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Pisot Numbers from {0, 1}-Polynomials

Published online by Cambridge University Press:  20 November 2018

Keshav Mukunda
Keshav Mukunda*
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6 e-mail: kmukunda@math.sfu.ca
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Abstract

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A Pisot number is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial — one with $\{0,\,1\}$-coefficients — and shows that they form a strictly increasing sequence with limit $\left( 1\,+\,\sqrt{5} \right)/2$. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials.

Keywords

11R0611R0911C08
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 1 , 01 March 2010 , pp. 140 - 152
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Bertin, M. J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M., and Schreiber, J. P., Pisot and Salem Numbers. Birkhäuser Verlag, Basel, 1992, pp. 119–139 Google Scholar
[2] Boyd, D. W., Pisot and Salem numbers in intervals of the real line. Math. Comp. 32(1978), no. 144, 1244–1260. doi:10.2307/2006349Google Scholar
[3] Campbell, D. M., Ferguson, H. R. P. and Forcade, R. W., Newman Polynomials on z = 1 . Indiana Univ. Math. J. 32(1983), no. 4, 517–525. doi:10.1512/iumj.1983.32.32037Google Scholar
[4] Dufresnoy, J. and Pisot, C., Étude de certaines fonctions méromorphes bornées sur le cercle unité. Application à un ensemble fermé d’entiers algébriques. Ann. Sci. école. Norm. Sup. 72(1955), 69–92.Google Scholar
[5] Flatto, L., Lagarias, J. C. and Poonen, B., The zeta function of the beta transformation. Ergodic Theory Dynam. Systems 14(1994), no. 2, 237–266.Google Scholar
[6] Marden, M., Geometry of Polynomials. Second edition. Mathematical Surveys 3. American Mathematical Society, Providence, RI, 1966, pp. 122–123.Google Scholar
[7] Mukunda, K., Littlewood Pisot numbers. J. Number Theory 117(2006), no. 1, 106–121. doi:10.1016/j.jnt.2005.05.009Google Scholar
[8] Odlyzko, A. M. and Poonen, B., Zeros of polynomials with 0, 1 coefficients. Enseign. Math. 39(1993), no. 3-4, 317–348.Google Scholar
[9] Salem, R., Algebraic Numbers and Fourier Analysis. D. C. Heath, Boston, 1963.Google Scholar
[10] Siegel, C. L., Algebraic integers whose conjugates lie in the unit circle. Duke Math. J. 11(1944), 597–602. doi:10.1215/S0012-7094-44-01152-XGoogle Scholar
[11] Solomyak, B., Conjugates of beta-numbers and the zero-free domain for a class of analytic functions. Proc. London Math. Soc. 68(1994), no. 3, 477–498. doi:10.1112/plms/s3-68.3.477Google Scholar