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

Open Mathematics (2013)

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

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topJames 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

top- [1] Beukers F., Heckman G., Monodromy for the hypergeometric function nF n−1, Invent. Math., 1989, 95(2), 325–354 http://dx.doi.org/10.1007/BF01393900 Zbl0663.30044
- [2] Bober J.W., Factorial ratios, hypergeometric series, and a family of step functions, J. Lond. Math. Soc., 2009, 79(2), 422–444 http://dx.doi.org/10.1112/jlms/jdn078 Zbl1195.11025
- [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
- [4] Boyd D.W., Pisot and Salem numbers in intervals of the real line, Math. Comp., 1978, 32(144), 1244–1260 http://dx.doi.org/10.1090/S0025-5718-1978-0491587-8 Zbl0395.12004
- [5] Brunotte H., On Garcia numbers, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 2009, 25(1), 9–16
- [6] Fisk S., A very short proof of Cauchy’s interlace theorem, Amer. Math. Monthly, 2005, 112(2), 118
- [7] Garsia A.M., Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc., 1962, 102(3), 409–432 http://dx.doi.org/10.1090/S0002-9947-1962-0137961-5 Zbl0103.36502
- [8] Hardy G.H., Littlewood J.E., Pólya G., Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1952
- [9] Hare K.G., Panju M., Some comments on Garsia numbers, Math. Comp., 2013, 82(282), 1197–1221 http://dx.doi.org/10.1090/S0025-5718-2012-02636-6 Zbl1275.11137
- [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] 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] 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] 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
- [14] Robertson M.I.S., On the theory of univalent functions, Ann. of Math., 1936, 37(2), 374–408 http://dx.doi.org/10.2307/1968451 Zbl62.0373.05
- [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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