomial of the form (where ). Some Newman polynomials are reducible over the rationals, and some are not. Some Newman polynomials have roots on the unit circle, and some do not. Defining, in a natural way, what we mean by the “proportion” of length k Newman polynomials with a given property, we prove that

  of length 3 Newman polynomials are reducible over the rationals,

  of length 3 Newman polynomials have roots on the unit circle,

  of length 4 Newman polynomials are reducible over the rationals,

  of length 4 Newman polynomials have roots on the unit circle.

We also show that certain plausible conjectures imply that the proportion of length 5 Newman polynomials with roots on the unit circle is .

Keywords: Polynomials; Factorization; Location of Zeros

About the article

Received: 2011-07-11

Accepted: 2011-12-14

Published Online: 2012-08-01

Published in Print: 2012-08-01


Citation Information: Integers, Volume 12, Issue 4, Pages 503–519, ISSN (Online) 1867-0652, ISSN (Print) 1867-0652, DOI: https://doi.org/10.1515/integers-2011-0120.

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© 2012 by Walter de Gruyter Berlin Boston. Copyright Clearance Center

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