Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year


IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

CiteScore 2017: 0.68

SCImago Journal Rank (SJR) 2017: 0.450
Source Normalized Impact per Paper (SNIP) 2017: 0.829

Mathematical Citation Quotient (MCQ) 2016: 0.23

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 11, Issue 5

Issues

Volume 13 (2015)

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

James McKee / Chris Smyth
  • School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, EH9 3JZ, Scotland, UK
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2013-03-14 | DOI: https://doi.org/10.2478/s11533-013-0209-9

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.

MSC: 11R06; 11C08

Keywords: Cyclotomic polynomials; Interlacing; Pisot polynomials; Disc-bionic

  • [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/BF01393900CrossrefGoogle Scholar

  • [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/jdn078CrossrefGoogle Scholar

  • [3] Boyd D.W., Small Salem numbers, Duke Math. J., 1977, 44(2), 315–328 http://dx.doi.org/10.1215/S0012-7094-77-04413-1CrossrefGoogle Scholar

  • [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/S0025-5718-1978-0491587-8CrossrefGoogle Scholar

  • [5] Brunotte H., On Garcia numbers, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 2009, 25(1), 9–16 Google Scholar

  • [6] Fisk S., A very short proof of Cauchy’s interlace theorem, Amer. Math. Monthly, 2005, 112(2), 118 Google Scholar

  • [7] Garsia A.M., Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc., 1962, 102(3), 409–432 http://dx.doi.org/10.1090/S0002-9947-1962-0137961-5CrossrefGoogle Scholar

  • [8] Hardy G.H., Littlewood J.E., Pólya G., Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1952 Google Scholar

  • [9] Hare K.G., Panju M., Some comments on Garsia numbers, Math. Comp., 2013, 82(282), 1197–1221 http://dx.doi.org/10.1090/S0025-5718-2012-02636-6CrossrefGoogle Scholar

  • [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/s10474-012-0225-4Web of ScienceCrossrefGoogle Scholar

  • [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/S0024609304003790CrossrefGoogle Scholar

  • [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/10.1080/10586458.2005.10128915CrossrefGoogle Scholar

  • [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-2011-051-2CrossrefGoogle Scholar

  • [14] Robertson M.I.S., On the theory of univalent functions, Ann. of Math., 1936, 37(2), 374–408 http://dx.doi.org/10.2307/1968451CrossrefGoogle Scholar

  • [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-XCrossrefGoogle Scholar

About the article

Published Online: 2013-03-14

Published in Print: 2013-05-01


Citation Information: Open Mathematics, Volume 11, Issue 5, Pages 882–899, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-013-0209-9.

Export Citation

© 2013 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in