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2013 | 11 | 5 | 882-899
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Single polynomials that correspond to pairs of cyclotomic polynomials with interlacing zeros

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
11
Numer
5
Strony
882-899
Opis fizyczny
Daty
wydano
2013-05-01
online
2013-03-14
Twórcy
autor
autor
Bibliografia
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-013-0209-9
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