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2014 Negative Pisot and Salem numbers as roots of Newman polynomials
Kevin G. Hare, Michael J. Mossinghoff
Rocky Mountain J. Math. 44(1): 113-138 (2014). DOI: 10.1216/RMJ-2014-44-1-113

Abstract

A Newman polynomial has all its coefficients in $\{0,1\}$ and constant term~$1$. It is known that every root of a Newman polynomial lies in the slit annulus $\{z\in\c: \tau^{-1} \lt |{z}| \lt \tau\}\setminus\r^+$, where $\tau$ denotes the golden ratio, but not every polynomial having all of its conjugates in this set divides a Newman polynomial. We show that every negative Pisot number in $(-\tau,-1)$ with no positive conjugates, and every negative Salem number in the same range obtained by using Salem's construction on small negative Pisot numbers, is satisfied by a Newman polynomial. We also construct a number of polynomials having all their conjugates in this slit annulus, but that do not divide any Newman polynomial. Finally, we determine all negative Salem numbers in $(-\tau,-1)$ with degree at most $20$, and verify that every one of these is satisfied by a Newman polynomial.

Citation

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Kevin G. Hare. Michael J. Mossinghoff. "Negative Pisot and Salem numbers as roots of Newman polynomials." Rocky Mountain J. Math. 44 (1) 113 - 138, 2014. https://doi.org/10.1216/RMJ-2014-44-1-113

Information

Published: 2014
First available in Project Euclid: 2 June 2014

zbMATH: 1294.11186
MathSciNet: MR3216012
Digital Object Identifier: 10.1216/RMJ-2014-44-1-113

Subjects:
Primary: 11R06
Secondary: 11C08 , 11Y40

Keywords: Newman polynomials , Pisot Numbers , Salem numbers

Rights: Copyright © 2014 Rocky Mountain Mathematics Consortium

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Vol.44 • No. 1 • 2014
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