Abstract
Using Schur’s algorithm for generating all Pisot numbers less than or equal to \({{\hat \theta }_{15}} \simeq 1.6183608 \ldots \), we prove that Inf S = θ 0, where ϑ 0 = 1.3247179572... satisfies the equation X 3 − X − 1 = 0, and that \(Inf S' = (\sqrt 5 + 1)/2\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Amara, Ensembles fermés de nombres algébriques, Ann. Scient. Ec. Norm., Sup. 3, 83, (1966), 215–270.
M.J. Bertin and M. Pathiaux-Delefosse, Conjecture de Lehmer et petits nombres de Salem, Queen’s Papers in Pure and Applied Mathematics, Kingston, Canada, 81, (1989).
D.W. Boyd, Small Salem numbers, Duke Math. J., 44, (1977), 315–328.
D.W. Boyd, Schur’s algorithm for bounded holomorphic functions, Bull. London Math. Soc, 11, (1979), 145–150.
D.W. Boyd, Pisot and Salem numbers in intervals of the real line, Math. Comp., 32, (1978), 1244–1260.
D.W. Boyd, Reciprocal polynomials having small measure, Math. Comp., 35, (1980), 1361–1377.
D.W. Boyd, Reciprocal polynomials having small measure II, Math. Comp., 53, (1989), no 187, 355–357.
D.C. Cantor and E.G. Straus, On a conjecture of Lehmer, Acta Arith., 42, (1982), 97–100.
C. Chamfy, Fonctions méromorphes dans le cercle unité et leurs séries de Taylor, Ann. Inst. Fourier, 8, (1958), 214–261.
E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith., 34, (1979), 391–401.
J. Dufresnoy and Ch. Pisot, Sur un ensemble fermé d’entiers algébriques, Ann. Scient. Ec. Norm., Sup. 3, 70, (1953), 105–133.
J. Dufresnoy and Ch. Pisot, Etude de certaines fonctions méromorphes bornées sur le cercle unité. Application à un ensemble fermé d’entiers algébriques, Ann. Sc. Ec. Norm., Sup. 3, 72, (1955), 69–92.
M. Grandet-Hugot, Ensembles fermés d’entiers algébriques, Ann. Sc. Ec. Norm., Sup. 3, 82, (1965), 1–35.
M. Langevin, Méthode de Fekete-Szegö et problème de Lehmer, C.R.A.S. Paris, 301, Série I, (1985), 463–466.
M. Langevin, Minorations de la maison et de la mesure de Mahler de certains entiers algébriques, C.R.A.S. Paris, 303, Série I, (1986), 523–526.
F. Lazami, Sur les éléments de S ⊃ [1, 2[, Séminaire Delange-Pisot-Poitou, 20 ième année, (1978–79), no 3.
D.H. Lehmer, Factorization of certain cyclotomic functions, Ann. Math., 34, (1933), 461–479.
F. Talmoudi, Sur les nombres de S ∩ [1,2[, C.R.A.S. Paris, 287, (1978).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Basel AG
About this chapter
Cite this chapter
Bertin, M.J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M., Schreiber, J.P. (1992). Small Pisot Numbers. In: Pisot and Salem Numbers. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8632-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8632-1_7
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9706-8
Online ISBN: 978-3-0348-8632-1
eBook Packages: Springer Book Archive