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CYCLOTOMIC FACTORS OF BORWEIN POLYNOMIALS

Part of: Algebraic number theory: global fields Real and complex fields Combinatorics General field theory

Published online by Cambridge University Press:  28 March 2019

BISWAJIT KOLEY Open the ORCID record for BISWAJIT KOLEY [Opens in a new window]  and
SATYANARAYANA REDDY ARIKATLA Open the ORCID record for SATYANARAYANA REDDY ARIKATLA [Opens in a new window]
BISWAJIT KOLEY
Affiliation:
Department of Mathematics, Shiv Nadar University, 201314, India email bk140@snu.edu.in
SATYANARAYANA REDDY ARIKATLA*
Affiliation:
Department of Mathematics, Shiv Nadar University, 201314, India email satyanarayana.reddy@snu.edu.in
*
✉satyanarayana.reddy@snu.edu.in
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Abstract

A cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{k}(x)$ is an essential cyclotomic factor of $f(x)\in \mathbb{Z}[x]$ if $\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$ and every prime divisor of $k$ is less than or equal to the number of terms of $f.$ We show that if a monic polynomial with coefficients from $\{-1,0,1\}$ has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers12(4) (2012), 503–519].

Keywords

Newman polynomialscyclotomic polynomialsirreducible polynomials

MSC classification

Primary: 11R09: Polynomials (irreducibility, etc.)
Secondary: 05A17: Partitions of integers 12D05: Polynomials 12E05: Polynomials (irreducibility, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 1 , August 2019 , pp. 41 - 47
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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