Single polynomials that correspond to pairs of cyclotomic polynomials with interlacing zeros

James McKee; Chris Smyth

Open Mathematics (2013)

  • Volume: 11, Issue: 5, page 882-899
  • ISSN: 2391-5455

Abstract

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We give a complete classification of all pairs of cyclotomic polynomials whose zeros interlace on the unit circle, making explicit a result essentially contained in work of Beukers and Heckman. We show that each such pair corresponds to a single polynomial from a certain special class of integer polynomials, the 2-reciprocal discbionic polynomials. We also show that each such pair also corresponds (in four different ways) to a single Pisot polynomial from a certain restricted class, the cyclogenic Pisot polynomials. We investigate properties of this class of Pisot polynomials.

How to cite

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James McKee, and Chris Smyth. "Single polynomials that correspond to pairs of cyclotomic polynomials with interlacing zeros." Open Mathematics 11.5 (2013): 882-899. <http://eudml.org/doc/269581>.

@article{JamesMcKee2013,
abstract = {We give a complete classification of all pairs of cyclotomic polynomials whose zeros interlace on the unit circle, making explicit a result essentially contained in work of Beukers and Heckman. We show that each such pair corresponds to a single polynomial from a certain special class of integer polynomials, the 2-reciprocal discbionic polynomials. We also show that each such pair also corresponds (in four different ways) to a single Pisot polynomial from a certain restricted class, the cyclogenic Pisot polynomials. We investigate properties of this class of Pisot polynomials.},
author = {James McKee, Chris Smyth},
journal = {Open Mathematics},
keywords = {Cyclotomic polynomials; Interlacing; Pisot polynomials; Disc-bionic; cyclotomic polynomials; interlacing; disc-bionic},
language = {eng},
number = {5},
pages = {882-899},
title = {Single polynomials that correspond to pairs of cyclotomic polynomials with interlacing zeros},
url = {http://eudml.org/doc/269581},
volume = {11},
year = {2013},
}

TY - JOUR
AU - James McKee
AU - Chris Smyth
TI - Single polynomials that correspond to pairs of cyclotomic polynomials with interlacing zeros
JO - Open Mathematics
PY - 2013
VL - 11
IS - 5
SP - 882
EP - 899
AB - We give a complete classification of all pairs of cyclotomic polynomials whose zeros interlace on the unit circle, making explicit a result essentially contained in work of Beukers and Heckman. We show that each such pair corresponds to a single polynomial from a certain special class of integer polynomials, the 2-reciprocal discbionic polynomials. We also show that each such pair also corresponds (in four different ways) to a single Pisot polynomial from a certain restricted class, the cyclogenic Pisot polynomials. We investigate properties of this class of Pisot polynomials.
LA - eng
KW - Cyclotomic polynomials; Interlacing; Pisot polynomials; Disc-bionic; cyclotomic polynomials; interlacing; disc-bionic
UR - http://eudml.org/doc/269581
ER -

References

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  3. [3] Boyd D.W., Small Salem numbers, Duke Math. J., 1977, 44(2), 315–328 http://dx.doi.org/10.1215/S0012-7094-77-04413-1 
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  5. [5] Brunotte H., On Garcia numbers, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 2009, 25(1), 9–16 
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  8. [8] Hardy G.H., Littlewood J.E., Pólya G., Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1952 
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  10. [10] Lalín M.N., Smyth C.J., Unimodularity of zeros of self-inversive polynomials, Acta Math. Hungar., 2013, 138(1–2), 85–101 http://dx.doi.org/10.1007/s10474-012-0225-4 Zbl1299.26037
  11. [11] McKee J., Smyth C.J., There are Salem numbers of every trace, Bull. London Math. Soc., 2005, 37(1), 25–36 http://dx.doi.org/10.1112/S0024609304003790 Zbl1166.11349
  12. [12] McKee J., Smyth C.J., Salem numbers, Pisot numbers, Mahler measure and graphs, Experiment. Math., 2005, 14(2), 211–229 http://dx.doi.org/10.1080/10586458.2005.10128915 Zbl1082.11066
  13. [13] McKee J., Smyth C.J., Salem numbers and Pisot numbers via interlacing, Canad. J. Math., 2012, 64(2), 345–367 http://dx.doi.org/10.4153/CJM-2011-051-2 Zbl1333.11097
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  15. [15] Siegel C.L., Algebraic integers whose conjugates lie in the unit circle, Duke Math. J., 1944, 11(3), 597–602 http://dx.doi.org/10.1215/S0012-7094-44-01152-X Zbl0063.07005

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